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## G = C62.6C23order 288 = 25·32

### 1st non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.6C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — C62.6C23
 Lower central C32 — C3×C6 — C62.6C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.6C23
G = < a,b,c,d,e | a6=b6=1, c2=e2=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=a3c, de=ed >

Subgroups: 634 in 169 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×8], C22, C22 [×4], S3 [×6], C6 [×6], C6 [×3], C2×C4, C2×C4 [×9], C23, C32, Dic3 [×4], Dic3 [×5], C12 [×9], D6 [×12], C2×C6 [×2], C2×C6, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×8], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C42⋊C2, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×5], C4×C12, C3×C4⋊C4, S3×C2×C4 [×2], C6.D6 [×4], C6×Dic3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C422S3, C4⋊C47S3, Dic32, C6.D12, Dic3⋊Dic3, Dic3×C12, C3×Dic3⋊C4, C6.11D12, C2×C6.D6, C62.6C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4○D4 [×2], C4×S3 [×4], C22×S3 [×2], C42⋊C2, S32, S3×C2×C4 [×2], C4○D12 [×2], D42S3, Q83S3, C2×S32, C422S3, C4⋊C47S3, D6.6D6, C4×S32, D6.3D6, C62.6C23

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G ··· 4L 4M 4N 6A ··· 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T 12U 12V order 1 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 18 18 2 2 4 2 2 3 3 3 3 6 ··· 6 18 18 2 ··· 2 4 4 4 2 2 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 4 4 4 4 4 4 4 type + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 S3 D6 D6 C4○D4 C4×S3 C4○D12 S32 D4⋊2S3 Q8⋊3S3 C2×S32 D6.6D6 C4×S32 D6.3D6 kernel C62.6C23 Dic32 C6.D12 Dic3⋊Dic3 Dic3×C12 C3×Dic3⋊C4 C6.11D12 C2×C6.D6 C6.D6 C4×Dic3 Dic3⋊C4 C2×Dic3 C2×C12 C3×C6 Dic3 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 4 2 4 8 8 1 1 1 1 2 2 2

Matrix representation of C62.6C23 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 1 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 5 3 0 0 0 0 5 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 1 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 11 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 0 8

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,5,5,0,0,0,0,3,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,1,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,11,12,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;`

C62.6C23 in GAP, Magma, Sage, TeX

`C_6^2._6C_2^3`
`% in TeX`

`G:=Group("C6^2.6C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,484);`
`// by ID`

`G=gap.SmallGroup(288,484);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,254,100,1356,9414]);`
`// Polycyclic`

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

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