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## G = C62.70C23order 288 = 25·32

### 65th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.70C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C62.70C23
 Lower central C32 — C3×C6 — C62.70C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.70C23
G = < a,b,c,d,e | a6=b6=1, c2=d2=a3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d >

Subgroups: 786 in 211 conjugacy classes, 68 normal (16 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C2×C4, C2×C4 [×13], C23, C32, Dic3 [×10], C12 [×4], C12 [×6], D6 [×18], C2×C6 [×2], C2×C6, C4⋊C4 [×4], C22×C4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×20], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C2×C4⋊C4, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×6], C62, Dic3⋊C4 [×4], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], S3×C2×C4 [×7], C6.D6 [×4], C6×Dic3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C4⋊C4 [×2], C62.C22 [×2], C3×C4⋊Dic3 [×2], C2×C6.D6 [×2], C2×C4×C3⋊S3, C62.70C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×6], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4×S3 [×4], C22×S3 [×2], C2×C4⋊C4, S32, S3×C2×C4 [×2], S3×D4 [×2], S3×Q8 [×2], C6.D6 [×2], C2×S32, S3×C4⋊C4 [×2], Dic3.D6, D6⋊D6, C2×C6.D6, C62.70C23

Smallest permutation representation of C62.70C23
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 34 4 31)(2 35 5 32)(3 36 6 33)(7 27 10 30)(8 28 11 25)(9 29 12 26)(13 38 16 41)(14 39 17 42)(15 40 18 37)(19 45 22 48)(20 46 23 43)(21 47 24 44)
(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 31 43 38)(8 32 44 39)(9 33 45 40)(10 34 46 41)(11 35 47 42)(12 36 48 37)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C ··· 4J 4K 4L 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I ··· 12P order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 9 9 9 9 2 2 4 2 2 6 ··· 6 18 18 2 ··· 2 4 4 4 4 ··· 4 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + - + + + + - + + image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D6 D6 C4×S3 S32 S3×D4 S3×Q8 C6.D6 C2×S32 Dic3.D6 D6⋊D6 kernel C62.70C23 C62.C22 C3×C4⋊Dic3 C2×C6.D6 C2×C4×C3⋊S3 C4×C3⋊S3 C4⋊Dic3 C2×C3⋊S3 C2×C3⋊S3 C2×Dic3 C2×C12 C12 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 2 1 8 2 2 2 4 2 8 1 2 2 2 1 2 2

Matrix representation of C62.70C23 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 12 0
,
 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

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

C62.70C23 in GAP, Magma, Sage, TeX

`C_6^2._{70}C_2^3`
`% in TeX`

`G:=Group("C6^2.70C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,548);`
`// by ID`

`G=gap.SmallGroup(288,548);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,422,219,100,1356,9414]);`
`// Polycyclic`

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

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