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## G = C62.115C23order 288 = 25·32

### 110th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.115C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C62.115C23
 Lower central C32 — C3×C6 — C62.115C23
 Upper central C1 — C22 — C23

Generators and relations for C62.115C23
G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=b3d >

Subgroups: 690 in 215 conjugacy classes, 70 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×6], C6 [×13], C2×C4 [×9], D4 [×4], C23, C23, C32, Dic3 [×2], Dic3 [×12], C12 [×4], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×4], C2×C6 [×15], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×2], C3×C6 [×3], C3×C6 [×2], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×16], C3⋊D4 [×4], C2×C12 [×3], C3×D4 [×4], C22×S3, C22×C6 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], C3⋊Dic3, S3×C6 [×2], S3×C6 [×2], C62, C62 [×2], C62 [×2], C4×Dic3 [×2], Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, S3×C2×C4, C22×Dic3 [×4], C2×C3⋊D4, C6×D4, S3×Dic3 [×2], C6×Dic3 [×3], C3×C3⋊D4 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6, C2×C62, Dic34D4, D4×Dic3, Dic32, D6⋊Dic3, Dic3⋊Dic3, C3×C6.D4, C2×S3×Dic3, C6×C3⋊D4, C22×C3⋊Dic3, C62.115C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], C23, Dic3 [×4], D6 [×6], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C4×D4, S32, S3×C2×C4, S3×D4 [×2], D42S3 [×2], C22×Dic3, S3×Dic3 [×2], C2×S32, Dic34D4, D4×Dic3, C2×S3×Dic3, D6.4D6, Dic3⋊D6, C62.115C23

Smallest permutation representation of C62.115C23
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 17 5 15 3 13)(2 18 6 16 4 14)(7 45 11 43 9 47)(8 46 12 44 10 48)(19 25 21 27 23 29)(20 26 22 28 24 30)(31 42 33 38 35 40)(32 37 34 39 36 41)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)
(1 19 4 22)(2 24 5 21)(3 23 6 20)(7 42 10 39)(8 41 11 38)(9 40 12 37)(13 29 16 26)(14 28 17 25)(15 27 18 30)(31 44 34 47)(32 43 35 46)(33 48 36 45)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 30)(20 25)(21 26)(22 27)(23 28)(24 29)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)(43 46)(44 47)(45 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), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,45,11,43,9,47)(8,46,12,44,10,48)(19,25,21,27,23,29)(20,26,22,28,24,30)(31,42,33,38,35,40)(32,37,34,39,36,41), (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (1,19,4,22)(2,24,5,21)(3,23,6,20)(7,42,10,39)(8,41,11,38)(9,40,12,37)(13,29,16,26)(14,28,17,25)(15,27,18,30)(31,44,34,47)(32,43,35,46)(33,48,36,45), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,30)(20,25)(21,26)(22,27)(23,28)(24,29)(31,41)(32,42)(33,37)(34,38)(35,39)(36,40)(43,46)(44,47)(45,48)>;`

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

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

48 conjugacy classes

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

48 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 Dic3 D6 D6 C4○D4 C4×S3 S32 S3×D4 D4⋊2S3 S3×Dic3 C2×S32 D6.4D6 Dic3⋊D6 kernel C62.115C23 Dic32 D6⋊Dic3 Dic3⋊Dic3 C3×C6.D4 C2×S3×Dic3 C6×C3⋊D4 C22×C3⋊Dic3 C3×C3⋊D4 C6.D4 C2×C3⋊D4 C3⋊Dic3 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C3×C6 C2×C6 C23 C6 C6 C22 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 2 3 4 1 2 2 4 1 2 2 2 1 2 2

Matrix representation of C62.115C23 in GL8(𝔽13)

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

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

C62.115C23 in GAP, Magma, Sage, TeX

`C_6^2._{115}C_2^3`
`% in TeX`

`G:=Group("C6^2.115C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,621);`
`// by ID`

`G=gap.SmallGroup(288,621);`
`# by ID`

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

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

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