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## G = C62.97C23order 288 = 25·32

### 92nd non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.97C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — C62.97C23
 Lower central C32 — C3×C6 — C62.97C23
 Upper central C1 — C22 — C23

Generators and relations for C62.97C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a3c, de=ed >

Subgroups: 474 in 169 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×2], C6 [×2], C6 [×4], C6 [×9], C2×C4 [×10], C23, C32, Dic3 [×4], Dic3 [×8], C12 [×6], C2×C6 [×2], C2×C6 [×4], C2×C6 [×9], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×Dic3 [×6], C2×C12 [×8], C22×C6 [×2], C22×C6, C42⋊C2, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3 [×2], C62, C62 [×2], C62 [×2], C4×Dic3 [×4], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C6.D4, C6.D4 [×3], C3×C22⋊C4, C22×Dic3, C22×C12, C6×Dic3 [×2], C6×Dic3 [×6], C2×C3⋊Dic3 [×2], C2×C62, C23.16D6, C23.26D6, Dic32 [×2], Dic3⋊Dic3 [×2], C3×C6.D4, C625C4, Dic3×C2×C6, C62.97C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], C22×C4, C4○D4 [×2], C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C42⋊C2, S32, S3×C2×C4, C4○D12 [×2], D42S3 [×2], C22×Dic3, S3×Dic3 [×2], C2×S32, C23.16D6, C23.26D6, C2×S3×Dic3, D6.3D6 [×2], C62.97C23

Smallest permutation representation of C62.97C23
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 47 11 45 9 43)(8 48 12 46 10 44)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 38 33 40 35 42)(32 39 34 41 36 37)
(1 39 15 36)(2 40 16 31)(3 41 17 32)(4 42 18 33)(5 37 13 34)(6 38 14 35)(7 27 45 21)(8 28 46 22)(9 29 47 23)(10 30 48 24)(11 25 43 19)(12 26 44 20)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(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 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 48)(8 43)(9 44)(10 45)(11 46)(12 47)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(31 37)(32 38)(33 39)(34 40)(35 41)(36 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), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,47,11,45,9,43)(8,48,12,46,10,44)(19,29,21,25,23,27)(20,30,22,26,24,28)(31,38,33,40,35,42)(32,39,34,41,36,37), (1,39,15,36)(2,40,16,31)(3,41,17,32)(4,42,18,33)(5,37,13,34)(6,38,14,35)(7,27,45,21)(8,28,46,22)(9,29,47,23)(10,30,48,24)(11,25,43,19)(12,26,44,20), (1,21,4,24)(2,20,5,23)(3,19,6,22)(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), (1,15)(2,16)(3,17)(4,18)(5,13)(6,14)(7,48)(8,43)(9,44)(10,45)(11,46)(12,47)(19,25)(20,26)(21,27)(22,28)(23,29)(24,30)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)>;`

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + - + + + - - + image C1 C2 C2 C2 C2 C2 C4 S3 S3 Dic3 D6 D6 C4○D4 C4×S3 C4○D12 S32 D4⋊2S3 S3×Dic3 C2×S32 D6.3D6 kernel C62.97C23 Dic32 Dic3⋊Dic3 C3×C6.D4 C62⋊5C4 Dic3×C2×C6 C6×Dic3 C6.D4 C22×Dic3 C2×Dic3 C2×Dic3 C22×C6 C3×C6 C2×C6 C6 C23 C6 C22 C22 C2 # reps 1 2 2 1 1 1 8 1 1 4 4 2 4 4 8 1 2 2 1 4

Matrix representation of C62.97C23 in GL6(𝔽13)

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

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

C62.97C23 in GAP, Magma, Sage, TeX

`C_6^2._{97}C_2^3`
`% in TeX`

`G:=Group("C6^2.97C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,603);`
`// by ID`

`G=gap.SmallGroup(288,603);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,254,1356,9414]);`
`// Polycyclic`

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

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