non-abelian, soluble, monomial
Aliases: C12.1S4, C22⋊Dic18, C23.1D18, C3.A4⋊Q8, C3.(A4⋊Q8), (C2×C6).Dic6, C6.16(C2×S4), C6.S4.C2, C4.1(C3.S4), (C22×C4).2D9, (C22×C12).2S3, (C22×C6).13D6, C2.3(C2×C3.S4), (C4×C3.A4).1C2, (C2×C3.A4).1C22, SmallGroup(288,332)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.1S4
G = < a,b,c,d,e | a12=b2=c2=1, d3=a4, e2=a6, ab=ba, ac=ca, ad=da, eae-1=a-1, dbd-1=ebe-1=bc=cb, dcd-1=b, ce=ec, ede-1=a8d2 >
Subgroups: 360 in 72 conjugacy classes, 18 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C2×C4, Q8, C23, C9, Dic3, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C18, Dic6, C2×Dic3, C2×C12, C22×C6, C22⋊Q8, Dic9, C36, C3.A4, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, Dic18, C2×C3.A4, C12.48D4, C6.S4, C4×C3.A4, C12.1S4
Quotients: C1, C2, C22, S3, Q8, D6, D9, Dic6, S4, D18, C2×S4, Dic18, C3.S4, A4⋊Q8, C2×C3.S4, C12.1S4
Character table of C12.1S4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 6 | 36 | 36 | 36 | 36 | 2 | 6 | 6 | 8 | 8 | 8 | 2 | 2 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ7 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -2 | -2 | -2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | 1 | 1 | 1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | 1 | 1 | 1 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | 1 | 1 | 1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ11 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | √3 | √3 | -√3 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ15 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | -√3 | -√3 | √3 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -√3 | √3 | -√3 | √3 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ43ζ98+ζ43ζ9 | ζ4ζ95-ζ4ζ94 | ζ43ζ98-ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | symplectic lifted from Dic18, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | √3 | -√3 | √3 | -√3 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ98-ζ9 | ζ43ζ98-ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ95-ζ4ζ94 | ζ4ζ97-ζ4ζ92 | symplectic lifted from Dic18, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -√3 | √3 | -√3 | √3 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | -ζ4ζ97+ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ43ζ98-ζ43ζ9 | ζ4ζ95-ζ4ζ94 | symplectic lifted from Dic18, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | √3 | -√3 | √3 | -√3 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ97-ζ92 | ζ4ζ97-ζ4ζ92 | ζ43ζ98-ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ95+ζ4ζ94 | symplectic lifted from Dic18, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | √3 | -√3 | √3 | -√3 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | -ζ4ζ95+ζ4ζ94 | ζ4ζ97-ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ43ζ98-ζ43ζ9 | symplectic lifted from Dic18, Schur index 2 |
| ρ21 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -√3 | √3 | -√3 | √3 | -ζ97-ζ92 | -ζ98-ζ9 | -ζ95-ζ94 | ζ4ζ95-ζ4ζ94 | -ζ4ζ97+ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ43ζ98-ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ43ζ98+ζ43ζ9 | symplectic lifted from Dic18, Schur index 2 |
| ρ22 | 3 | 3 | -1 | -1 | 3 | 3 | -1 | 1 | -1 | 1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ23 | 3 | 3 | -1 | -1 | 3 | -3 | 1 | -1 | -1 | 1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ24 | 3 | 3 | -1 | -1 | 3 | 3 | -1 | -1 | 1 | -1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ25 | 3 | 3 | -1 | -1 | 3 | -3 | 1 | 1 | 1 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ26 | 6 | 6 | -2 | -2 | -3 | 6 | -2 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ27 | 6 | 6 | -2 | -2 | -3 | -6 | 2 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C3.S4 |
| ρ28 | 6 | -6 | -2 | 2 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from A4⋊Q8, Schur index 2 |
| ρ29 | 6 | -6 | -2 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | 0 | 0 | 3√3 | -3√3 | -√3 | √3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ30 | 6 | -6 | -2 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | 0 | 0 | -3√3 | 3√3 | √3 | -√3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 72 points
