Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,15}

Atlas Canonical Name {4,4,15}*1920a

Overview

Group
SmallGroup(1920,237638)
Rank
4
Schläfli Type
{4,4,15}
Vertices, edges, …
16, 32, 120, 15
Order of s0s1s2s3
30
Order of s0s1s2s3s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

5-fold

8-fold

10-fold

40-fold

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=<(s1*s0*s1*s2)^2> of order 2

15 facets

  • 5 of 2-fold non-regular quotient of {4,4}*128
  • 10 of 2-fold non-regular quotient of {4,4}*128

8 vertex figures

P/N, where N=<(s1*s0*s2)^3*s1*s2> of order 2

15 facets

  • 15 of 2-fold non-regular quotient of {4,4}*128

8 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2, (s0*s1)^2*s2*s1*s0*s1*s2> of order 4

15 facets

  • 5 of 4-fold non-regular quotient of {4,4}*128
  • 10 of 4-fold non-regular quotient of {4,4}*128

4 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s1*s0*s2*s1*s0*s1*s2*s1> of order 4

15 facets

  • 15 of 4-fold non-regular quotient of {4,4}*128

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 22)( 18, 21)( 19, 24)( 20, 23)( 25, 32)( 26, 31)( 27, 30)( 28, 29)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 54)( 50, 53)( 51, 56)( 52, 55)( 57, 64)( 58, 63)( 59, 62)( 60, 61)( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 86)( 82, 85)( 83, 88)( 84, 87)( 89, 96)( 90, 95)( 91, 94)( 92, 93)(105,108)(106,107)(109,112)(110,111)(113,118)(114,117)(115,120)(116,119)(121,128)(122,127)(123,126)(124,125)(137,140)(138,139)(141,144)(142,143)(145,150)(146,149)(147,152)(148,151)(153,160)(154,159)(155,158)(156,157);;
s1 := (  1, 17)(  2, 18)(  3, 20)(  4, 19)(  5, 21)(  6, 22)(  7, 24)(  8, 23)(  9, 25)( 10, 26)( 11, 28)( 12, 27)( 13, 29)( 14, 30)( 15, 32)( 16, 31)( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 37, 53)( 38, 54)( 39, 56)( 40, 55)( 41, 57)( 42, 58)( 43, 60)( 44, 59)( 45, 61)( 46, 62)( 47, 64)( 48, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 93)( 78, 94)( 79, 96)( 80, 95)( 97,113)( 98,114)( 99,116)(100,115)(101,117)(102,118)(103,120)(104,119)(105,121)(106,122)(107,124)(108,123)(109,125)(110,126)(111,128)(112,127)(129,145)(130,146)(131,148)(132,147)(133,149)(134,150)(135,152)(136,151)(137,153)(138,154)(139,156)(140,155)(141,157)(142,158)(143,160)(144,159);;
s2 := (  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33,129)( 34,130)( 35,131)( 36,132)( 37,135)( 38,136)( 39,133)( 40,134)( 41,137)( 42,138)( 43,139)( 44,140)( 45,143)( 46,144)( 47,141)( 48,142)( 49,153)( 50,154)( 51,155)( 52,156)( 53,159)( 54,160)( 55,157)( 56,158)( 57,145)( 58,146)( 59,147)( 60,148)( 61,151)( 62,152)( 63,149)( 64,150)( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,103)( 70,104)( 71,101)( 72,102)( 73,105)( 74,106)( 75,107)( 76,108)( 77,111)( 78,112)( 79,109)( 80,110)( 81,121)( 82,122)( 83,123)( 84,124)( 85,127)( 86,128)( 87,125)( 88,126)( 89,113)( 90,114)( 91,115)( 92,116)( 93,119)( 94,120)( 95,117)( 96,118);;
s3 := (  1, 33)(  2, 34)(  3, 39)(  4, 40)(  5, 37)(  6, 38)(  7, 35)(  8, 36)(  9, 57)( 10, 58)( 11, 63)( 12, 64)( 13, 61)( 14, 62)( 15, 59)( 16, 60)( 17, 49)( 18, 50)( 19, 55)( 20, 56)( 21, 53)( 22, 54)( 23, 51)( 24, 52)( 25, 41)( 26, 42)( 27, 47)( 28, 48)( 29, 45)( 30, 46)( 31, 43)( 32, 44)( 65,129)( 66,130)( 67,135)( 68,136)( 69,133)( 70,134)( 71,131)( 72,132)( 73,153)( 74,154)( 75,159)( 76,160)( 77,157)( 78,158)( 79,155)( 80,156)( 81,145)( 82,146)( 83,151)( 84,152)( 85,149)( 86,150)( 87,147)( 88,148)( 89,137)( 90,138)( 91,143)( 92,144)( 93,141)( 94,142)( 95,139)( 96,140)( 99,103)(100,104)(105,121)(106,122)(107,127)(108,128)(109,125)(110,126)(111,123)(112,124)(115,119)(116,120);;
