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## G = C62.38C23order 288 = 25·32

### 33rd non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.38C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C62.38C23
 Lower central C32 — C62 — C62.38C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.38C23
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, dcd-1=b3c, ece-1=a3b3c, de=ed >

Subgroups: 570 in 137 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3 [×2], C3, C4 [×6], C22, C22 [×3], S3 [×4], C6 [×6], C6 [×3], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×7], C12 [×7], D6 [×10], C2×C6 [×2], C2×C6, C42, C22⋊C4 [×3], C4⋊C4 [×3], C3⋊S3, C3×C6 [×3], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C422C2, C3×Dic3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×3], C62, C4×Dic3, Dic3⋊C4, Dic3⋊C4 [×3], C4⋊Dic3, D6⋊C4 [×7], C4×C12, C3×C4⋊C4, C6×Dic3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C423S3, C4⋊C4⋊S3, C6.D12 [×2], Dic3⋊Dic3, C62.C22, Dic3×C12, C3×Dic3⋊C4, C6.11D12, C62.38C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], C23, D6 [×6], C4○D4 [×3], C22×S3 [×2], C422C2, S32, C4○D12 [×4], D42S3, Q83S3, C2×S32, C423S3, C4⋊C4⋊S3, D6.D6, D6.6D6, D6.3D6, C62.38C23

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 6A ··· 6F 6G 6H 6I 12A 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T 12U 12V order 1 2 2 2 2 3 3 3 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 36 2 2 4 2 2 6 6 6 6 12 12 36 2 ··· 2 4 4 4 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

48 irreducible representations

 dim 1 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 C2 C2 S3 S3 D6 D6 C4○D4 C4○D12 S32 D4⋊2S3 Q8⋊3S3 C2×S32 D6.D6 D6.6D6 D6.3D6 kernel C62.38C23 C6.D12 Dic3⋊Dic3 C62.C22 Dic3×C12 C3×Dic3⋊C4 C6.11D12 C4×Dic3 Dic3⋊C4 C2×Dic3 C2×C12 C3×C6 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 1 1 4 2 6 16 1 1 1 1 2 2 2

Matrix representation of C62.38C23 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
,
 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 12 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 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 3 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 0 1 0 0 0 0 0 0 1 0
,
 5 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 11 5 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

`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],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,11,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;`

C62.38C23 in GAP, Magma, Sage, TeX

`C_6^2._{38}C_2^3`
`% in TeX`

`G:=Group("C6^2.38C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,516);`
`// by ID`

`G=gap.SmallGroup(288,516);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,590,100,1356,9414]);`
`// 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,d*c*d^-1=b^3*c,e*c*e^-1=a^3*b^3*c,d*e=e*d>;`
`// generators/relations`

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