Polytope of Type {4,8,14} with group SmallGroup(1792,323566)

Overview

Group: SmallGroup(1792,323566)
Rank: 4
Schläfli Type: {4,8,14}
Vertices, edges, …: 8, 32, 112, 14
Order of s₀s₁s₂s₃: 28
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{4,4,14}*896

4-fold

{4,4,14}*448

7-fold

{4,8,2}*256b

8-fold

14-fold

{4,4,2}*128

16-fold

28-fold

{4,4,2}*64

32-fold

{2,2,7}*56

56-fold

112-fold

{2,2,2}*16

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1*s2*s1)^2> of order 2

14 facets

14 of 2-fold non-regular quotient of {4,8}*128b

4 vertex figures

4 of {8,14}*224

Representations

Permutation Representation (GAP)

s0 := ( 29, 43)( 30, 44)( 31, 45)( 32, 46)( 33, 47)( 34, 48)( 35, 49)( 36, 50)( 37, 51)( 38, 52)( 39, 53)( 40, 54)( 41, 55)( 42, 56)( 85, 99)( 86,100)( 87,101)( 88,102)( 89,103)( 90,104)( 91,105)( 92,106)( 93,107)( 94,108)( 95,109)( 96,110)( 97,111)( 98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196);;
s1 := ( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 43, 50)( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 71, 78)( 72, 79)( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,218)(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217);;
s2 := (  1,197)(  2,203)(  3,202)(  4,201)(  5,200)(  6,199)(  7,198)(  8,204)(  9,210)( 10,209)( 11,208)( 12,207)( 13,206)( 14,205)( 15,211)( 16,217)( 17,216)( 18,215)( 19,214)( 20,213)( 21,212)( 22,218)( 23,224)( 24,223)( 25,222)( 26,221)( 27,220)( 28,219)( 29,183)( 30,189)( 31,188)( 32,187)( 33,186)( 34,185)( 35,184)( 36,190)( 37,196)( 38,195)( 39,194)( 40,193)( 41,192)( 42,191)( 43,169)( 44,175)( 45,174)( 46,173)( 47,172)( 48,171)( 49,170)( 50,176)( 51,182)( 52,181)( 53,180)( 54,179)( 55,178)( 56,177)( 57,141)( 58,147)( 59,146)( 60,145)( 61,144)( 62,143)( 63,142)( 64,148)( 65,154)( 66,153)( 67,152)( 68,151)( 69,150)( 70,149)( 71,155)( 72,161)( 73,160)( 74,159)( 75,158)( 76,157)( 77,156)( 78,162)( 79,168)( 80,167)( 81,166)( 82,165)( 83,164)( 84,163)( 85,127)( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)( 94,139)( 95,138)( 96,137)( 97,136)( 98,135)( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(106,120)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121);;
s3 := (  1, 58)(  2, 57)(  3, 63)(  4, 62)(  5, 61)(  6, 60)(  7, 59)(  8, 65)(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 72)( 16, 71)( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 79)( 23, 78)( 24, 84)( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 86)( 30, 85)( 31, 91)( 32, 90)( 33, 89)( 34, 88)( 35, 87)( 36, 93)( 37, 92)( 38, 98)( 39, 97)( 40, 96)( 41, 95)( 42, 94)( 43,100)( 44, 99)( 45,105)( 46,104)( 47,103)( 48,102)( 49,101)( 50,107)( 51,106)( 52,112)( 53,111)( 54,110)( 55,109)( 56,108)(113,170)(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)(120,177)(121,176)(122,182)(123,181)(124,180)(125,179)(126,178)(127,184)(128,183)(129,189)(130,188)(131,187)(132,186)(133,185)(134,191)(135,190)(136,196)(137,195)(138,194)(139,193)(140,192)(141,198)(142,197)(143,203)(144,202)(145,201)(146,200)(147,199)(148,205)(149,204)(150,210)(151,209)(152,208)(153,207)(154,206)(155,212)(156,211)(157,217)(158,216)(159,215)(160,214)(161,213)(162,219)(163,218)(164,224)(165,223)(166,222)(167,221)(168,220);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(224)!