Generators in S72
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 61)(59 62)(60 63)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 46)(14 47)(15 48)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(49 70)(50 71)(51 72)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)
(1 55 48 5 59 40 9 51 44)(2 56 37 6 60 41 10 52 45)(3 57 38 7 49 42 11 53 46)(4 58 39 8 50 43 12 54 47)(13 36 72 17 28 64 21 32 68)(14 25 61 18 29 65 22 33 69)(15 26 62 19 30 66 23 34 70)(16 27 63 20 31 67 24 35 71)
(1 31 7 25)(2 30 8 36)(3 29 9 35)(4 28 10 34)(5 27 11 33)(6 26 12 32)(13 56 19 50)(14 55 20 49)(15 54 21 60)(16 53 22 59)(17 52 23 58)(18 51 24 57)(37 62 43 68)(38 61 44 67)(39 72 45 66)(40 71 46 65)(41 70 47 64)(42 69 48 63)
G:=sub<Sym(72)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,61)(59,62)(60,63), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (1,55,48,5,59,40,9,51,44)(2,56,37,6,60,41,10,52,45)(3,57,38,7,49,42,11,53,46)(4,58,39,8,50,43,12,54,47)(13,36,72,17,28,64,21,32,68)(14,25,61,18,29,65,22,33,69)(15,26,62,19,30,66,23,34,70)(16,27,63,20,31,67,24,35,71), (1,31,7,25)(2,30,8,36)(3,29,9,35)(4,28,10,34)(5,27,11,33)(6,26,12,32)(13,56,19,50)(14,55,20,49)(15,54,21,60)(16,53,22,59)(17,52,23,58)(18,51,24,57)(37,62,43,68)(38,61,44,67)(39,72,45,66)(40,71,46,65)(41,70,47,64)(42,69,48,63)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,61)(59,62)(60,63), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,46)(14,47)(15,48)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(49,70)(50,71)(51,72)(52,61)(53,62)(54,63)(55,64)(56,65)(57,66)(58,67)(59,68)(60,69), (1,55,48,5,59,40,9,51,44)(2,56,37,6,60,41,10,52,45)(3,57,38,7,49,42,11,53,46)(4,58,39,8,50,43,12,54,47)(13,36,72,17,28,64,21,32,68)(14,25,61,18,29,65,22,33,69)(15,26,62,19,30,66,23,34,70)(16,27,63,20,31,67,24,35,71), (1,31,7,25)(2,30,8,36)(3,29,9,35)(4,28,10,34)(5,27,11,33)(6,26,12,32)(13,56,19,50)(14,55,20,49)(15,54,21,60)(16,53,22,59)(17,52,23,58)(18,51,24,57)(37,62,43,68)(38,61,44,67)(39,72,45,66)(40,71,46,65)(41,70,47,64)(42,69,48,63) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,25),(11,26),(12,27),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,61),(59,62),(60,63)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,46),(14,47),(15,48),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(49,70),(50,71),(51,72),(52,61),(53,62),(54,63),(55,64),(56,65),(57,66),(58,67),(59,68),(60,69)], [(1,55,48,5,59,40,9,51,44),(2,56,37,6,60,41,10,52,45),(3,57,38,7,49,42,11,53,46),(4,58,39,8,50,43,12,54,47),(13,36,72,17,28,64,21,32,68),(14,25,61,18,29,65,22,33,69),(15,26,62,19,30,66,23,34,70),(16,27,63,20,31,67,24,35,71)], [(1,31,7,25),(2,30,8,36),(3,29,9,35),(4,28,10,34),(5,27,11,33),(6,26,12,32),(13,56,19,50),(14,55,20,49),(15,54,21,60),(16,53,22,59),(17,52,23,58),(18,51,24,57),(37,62,43,68),(38,61,44,67),(39,72,45,66),(40,71,46,65),(41,70,47,64),(42,69,48,63)]])
Matrix representation of C12.1S4 ►in GL5(𝔽37)
| 31 | 4 | 0 | 0 | 0 |
| 4 | 28 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 36 |
| 10 | 27 | 0 | 0 | 0 |
| 27 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 23 | 14 | 0 | 0 | 0 |
| 15 | 14 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(37))| [31,4,0,0,0,4,28,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36],[10,27,0,0,0,27,36,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[23,15,0,0,0,14,14,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
C12.1S4 in GAP, Magma, Sage, TeX
C_{12}._1S_4 % in TeX
G:=Group("C12.1S4"); // GroupNames label
G:=SmallGroup(288,332);
// by ID
G=gap.SmallGroup(288,332);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,28,85,36,1123,192,1684,6053,782,3534,1350]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^2=c^2=1,d^3=a^4,e^2=a^6,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,d*b*d^-1=e*b*e^-1=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e^-1=a^8*d^2>;
// generators/relations
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