Aliases: C12.4S4, Q8.5D18, C4○D4⋊2D9, Q8⋊D9⋊2C2, C6.24(C2×S4), Q8⋊C9⋊2C22, C4.3(C3.S4), C3.(C4.3S4), Q8.C18⋊1C2, (C3×Q8).13D6, C2.10(C2×C3.S4), (C3×C4○D4).4S3, SmallGroup(288,340)
Series: Derived ►Chief ►Lower central ►Upper central
| Q8⋊C9 — C12.4S4 |
Generators and relations for C12.4S4
G = < a,b,c,d,e | a12=e2=1, b2=c2=a6, d3=a4, ab=ba, ac=ca, ad=da, eae=a-1, cbc-1=a6b, dbd-1=a6bc, ebe=bc, dcd-1=b, ece=a6c, ede=a8d2 >
Subgroups: 511 in 71 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, C23, C9, C12, C12, D6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, D9, C18, C3⋊C8, D12, C2×C12, C3×D4, C3×Q8, C22×S3, C8⋊C22, C36, D18, C4.Dic3, D4⋊S3, Q8⋊2S3, C2×D12, C3×C4○D4, Q8⋊C9, D36, D4⋊D6, Q8⋊D9, Q8.C18, C12.4S4
Quotients: C1, C2, C22, S3, D6, D9, S4, D18, C2×S4, C3.S4, C4.3S4, C2×C3.S4, C12.4S4
Character table of C12.4S4
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 6A | 6B | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 18A | 18B | 18C | 36A | 36B | 36C | 36D | 36E | 36F | |
| size | 1 | 1 | 6 | 36 | 36 | 2 | 2 | 6 | 2 | 12 | 36 | 36 | 8 | 8 | 8 | 2 | 2 | 12 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ8 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -1 | 1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | 1 | 1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ10 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -1 | 1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | 1 | 1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | -1 | 1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | 1 | 1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ13 | 3 | 3 | -1 | 1 | 1 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | 1 | -1 | 1 | 3 | -3 | -1 | 3 | 1 | 1 | -1 | 0 | 0 | 0 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ15 | 3 | 3 | 1 | 1 | -1 | 3 | -3 | -1 | 3 | 1 | -1 | 1 | 0 | 0 | 0 | -3 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | -1 | -1 | -1 | 3 | 3 | -1 | 3 | -1 | 1 | 1 | 0 | 0 | 0 | 3 | 3 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4.3S4 |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -√3 | √3 | -√3 | √3 | -√3 | √3 | orthogonal lifted from C4.3S4 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | √3 | -√3 | √3 | -√3 | √3 | -√3 | orthogonal lifted from C4.3S4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -2√3 | 2√3 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ4ζ95-ζ4ζ94 | -ζ4ζ95+ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | ζ43ζ98-ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ97-ζ4ζ92 | orthogonal faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | 2√3 | -2√3 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ43ζ98-ζ43ζ9 | -ζ43ζ98+ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ97+ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ4ζ95-ζ4ζ94 | orthogonal faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -2√3 | 2√3 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ43ζ98+ζ43ζ9 | ζ43ζ98-ζ43ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ97-ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ4ζ95+ζ4ζ94 | orthogonal faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | 2√3 | -2√3 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ4ζ95+ζ4ζ94 | ζ4ζ95-ζ4ζ94 | ζ43ζ98-ζ43ζ9 | -ζ43ζ98+ζ43ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ97+ζ4ζ92 | orthogonal faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | 2√3 | -2√3 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ4ζ97-ζ4ζ92 | -ζ4ζ97+ζ4ζ92 | -ζ4ζ95+ζ4ζ94 | ζ4ζ95-ζ4ζ94 | ζ43ζ98-ζ43ζ9 | -ζ43ζ98+ζ43ζ9 | orthogonal faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -2√3 | 2√3 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ4ζ97+ζ4ζ92 | ζ4ζ97-ζ4ζ92 | ζ4ζ95-ζ4ζ94 | -ζ4ζ95+ζ4ζ94 | -ζ43ζ98+ζ43ζ9 | ζ43ζ98-ζ43ζ9 | orthogonal faithful |
| ρ26 | 6 | 6 | -2 | 0 | 0 | -3 | 6 | -2 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | -3 | -3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3.S4 |
| ρ27 | 6 | 6 | 2 | 0 | 0 | -3 | -6 | -2 | -3 | -1 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×C3.S4 |
►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 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 22 19 16)(14 23 20 17)(15 24 21 18)(37 59 43 53)(38 60 44 54)(39 49 45 55)(40 50 46 56)(41 51 47 57)(42 52 48 58)(61 64 67 70)(62 65 68 71)(63 66 69 72)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 67 19 61)(14 68 20 62)(15 69 21 63)(16 70 22 64)(17 71 23 65)(18 72 24 66)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 56 43 50)(38 57 44 51)(39 58 45 52)(40 59 46 53)(41 60 47 54)(42 49 48 55)