( 29, 43)( 30, 44)( 31, 45)( 32, 46)( 33, 47)( 34, 48)( 35, 49)( 36, 50)( 37, 51)( 38, 52)( 39, 53)( 40, 54)( 41, 55)( 42, 56)( 85, 99)( 86,100)( 87,101)( 88,102)( 89,103)( 90,104)( 91,105)( 92,106)( 93,107)( 94,108)( 95,109)( 96,110)( 97,111)( 98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196);
s1 := Sym(224)!( 15, 22)( 16, 23)( 17, 24)( 18, 25)( 19, 26)( 20, 27)( 21, 28)( 43, 50)( 44, 51)( 45, 52)( 46, 53)( 47, 54)( 48, 55)( 49, 56)( 71, 78)( 72, 79)( 73, 80)( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,218)(184,219)(185,220)(186,221)(187,222)(188,223)(189,224)(190,211)(191,212)(192,213)(193,214)(194,215)(195,216)(196,217);
s2 := Sym(224)!(  1,197)(  2,203)(  3,202)(  4,201)(  5,200)(  6,199)(  7,198)(  8,204)(  9,210)( 10,209)( 11,208)( 12,207)( 13,206)( 14,205)( 15,211)( 16,217)( 17,216)( 18,215)( 19,214)( 20,213)( 21,212)( 22,218)( 23,224)( 24,223)( 25,222)( 26,221)( 27,220)( 28,219)( 29,183)( 30,189)( 31,188)( 32,187)( 33,186)( 34,185)( 35,184)( 36,190)( 37,196)( 38,195)( 39,194)( 40,193)( 41,192)( 42,191)( 43,169)( 44,175)( 45,174)( 46,173)( 47,172)( 48,171)( 49,170)( 50,176)( 51,182)( 52,181)( 53,180)( 54,179)( 55,178)( 56,177)( 57,141)( 58,147)( 59,146)( 60,145)( 61,144)( 62,143)( 63,142)( 64,148)( 65,154)( 66,153)( 67,152)( 68,151)( 69,150)( 70,149)( 71,155)( 72,161)( 73,160)( 74,159)( 75,158)( 76,157)( 77,156)( 78,162)( 79,168)( 80,167)( 81,166)( 82,165)( 83,164)( 84,163)( 85,127)( 86,133)( 87,132)( 88,131)( 89,130)( 90,129)( 91,128)( 92,134)( 93,140)( 94,139)( 95,138)( 96,137)( 97,136)( 98,135)( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(106,120)(107,126)(108,125)(109,124)(110,123)(111,122)(112,121);
s3 := Sym(224)!(  1, 58)(  2, 57)(  3, 63)(  4, 62)(  5, 61)(  6, 60)(  7, 59)(  8, 65)(  9, 64)( 10, 70)( 11, 69)( 12, 68)( 13, 67)( 14, 66)( 15, 72)( 16, 71)( 17, 77)( 18, 76)( 19, 75)( 20, 74)( 21, 73)( 22, 79)( 23, 78)( 24, 84)( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 86)( 30, 85)( 31, 91)( 32, 90)( 33, 89)( 34, 88)( 35, 87)( 36, 93)( 37, 92)( 38, 98)( 39, 97)( 40, 96)( 41, 95)( 42, 94)( 43,100)( 44, 99)( 45,105)( 46,104)( 47,103)( 48,102)( 49,101)( 50,107)( 51,106)( 52,112)( 53,111)( 54,110)( 55,109)( 56,108)(113,170)(114,169)(115,175)(116,174)(117,173)(118,172)(119,171)(120,177)(121,176)(122,182)(123,181)(124,180)(125,179)(126,178)(127,184)(128,183)(129,189)(130,188)(131,187)(132,186)(133,185)(134,191)(135,190)(136,196)(137,195)(138,194)(139,193)(140,192)(141,198)(142,197)(143,203)(144,202)(145,201)(146,200)(147,199)(148,205)(149,204)(150,210)(151,209)(152,208)(153,207)(154,206)(155,212)(156,211)(157,217)(158,216)(159,215)(160,214)(161,213)(162,219)(163,218)(164,224)(165,223)(166,222)(167,221)(168,220);
poly := sub<Sym(224)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References

None.

to this polytope.