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## G = C62.58C23order 288 = 25·32

### 53rd non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.58C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C62.58C23
 Lower central C32 — C62 — C62.58C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.58C23
G = < a,b,c,d,e | a6=b6=1, c2=d2=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=a3d >

Subgroups: 666 in 165 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×7], C22, C22 [×4], S3 [×6], C6 [×6], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, C32, Dic3 [×2], Dic3 [×6], C12 [×8], D6 [×12], C2×C6 [×2], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3⋊S3 [×2], C3×C6 [×3], Dic6 [×2], C4×S3 [×4], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C3×Q8 [×2], C22×S3 [×3], C22⋊Q8, C3×Dic3 [×2], C3×Dic3 [×3], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, Dic3⋊C4, Dic3⋊C4 [×3], C4⋊Dic3, D6⋊C4 [×5], C3×C4⋊C4, C2×Dic6, S3×C2×C4 [×2], C6×Q8, C6.D6 [×2], C3×Dic6 [×2], C6×Dic3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊Q8, D63Q8, C6.D12, Dic3⋊Dic3, C62.C22, C3×Dic3⋊C4, C6.11D12, C2×C6.D6, C6×Dic6, C62.58C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, C3⋊D4 [×2], C22×S3 [×2], C22⋊Q8, S32, C4○D12, S3×D4, S3×Q8 [×2], Q83S3, C2×C3⋊D4, C2×S32, D6⋊Q8, D63Q8, Dic3.D6, D6.6D6, S3×C3⋊D4, C62.58C23

Smallest permutation representation of C62.58C23
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 41 16 34)(2 42 17 35)(3 37 18 36)(4 38 13 31)(5 39 14 32)(6 40 15 33)(7 20 43 27)(8 21 44 28)(9 22 45 29)(10 23 46 30)(11 24 47 25)(12 19 48 26)
(1 41 16 34)(2 40 17 33)(3 39 18 32)(4 38 13 31)(5 37 14 36)(6 42 15 35)(7 30 43 23)(8 29 44 22)(9 28 45 21)(10 27 46 20)(11 26 47 19)(12 25 48 24)
(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 41 43 34)(8 42 44 35)(9 37 45 36)(10 38 46 31)(11 39 47 32)(12 40 48 33)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 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 4 6 6 6 6 12 12 36 2 ··· 2 4 4 4 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + - + + + + - + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 Q8 D6 D6 C4○D4 C3⋊D4 C4○D12 S32 S3×D4 S3×Q8 Q8⋊3S3 C2×S32 Dic3.D6 D6.6D6 S3×C3⋊D4 kernel C62.58C23 C6.D12 Dic3⋊Dic3 C62.C22 C3×Dic3⋊C4 C6.11D12 C2×C6.D6 C6×Dic6 Dic3⋊C4 C2×Dic6 C3×Dic3 C2×C3⋊S3 C2×Dic3 C2×C12 C3×C6 Dic3 C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 2 4 2 2 4 4 1 1 2 1 1 2 2 2

Matrix representation of C62.58C23 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 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
,
 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 5 0 0 0 0 0 0 0 0 8 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 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 0 0 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 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 1
,
 0 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 0 1 0 0 0 0 0 0 1 0

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

C62.58C23 in GAP, Magma, Sage, TeX

`C_6^2._{58}C_2^3`
`% in TeX`

`G:=Group("C6^2.58C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,536);`
`// by ID`

`G=gap.SmallGroup(288,536);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,422,135,100,1356,9414]);`
`// Polycyclic`

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

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