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## G = C12.1S4order 288 = 25·32

### 1st non-split extension by C12 of S4 acting via S4/A4=C2

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

 Derived series C1 — C22 — C2×C3.A4 — C12.1S4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4 — C6.S4 — C12.1S4
 Lower central C3.A4 — C2×C3.A4 — C12.1S4
 Upper central C1 — C2 — C4

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 [×2], C3, C4, C4 [×5], C22, C22 [×2], C6, C6 [×2], C2×C4 [×6], Q8 [×2], C23, C9, Dic3 [×4], C12, C12, C2×C6, C2×C6 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C18, Dic6 [×2], C2×Dic3 [×4], C2×C12 [×2], C22×C6, C22⋊Q8, Dic9 [×2], C36, C3.A4, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C2×Dic6, C22×C12, Dic18, C2×C3.A4, C12.48D4, C6.S4 [×2], C4×C3.A4, C12.1S4
Quotients: C1, C2 [×3], 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

Smallest permutation representation of C12.1S4
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 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(33 37)(34 38)(35 39)(36 40)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)
(1 71 47 5 63 39 9 67 43)(2 72 48 6 64 40 10 68 44)(3 61 37 7 65 41 11 69 45)(4 62 38 8 66 42 12 70 46)(13 53 31 17 57 35 21 49 27)(14 54 32 18 58 36 22 50 28)(15 55 33 19 59 25 23 51 29)(16 56 34 20 60 26 24 52 30)
(1 22 7 16)(2 21 8 15)(3 20 9 14)(4 19 10 13)(5 18 11 24)(6 17 12 23)(25 64 31 70)(26 63 32 69)(27 62 33 68)(28 61 34 67)(29 72 35 66)(30 71 36 65)(37 56 43 50)(38 55 44 49)(39 54 45 60)(40 53 46 59)(41 52 47 58)(42 51 48 57)```

`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), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72), (1,71,47,5,63,39,9,67,43)(2,72,48,6,64,40,10,68,44)(3,61,37,7,65,41,11,69,45)(4,62,38,8,66,42,12,70,46)(13,53,31,17,57,35,21,49,27)(14,54,32,18,58,36,22,50,28)(15,55,33,19,59,25,23,51,29)(16,56,34,20,60,26,24,52,30), (1,22,7,16)(2,21,8,15)(3,20,9,14)(4,19,10,13)(5,18,11,24)(6,17,12,23)(25,64,31,70)(26,63,32,69)(27,62,33,68)(28,61,34,67)(29,72,35,66)(30,71,36,65)(37,56,43,50)(38,55,44,49)(39,54,45,60)(40,53,46,59)(41,52,47,58)(42,51,48,57)>;`

`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), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,61)(56,62)(57,63)(58,64)(59,65)(60,66), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(33,37)(34,38)(35,39)(36,40)(49,61)(50,62)(51,63)(52,64)(53,65)(54,66)(55,67)(56,68)(57,69)(58,70)(59,71)(60,72), (1,71,47,5,63,39,9,67,43)(2,72,48,6,64,40,10,68,44)(3,61,37,7,65,41,11,69,45)(4,62,38,8,66,42,12,70,46)(13,53,31,17,57,35,21,49,27)(14,54,32,18,58,36,22,50,28)(15,55,33,19,59,25,23,51,29)(16,56,34,20,60,26,24,52,30), (1,22,7,16)(2,21,8,15)(3,20,9,14)(4,19,10,13)(5,18,11,24)(6,17,12,23)(25,64,31,70)(26,63,32,69)(27,62,33,68)(28,61,34,67)(29,72,35,66)(30,71,36,65)(37,56,43,50)(38,55,44,49)(39,54,45,60)(40,53,46,59)(41,52,47,58)(42,51,48,57) );`

`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)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,61),(56,62),(57,63),(58,64),(59,65),(60,66)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(33,37),(34,38),(35,39),(36,40),(49,61),(50,62),(51,63),(52,64),(53,65),(54,66),(55,67),(56,68),(57,69),(58,70),(59,71),(60,72)], [(1,71,47,5,63,39,9,67,43),(2,72,48,6,64,40,10,68,44),(3,61,37,7,65,41,11,69,45),(4,62,38,8,66,42,12,70,46),(13,53,31,17,57,35,21,49,27),(14,54,32,18,58,36,22,50,28),(15,55,33,19,59,25,23,51,29),(16,56,34,20,60,26,24,52,30)], [(1,22,7,16),(2,21,8,15),(3,20,9,14),(4,19,10,13),(5,18,11,24),(6,17,12,23),(25,64,31,70),(26,63,32,69),(27,62,33,68),(28,61,34,67),(29,72,35,66),(30,71,36,65),(37,56,43,50),(38,55,44,49),(39,54,45,60),(40,53,46,59),(41,52,47,58),(42,51,48,57)])`

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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