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## G = C62.112C23order 288 = 25·32

### 107th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.112C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×S3×Dic3 — C62.112C23
 Lower central C32 — C62 — C62.112C23
 Upper central C1 — C22 — C23

Generators and relations for C62.112C23
G = < a,b,c,d,e | a6=b6=d2=e2=1, c2=b3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=a3d >

Subgroups: 762 in 205 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×3], C6 [×6], C6 [×10], C2×C4 [×6], D4 [×6], C23, C23 [×2], C32, Dic3 [×2], Dic3 [×7], C12 [×3], D6 [×2], D6 [×5], C2×C6 [×2], C2×C6 [×18], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×3], C3×C6 [×3], C3×C6, C4×S3 [×2], C2×Dic3 [×2], C2×Dic3 [×8], C3⋊D4 [×8], C2×C12 [×2], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C22×C6 [×3], C4⋊D4, C3×Dic3 [×2], C3×Dic3, C3⋊Dic3 [×2], S3×C6 [×2], S3×C6 [×5], C62, C62 [×3], Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4 [×4], S3×C2×C4, C22×Dic3, C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C6×D4 [×2], S3×Dic3 [×2], D6⋊S3 [×2], C6×Dic3 [×2], C3×C3⋊D4 [×4], C2×C3⋊Dic3 [×2], S3×C2×C6 [×2], C2×C62, D63D4, C23.14D6, D6⋊Dic3, Dic3⋊Dic3, C625C4, C2×S3×Dic3, C2×D6⋊S3, C6×C3⋊D4 [×2], C62.112C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, C3⋊D4 [×4], C22×S3 [×2], C4⋊D4, S32, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4 [×2], C2×S32, D63D4, C23.14D6, D6.4D6, S3×C3⋊D4 [×2], C62.112C23

Smallest permutation representation of C62.112C23
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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 26)(14 25)(15 30)(16 29)(17 28)(18 27)(31 47)(32 46)(33 45)(34 44)(35 43)(36 48)
(7 48)(8 43)(9 44)(10 45)(11 46)(12 47)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)```

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

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

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

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 6T 6U 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 1 1 4 6 6 12 2 2 4 6 6 12 18 18 36 2 ··· 2 4 ··· 4 12 12 12 12 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 C3⋊D4 S32 S3×D4 D4⋊2S3 C2×S32 D6.4D6 S3×C3⋊D4 kernel C62.112C23 D6⋊Dic3 Dic3⋊Dic3 C62⋊5C4 C2×S3×Dic3 C2×D6⋊S3 C6×C3⋊D4 C2×C3⋊D4 C3×Dic3 S3×C6 C2×Dic3 C22×S3 C22×C6 C3×C6 Dic3 D6 C23 C6 C6 C22 C2 C2 # reps 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 1 2 2 1 2 4

Matrix representation of C62.112C23 in GL8(𝔽13)

 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 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 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 8 0 0 0 0 0 0 8 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 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
,
 0 1 0 0 0 0 0 0 1 0 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 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 12 12
,
 1 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 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

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

C62.112C23 in GAP, Magma, Sage, TeX

`C_6^2._{112}C_2^3`
`% in TeX`

`G:=Group("C6^2.112C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,618);`
`// by ID`

`G=gap.SmallGroup(288,618);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,422,135,1356,9414]);`
`// Polycyclic`

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

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