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## G = C62.99C23order 288 = 25·32

### 94th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.99C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — C62.99C23
 Lower central C32 — C3×C6 — C62.99C23
 Upper central C1 — C22 — C23

Generators and relations for C62.99C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=a3, d2=a3b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=b3d >

Subgroups: 530 in 179 conjugacy classes, 60 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×2], C6 [×6], C6 [×11], C2×C4 [×10], C23, C32, Dic3 [×16], C12 [×4], C2×C6 [×6], C2×C6 [×11], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3 [×4], C2×Dic3 [×18], C2×C12 [×4], C22×C6 [×2], C22×C6, C42⋊C2, C3×Dic3 [×4], C3⋊Dic3 [×4], C62, C62 [×2], C62 [×2], C4×Dic3 [×4], Dic3⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C22×Dic3 [×3], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×4], C2×C62, C23.16D6 [×2], Dic32 [×2], C62.C22 [×2], C3×C6.D4 [×2], C22×C3⋊Dic3, C62.99C23
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 [×4], C6.D6 [×2], C2×S32, C23.16D6 [×2], D6.4D6 [×2], C2×C6.D6, C62.99C23

Smallest permutation representation of C62.99C23
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 33 4 36)(2 34 5 31)(3 35 6 32)(7 27 10 30)(8 28 11 25)(9 29 12 26)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)
(1 27 18 24)(2 26 13 23)(3 25 14 22)(4 30 15 21)(5 29 16 20)(6 28 17 19)(7 42 48 36)(8 41 43 35)(9 40 44 34)(10 39 45 33)(11 38 46 32)(12 37 47 31)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(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)(7,47,11,45,9,43)(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), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,27,10,30)(8,28,11,25)(9,29,12,26)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45), (1,27,18,24)(2,26,13,23)(3,25,14,22)(4,30,15,21)(5,29,16,20)(6,28,17,19)(7,42,48,36)(8,41,43,35)(9,40,44,34)(10,39,45,33)(11,38,46,32)(12,37,47,31), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(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)(7,47,11,45,9,43)(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), (1,33,4,36)(2,34,5,31)(3,35,6,32)(7,27,10,30)(8,28,11,25)(9,29,12,26)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45), (1,27,18,24)(2,26,13,23)(3,25,14,22)(4,30,15,21)(5,29,16,20)(6,28,17,19)(7,42,48,36)(8,41,43,35)(9,40,44,34)(10,39,45,33)(11,38,46,32)(12,37,47,31), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(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),(7,47,11,45,9,43),(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)], [(1,33,4,36),(2,34,5,31),(3,35,6,32),(7,27,10,30),(8,28,11,25),(9,29,12,26),(13,40,16,37),(14,41,17,38),(15,42,18,39),(19,46,22,43),(20,47,23,44),(21,48,24,45)], [(1,27,18,24),(2,26,13,23),(3,25,14,22),(4,30,15,21),(5,29,16,20),(6,28,17,19),(7,42,48,36),(8,41,43,35),(9,40,44,34),(10,39,45,33),(11,38,46,32),(12,37,47,31)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42),(43,46),(44,47),(45,48)])`

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 S3 D6 D6 C4○D4 C4×S3 S32 D4⋊2S3 C6.D6 C2×S32 D6.4D6 kernel C62.99C23 Dic32 C62.C22 C3×C6.D4 C22×C3⋊Dic3 C2×C3⋊Dic3 C6.D4 C2×Dic3 C22×C6 C3×C6 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 2 2 1 8 2 4 2 4 8 1 4 2 1 4

Matrix representation of C62.99C23 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C62.99C23 in GAP, Magma, Sage, TeX

`C_6^2._{99}C_2^3`
`% in TeX`

`G:=Group("C6^2.99C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,605);`
`// by ID`

`G=gap.SmallGroup(288,605);`
`# by ID`

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

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