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## G = C62.113C23order 288 = 25·32

### 108th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.113C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C62.113C23
 Lower central C32 — C62 — C62.113C23
 Upper central C1 — C22 — C23

Generators and relations for C62.113C23
G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3b3c, ede=b3d >

Subgroups: 930 in 215 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×6], C6 [×6], C6 [×9], C2×C4 [×6], D4 [×6], C23, C23 [×2], C32, Dic3 [×9], C12 [×3], D6 [×2], D6 [×12], C2×C6 [×2], C2×C6 [×15], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3⋊S3, C3×C6 [×3], C3×C6, C4×S3 [×2], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×5], C3⋊D4 [×12], C2×C12 [×3], C3×D4 [×2], C22×S3, C22×S3 [×3], C22×C6 [×2], C22×C6 [×2], C4⋊D4, C3×Dic3 [×3], C3⋊Dic3 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×3], C62, C62 [×3], Dic3⋊C4 [×2], D6⋊C4 [×2], C6.D4, C3×C22⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4 [×4], C6×D4, S3×Dic3 [×2], C3⋊D12 [×2], C6×Dic3 [×3], C3×C3⋊D4 [×2], C2×C3⋊Dic3, C327D4 [×2], S3×C2×C6, C22×C3⋊S3, C2×C62, Dic3⋊D4, C23.14D6, C6.D12, C62.C22, C3×C6.D4, C2×S3×Dic3, C2×C3⋊D12, C6×C3⋊D4, C2×C327D4, C62.113C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3 [×2], C4⋊D4, S32, C4○D12, S3×D4 [×3], D42S3, C2×C3⋊D4, C2×S32, Dic3⋊D4, C23.14D6, D6.3D6, S3×C3⋊D4, Dic3⋊D6, C62.113C23

Smallest permutation representation of C62.113C23
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 44)(2 45)(3 46)(4 47)(5 48)(6 43)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)(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 35 4 32)(2 34 5 31)(3 33 6 36)(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 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(13 26)(14 27)(15 28)(16 29)(17 30)(18 25)(37 47)(38 48)(39 43)(40 44)(41 45)(42 46)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6Q 6R 6S 12A ··· 12F order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 12 ··· 12 size 1 1 1 1 4 6 6 36 2 2 4 6 6 12 12 18 18 2 ··· 2 4 ··· 4 12 12 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 C4○D12 S32 S3×D4 D4⋊2S3 C2×S32 D6.3D6 S3×C3⋊D4 Dic3⋊D6 kernel C62.113C23 C6.D12 C62.C22 C3×C6.D4 C2×S3×Dic3 C2×C3⋊D12 C6×C3⋊D4 C2×C32⋊7D4 C6.D4 C2×C3⋊D4 C3⋊Dic3 S3×C6 C2×Dic3 C22×S3 C22×C6 C3×C6 D6 C6 C23 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 2 3 1 2 2 4 4 1 3 1 1 2 2 2

Matrix representation of C62.113C23 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 0 1 0 0 0 0 0 0 12 12
,
 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 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
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 5 2 0 0 0 0 0 0 1 8 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 1 0 0 0 0 0 0 0 0 1
,
 6 3 0 0 0 0 0 0 5 7 0 0 0 0 0 0 0 0 8 11 0 0 0 0 0 0 12 5 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 12 12
,
 6 3 0 0 0 0 0 0 10 7 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 5 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 1 0 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[6,5,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,8,12,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[6,10,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;`

C62.113C23 in GAP, Magma, Sage, TeX

`C_6^2._{113}C_2^3`
`% in TeX`

`G:=Group("C6^2.113C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,619);`
`// by ID`

`G=gap.SmallGroup(288,619);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,1356,9414]);`
`// Polycyclic`

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

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