(1 50 16 5 54 20 9 58 24)(2 51 17 6 55 21 10 59 13)(3 52 18 7 56 22 11 60 14)(4 53 19 8 57 23 12 49 15)(25 43 67 29 47 71 33 39 63)(26 44 68 30 48 72 34 40 64)(27 45 69 31 37 61 35 41 65)(28 46 70 32 38 62 36 42 66)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 49)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)(37 72)(38 71)(39 70)(40 69)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)
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), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,22,19,16)(14,23,20,17)(15,24,21,18)(37,59,43,53)(38,60,44,54)(39,49,45,55)(40,50,46,56)(41,51,47,57)(42,52,48,58)(61,64,67,70)(62,65,68,71)(63,66,69,72), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,67,19,61)(14,68,20,62)(15,69,21,63)(16,70,22,64)(17,71,23,65)(18,72,24,66)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,56,43,50)(38,57,44,51)(39,58,45,52)(40,59,46,53)(41,60,47,54)(42,49,48,55), (1,50,16,5,54,20,9,58,24)(2,51,17,6,55,21,10,59,13)(3,52,18,7,56,22,11,60,14)(4,53,19,8,57,23,12,49,15)(25,43,67,29,47,71,33,39,63)(26,44,68,30,48,72,34,40,64)(27,45,69,31,37,61,35,41,65)(28,46,70,32,38,62,36,42,66), (2,12)(3,11)(4,10)(5,9)(6,8)(13,49)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)>;
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), (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,22,19,16)(14,23,20,17)(15,24,21,18)(37,59,43,53)(38,60,44,54)(39,49,45,55)(40,50,46,56)(41,51,47,57)(42,52,48,58)(61,64,67,70)(62,65,68,71)(63,66,69,72), (1,10,7,4)(2,11,8,5)(3,12,9,6)(13,67,19,61)(14,68,20,62)(15,69,21,63)(16,70,22,64)(17,71,23,65)(18,72,24,66)(25,28,31,34)(26,29,32,35)(27,30,33,36)(37,56,43,50)(38,57,44,51)(39,58,45,52)(40,59,46,53)(41,60,47,54)(42,49,48,55), (1,50,16,5,54,20,9,58,24)(2,51,17,6,55,21,10,59,13)(3,52,18,7,56,22,11,60,14)(4,53,19,8,57,23,12,49,15)(25,43,67,29,47,71,33,39,63)(26,44,68,30,48,72,34,40,64)(27,45,69,31,37,61,35,41,65)(28,46,70,32,38,62,36,42,66), (2,12)(3,11)(4,10)(5,9)(6,8)(13,49)(14,60)(15,59)(16,58)(17,57)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)(37,72)(38,71)(39,70)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61) );
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)], [(1,31,7,25),(2,32,8,26),(3,33,9,27),(4,34,10,28),(5,35,11,29),(6,36,12,30),(13,22,19,16),(14,23,20,17),(15,24,21,18),(37,59,43,53),(38,60,44,54),(39,49,45,55),(40,50,46,56),(41,51,47,57),(42,52,48,58),(61,64,67,70),(62,65,68,71),(63,66,69,72)], [(1,10,7,4),(2,11,8,5),(3,12,9,6),(13,67,19,61),(14,68,20,62),(15,69,21,63),(16,70,22,64),(17,71,23,65),(18,72,24,66),(25,28,31,34),(26,29,32,35),(27,30,33,36),(37,56,43,50),(38,57,44,51),(39,58,45,52),(40,59,46,53),(41,60,47,54),(42,49,48,55)], [(1,50,16,5,54,20,9,58,24),(2,51,17,6,55,21,10,59,13),(3,52,18,7,56,22,11,60,14),(4,53,19,8,57,23,12,49,15),(25,43,67,29,47,71,33,39,63),(26,44,68,30,48,72,34,40,64),(27,45,69,31,37,61,35,41,65),(28,46,70,32,38,62,36,42,66)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,49),(14,60),(15,59),(16,58),(17,57),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33),(37,72),(38,71),(39,70),(40,69),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61)]])
Matrix representation of C12.4S4 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 2 |
| 0 | 0 | 0 | 72 | 2 | 0 |
| 0 | 0 | 0 | 72 | 1 | 0 |
| 0 | 0 | 72 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 71 | 0 |
| 0 | 0 | 1 | 0 | 0 | 71 |
| 0 | 0 | 1 | 0 | 0 | 72 |
| 0 | 0 | 0 | 1 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 2 |
| 0 | 0 | 0 | 1 | 71 | 0 |
| 0 | 0 | 0 | 1 | 72 | 0 |
| 0 | 0 | 72 | 0 | 0 | 1 |
| 45 | 42 | 0 | 0 | 0 | 0 |
| 31 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 47 | 62 | 11 |
| 0 | 0 | 37 | 37 | 10 | 10 |
| 0 | 0 | 68 | 68 | 47 | 47 |
| 0 | 0 | 31 | 42 | 36 | 37 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 71 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 1 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,72,0,0,0,72,72,0,0,0,0,2,1,0,0,0,2,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,1,0,0,1,0,0,71,0,0,72,0,0,0,71,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,72,0,0,0,1,1,0,0,0,0,71,72,0,0,0,2,0,0,1],[45,31,0,0,0,0,42,3,0,0,0,0,0,0,26,37,68,31,0,0,47,37,68,42,0,0,62,10,47,36,0,0,11,10,47,37],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,71,72,0,0,0,0,0,0,72] >;
C12.4S4 in GAP, Magma, Sage, TeX
C_{12}._4S_4 % in TeX
G:=Group("C12.4S4"); // GroupNames label
G:=SmallGroup(288,340);
// by ID
G=gap.SmallGroup(288,340);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,1016,422,142,675,2524,1908,172,1517,1153,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=e^2=1,b^2=c^2=a^6,d^3=a^4,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c^-1=a^6*b,d*b*d^-1=a^6*b*c,e*b*e=b*c,d*c*d^-1=b,e*c*e=a^6*c,e*d*e=a^8*d^2>;
// generators/relations
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