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## G = C62.94C23order 288 = 25·32

### 89th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.94C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — C62.94C23
 Lower central C32 — C3×C6 — C62.94C23
 Upper central C1 — C22 — C23

Generators and relations for C62.94C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a3b3c, de=ed >

Subgroups: 802 in 215 conjugacy classes, 64 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×6], S3 [×6], C6 [×6], C6 [×9], C2×C4 [×9], D4 [×4], C23, C23, C32, Dic3 [×2], Dic3 [×9], C12 [×5], D6 [×12], C2×C6 [×2], C2×C6 [×4], C2×C6 [×9], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3⋊S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×4], C2×Dic3 [×4], C2×Dic3 [×5], C3⋊D4 [×12], C2×C12 [×6], C22×S3 [×3], C22×C6 [×2], C22×C6, C4×D4, C3×Dic3 [×2], C3×Dic3 [×3], C3⋊Dic3 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×2], C62 [×2], C4×Dic3 [×2], Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4, C3×C22⋊C4, S3×C2×C4 [×2], C22×Dic3, C2×C3⋊D4 [×3], C22×C12, C6.D6 [×2], C6×Dic3 [×4], C6×Dic3 [×2], C2×C3⋊Dic3, C327D4 [×4], C22×C3⋊S3, C2×C62, Dic34D4, C4×C3⋊D4, Dic32, C6.D12, C62.C22, C3×C6.D4, C2×C6.D6, Dic3×C2×C6, C2×C327D4, C62.94C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×4], C3⋊D4 [×2], C22×S3 [×2], C4×D4, S32, S3×C2×C4 [×2], C4○D12, S3×D4, D42S3, C2×C3⋊D4, C6.D6 [×2], C2×S32, Dic34D4, C4×C3⋊D4, D6.3D6, C2×C6.D6, S3×C3⋊D4, C62.94C23

Smallest permutation representation of C62.94C23
On 48 points
Generators in S48
```(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)
(1 18 5 16 3 14)(2 13 6 17 4 15)(7 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)
(1 10 16 48)(2 11 17 43)(3 12 18 44)(4 7 13 45)(5 8 14 46)(6 9 15 47)(19 34 28 39)(20 35 29 40)(21 36 30 41)(22 31 25 42)(23 32 26 37)(24 33 27 38)
(1 33 4 36)(2 32 5 35)(3 31 6 34)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 38)(8 39)(9 40)(10 41)(11 42)(12 37)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)```

`G:=sub<Sym(48)| (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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,43,9,45,11,47)(8,44,10,46,12,48)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39), (1,10,16,48)(2,11,17,43)(3,12,18,44)(4,7,13,45)(5,8,14,46)(6,9,15,47)(19,34,28,39)(20,35,29,40)(21,36,30,41)(22,31,25,42)(23,32,26,37)(24,33,27,38), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45), (1,24)(2,19)(3,20)(4,21)(5,22)(6,23)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)>;`

`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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,43,9,45,11,47)(8,44,10,46,12,48)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39), (1,10,16,48)(2,11,17,43)(3,12,18,44)(4,7,13,45)(5,8,14,46)(6,9,15,47)(19,34,28,39)(20,35,29,40)(21,36,30,41)(22,31,25,42)(23,32,26,37)(24,33,27,38), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45), (1,24)(2,19)(3,20)(4,21)(5,22)(6,23)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48) );`

`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)], [(1,18,5,16,3,14),(2,13,6,17,4,15),(7,43,9,45,11,47),(8,44,10,46,12,48),(19,30,23,28,21,26),(20,25,24,29,22,27),(31,40,33,42,35,38),(32,41,34,37,36,39)], [(1,10,16,48),(2,11,17,43),(3,12,18,44),(4,7,13,45),(5,8,14,46),(6,9,15,47),(19,34,28,39),(20,35,29,40),(21,36,30,41),(22,31,25,42),(23,32,26,37),(24,33,27,38)], [(1,33,4,36),(2,32,5,35),(3,31,6,34),(7,30,10,27),(8,29,11,26),(9,28,12,25),(13,41,16,38),(14,40,17,37),(15,39,18,42),(19,44,22,47),(20,43,23,46),(21,48,24,45)], [(1,24),(2,19),(3,20),(4,21),(5,22),(6,23),(7,38),(8,39),(9,40),(10,41),(11,42),(12,37),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)])`

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E ··· 4J 4K 4L 6A ··· 6J 6K ··· 6S 12A ··· 12H 12I 12J 12K 12L order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 ··· 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 18 18 2 2 4 3 3 3 3 6 ··· 6 18 18 2 ··· 2 4 ··· 4 6 ··· 6 12 12 12 12

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 S3 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 S32 S3×D4 D4⋊2S3 C6.D6 C2×S32 D6.3D6 S3×C3⋊D4 kernel C62.94C23 Dic32 C6.D12 C62.C22 C3×C6.D4 C2×C6.D6 Dic3×C2×C6 C2×C32⋊7D4 C32⋊7D4 C6.D4 C22×Dic3 C3×Dic3 C2×Dic3 C22×C6 C3×C6 Dic3 C2×C6 C6 C23 C6 C6 C22 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 2 4 2 2 4 8 4 1 1 1 2 1 2 2

Matrix representation of C62.94C23 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 1 1
,
 1 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 7 10 0 0 0 0 8 6 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 3 0 0 0 0 10 7 0 0 0 0 0 0 12 0 0 0 0 0 0 12

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,7,8,0,0,0,0,10,6,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,10,0,0,0,0,3,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;`

C62.94C23 in GAP, Magma, Sage, TeX

`C_6^2._{94}C_2^3`
`% in TeX`

`G:=Group("C6^2.94C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,600);`
`// by ID`

`G=gap.SmallGroup(288,600);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=b^3,d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^3*b^3*c,d*e=e*d>;`
`// generators/relations`

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