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## G = C62.23C23order 288 = 25·32

### 18th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.23C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.23C23
 Lower central C32 — C62 — C62.23C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.23C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=b3, e2=a3b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a3c, ece-1=b3c, ede-1=b3d >

Subgroups: 786 in 183 conjugacy classes, 46 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×9], C6 [×6], C6 [×4], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C32, Dic3 [×7], C12 [×7], D6 [×17], C2×C6 [×2], C2×C6 [×4], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3, C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×8], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×3], C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×S3 [×3], C22×C6, C22.D4, C3×Dic3 [×3], C3⋊Dic3, C3×C12, S3×C6 [×3], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4 [×3], C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4 [×3], C2×D12, C2×C3⋊D4, C3⋊D12 [×2], C6×Dic3 [×3], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C23.9D6, D6.D4, D6⋊Dic3, C6.D12, Dic3⋊Dic3, C3×Dic3⋊C4, C3×D6⋊C4, C2×C3⋊D12, C2×C4×C3⋊S3, C62.23C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], C22×S3 [×2], C22.D4, S32, C4○D12 [×2], S3×D4 [×2], D42S3, Q83S3, C2×S32, C23.9D6, D6.D4, D12⋊S3, D6.D6, Dic3⋊D6, C62.23C23

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C4○D4 C4○D12 S32 S3×D4 D4⋊2S3 Q8⋊3S3 C2×S32 D12⋊S3 D6.D6 Dic3⋊D6 kernel C62.23C23 D6⋊Dic3 C6.D12 Dic3⋊Dic3 C3×Dic3⋊C4 C3×D6⋊C4 C2×C3⋊D12 C2×C4×C3⋊S3 Dic3⋊C4 D6⋊C4 C2×C3⋊S3 C2×Dic3 C2×C12 C22×S3 C3×C6 C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 3 2 1 4 8 1 2 1 1 1 2 2 2

Matrix representation of C62.23C23 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 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
,
 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 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
,
 8 5 0 0 0 0 0 0 3 5 0 0 0 0 0 0 0 0 0 12 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 7 3 0 0 0 0 0 0 10 6
,
 12 0 0 0 0 0 0 0 11 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 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 7 0 0 0 0 0 0 6 10
,
 8 0 0 0 0 0 0 0 0 8 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 0 1 0 0 0 0 0 0 1 0

`G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[8,3,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6],[12,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,7,10],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;`

C62.23C23 in GAP, Magma, Sage, TeX

`C_6^2._{23}C_2^3`
`% in TeX`

`G:=Group("C6^2.23C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,501);`
`// by ID`

`G=gap.SmallGroup(288,501);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,422,219,142,1356,9414]);`
`// Polycyclic`

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

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