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*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s1*s2, 
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*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(160)!(  9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 22)( 18, 21)( 19, 24)( 20, 23)( 25, 32)( 26, 31)( 27, 30)( 28, 29)( 41, 44)( 42, 43)( 45, 48)( 46, 47)( 49, 54)( 50, 53)( 51, 56)( 52, 55)( 57, 64)( 58, 63)( 59, 62)( 60, 61)( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 86)( 82, 85)( 83, 88)( 84, 87)( 89, 96)( 90, 95)( 91, 94)( 92, 93)(105,108)(106,107)(109,112)(110,111)(113,118)(114,117)(115,120)(116,119)(121,128)(122,127)(123,126)(124,125)(137,140)(138,139)(141,144)(142,143)(145,150)(146,149)(147,152)(148,151)(153,160)(154,159)(155,158)(156,157);
s1 := Sym(160)!(  1, 17)(  2, 18)(  3, 20)(  4, 19)(  5, 21)(  6, 22)(  7, 24)(  8, 23)(  9, 25)( 10, 26)( 11, 28)( 12, 27)( 13, 29)( 14, 30)( 15, 32)( 16, 31)( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 37, 53)( 38, 54)( 39, 56)( 40, 55)( 41, 57)( 42, 58)( 43, 60)( 44, 59)( 45, 61)( 46, 62)( 47, 64)( 48, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 93)( 78, 94)( 79, 96)( 80, 95)( 97,113)( 98,114)( 99,116)(100,115)(101,117)(102,118)(103,120)(104,119)(105,121)(106,122)(107,124)(108,123)(109,125)(110,126)(111,128)(112,127)(129,145)(130,146)(131,148)(132,147)(133,149)(134,150)(135,152)(136,151)(137,153)(138,154)(139,156)(140,155)(141,157)(142,158)(143,160)(144,159);
s2 := Sym(160)!(  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33,129)( 34,130)( 35,131)( 36,132)( 37,135)( 38,136)( 39,133)( 40,134)( 41,137)( 42,138)( 43,139)( 44,140)( 45,143)( 46,144)( 47,141)( 48,142)( 49,153)( 50,154)( 51,155)( 52,156)( 53,159)( 54,160)( 55,157)( 56,158)( 57,145)( 58,146)( 59,147)( 60,148)( 61,151)( 62,152)( 63,149)( 64,150)( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,103)( 70,104)( 71,101)( 72,102)( 73,105)( 74,106)( 75,107)( 76,108)( 77,111)( 78,112)( 79,109)( 80,110)( 81,121)( 82,122)( 83,123)( 84,124)( 85,127)( 86,128)( 87,125)( 88,126)( 89,113)( 90,114)( 91,115)( 92,116)( 93,119)( 94,120)( 95,117)( 96,118);
s3 := Sym(160)!(  1, 33)(  2, 34)(  3, 39)(  4, 40)(  5, 37)(  6, 38)(  7, 35)(  8, 36)(  9, 57)( 10, 58)( 11, 63)( 12, 64)( 13, 61)( 14, 62)( 15, 59)( 16, 60)( 17, 49)( 18, 50)( 19, 55)( 20, 56)( 21, 53)( 22, 54)( 23, 51)( 24, 52)( 25, 41)( 26, 42)( 27, 47)( 28, 48)( 29, 45)( 30, 46)( 31, 43)( 32, 44)( 65,129)( 66,130)( 67,135)( 68,136)( 69,133)( 70,134)( 71,131)( 72,132)( 73,153)( 74,154)( 75,159)( 76,160)( 77,157)( 78,158)( 79,155)( 80,156)( 81,145)( 82,146)( 83,151)( 84,152)( 85,149)( 86,150)( 87,147)( 88,148)( 89,137)( 90,138)( 91,143)( 92,144)( 93,141)( 94,142)( 95,139)( 96,140)( 99,103)(100,104)(105,121)(106,122)(107,127)(108,128)(109,125)(110,126)(111,123)(112,124)(115,119)(116,120);
poly := sub<Sym(160)|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*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s1*s2, 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*s2*s3 >;

References

None.

to this polytope.