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## G = C62.19C23order 288 = 25·32

### 14th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.19C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — C62.19C23
 Lower central C32 — C3×C6 — C62.19C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.19C23
G = < a,b,c,d,e | a6=b6=1, c2=a3, d2=a3b3, 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: 658 in 179 conjugacy classes, 60 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×8], C6 [×6], C6 [×3], C2×C4, C2×C4 [×9], C23, C32, Dic3 [×10], C12 [×4], C12 [×6], D6 [×14], C2×C6 [×2], C2×C6, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×12], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C42⋊C2, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4×Dic3 [×4], C4⋊Dic3 [×2], D6⋊C4 [×4], C3×C4⋊C4 [×2], S3×C2×C4 [×3], C6×Dic3 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C4⋊C47S3 [×2], Dic32 [×2], C6.D12 [×2], C3×C4⋊Dic3 [×2], C2×C4×C3⋊S3, C62.19C23
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], D42S3 [×2], Q83S3 [×2], C6.D6 [×2], C2×S32, C4⋊C47S3 [×2], D12⋊S3 [×2], C2×C6.D6, C62.19C23

Smallest permutation representation of C62.19C23
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 43 4 46)(2 44 5 47)(3 45 6 48)(7 13 10 16)(8 14 11 17)(9 15 12 18)(19 37 22 40)(20 38 23 41)(21 39 24 42)(25 35 28 32)(26 36 29 33)(27 31 30 34)
(1 38 13 34)(2 37 14 33)(3 42 15 32)(4 41 16 31)(5 40 17 36)(6 39 18 35)(7 30 46 20)(8 29 47 19)(9 28 48 24)(10 27 43 23)(11 26 44 22)(12 25 45 21)
(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), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,13,10,16)(8,14,11,17)(9,15,12,18)(19,37,22,40)(20,38,23,41)(21,39,24,42)(25,35,28,32)(26,36,29,33)(27,31,30,34), (1,38,13,34)(2,37,14,33)(3,42,15,32)(4,41,16,31)(5,40,17,36)(6,39,18,35)(7,30,46,20)(8,29,47,19)(9,28,48,24)(10,27,43,23)(11,26,44,22)(12,25,45,21), (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), (1,43,4,46)(2,44,5,47)(3,45,6,48)(7,13,10,16)(8,14,11,17)(9,15,12,18)(19,37,22,40)(20,38,23,41)(21,39,24,42)(25,35,28,32)(26,36,29,33)(27,31,30,34), (1,38,13,34)(2,37,14,33)(3,42,15,32)(4,41,16,31)(5,40,17,36)(6,39,18,35)(7,30,46,20)(8,29,47,19)(9,28,48,24)(10,27,43,23)(11,26,44,22)(12,25,45,21), (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)], [(1,43,4,46),(2,44,5,47),(3,45,6,48),(7,13,10,16),(8,14,11,17),(9,15,12,18),(19,37,22,40),(20,38,23,41),(21,39,24,42),(25,35,28,32),(26,36,29,33),(27,31,30,34)], [(1,38,13,34),(2,37,14,33),(3,42,15,32),(4,41,16,31),(5,40,17,36),(6,39,18,35),(7,30,46,20),(8,29,47,19),(9,28,48,24),(10,27,43,23),(11,26,44,22),(12,25,45,21)], [(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 3A 3B 3C 4A 4B 4C ··· 4J 4K 4L 4M 4N 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I ··· 12P order 1 2 2 2 2 2 3 3 3 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 18 18 2 2 4 2 2 6 ··· 6 9 9 9 9 2 ··· 2 4 4 4 4 ··· 4 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + - + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 C4○D4 C4×S3 S32 D4⋊2S3 Q8⋊3S3 C6.D6 C2×S32 D12⋊S3 kernel C62.19C23 Dic32 C6.D12 C3×C4⋊Dic3 C2×C4×C3⋊S3 C4×C3⋊S3 C4⋊Dic3 C2×Dic3 C2×C12 C3×C6 C12 C2×C4 C6 C6 C4 C22 C2 # reps 1 2 2 2 1 8 2 4 2 4 8 1 2 2 2 1 4

Matrix representation of C62.19C23 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 0 1 0 0 0 0 0 0 12 12
,
 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 8 3 0 0 0 0 0 0 5 5 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 12 0 0 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 11 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 12 12
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 8 8 0 0 0 0 0 0 0 0 8 0 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

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

C62.19C23 in GAP, Magma, Sage, TeX

`C_6^2._{19}C_2^3`
`% in TeX`

`G:=Group("C6^2.19C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,497);`
`// by ID`

`G=gap.SmallGroup(288,497);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,176,422,219,100,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=a^3,d^2=a^3*b^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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