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## G = C62.65C23order 288 = 25·32

### 60th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.65C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C62.65C23
 Lower central C32 — C62 — C62.65C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.65C23
G = < a,b,c,d,e | a6=b6=1, c2=d2=b3, e2=a3, 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: 698 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 [×4], C4×S3 [×4], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], 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, C4⋊Dic3, C4⋊Dic3, D6⋊C4 [×5], C3×C4⋊C4 [×2], C2×Dic6 [×2], S3×C2×C4 [×2], C6.D6 [×2], C322Q8 [×2], C6×Dic3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, D6⋊Q8, C4.D12, C6.D12, Dic3⋊Dic3, C3×Dic3⋊C4, C3×C4⋊Dic3, C6.11D12, C2×C6.D6, C2×C322Q8, C62.65C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×2], C23, D6 [×6], C2×D4, C2×Q8, C4○D4, D12 [×2], C22×S3 [×2], C22⋊Q8, S32, C2×D12, C4○D12, S3×D4, D42S3, S3×Q8 [×2], C2×S32, D6⋊Q8, C4.D12, Dic3.D6, S3×D12, D6.3D6, C62.65C23

Smallest permutation representation of C62.65C23
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 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)
(1 41 16 36)(2 42 17 31)(3 37 18 32)(4 38 13 33)(5 39 14 34)(6 40 15 35)(7 24 45 27)(8 19 46 28)(9 20 47 29)(10 21 48 30)(11 22 43 25)(12 23 44 26)
(1 41 16 36)(2 40 17 35)(3 39 18 34)(4 38 13 33)(5 37 14 32)(6 42 15 31)(7 30 45 21)(8 29 46 20)(9 28 47 19)(10 27 48 24)(11 26 43 23)(12 25 44 22)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 41 10 38)(8 42 11 39)(9 37 12 40)(13 27 16 30)(14 28 17 25)(15 29 18 26)(31 43 34 46)(32 44 35 47)(33 45 36 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), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,43,9,45,11,47)(8,44,10,46,12,48)(19,30,23,28,21,26)(20,25,24,29,22,27)(31,40,33,42,35,38)(32,41,34,37,36,39), (1,41,16,36)(2,42,17,31)(3,37,18,32)(4,38,13,33)(5,39,14,34)(6,40,15,35)(7,24,45,27)(8,19,46,28)(9,20,47,29)(10,21,48,30)(11,22,43,25)(12,23,44,26), (1,41,16,36)(2,40,17,35)(3,39,18,34)(4,38,13,33)(5,37,14,32)(6,42,15,31)(7,30,45,21)(8,29,46,20)(9,28,47,19)(10,27,48,24)(11,26,43,23)(12,25,44,22), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,41,10,38)(8,42,11,39)(9,37,12,40)(13,27,16,30)(14,28,17,25)(15,29,18,26)(31,43,34,46)(32,44,35,47)(33,45,36,48)>;`

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

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

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 D12 C4○D12 S32 S3×D4 D4⋊2S3 S3×Q8 C2×S32 Dic3.D6 S3×D12 D6.3D6 kernel C62.65C23 C6.D12 Dic3⋊Dic3 C3×Dic3⋊C4 C3×C4⋊Dic3 C6.11D12 C2×C6.D6 C2×C32⋊2Q8 Dic3⋊C4 C4⋊Dic3 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 1 2 1 2 2 2

Matrix representation of C62.65C23 in GL8(𝔽13)

 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 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 1 0 0 0 0 0 0 12 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 12 0 0 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 0 1 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 12 0 0 0 0 0 0 0 0 1 9 0 0 0 0 0 0 7 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 1 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 1 9 0 0 0 0 0 0 7 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 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 3 0 0 0 0 0 0 8 1

`G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,9,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,9,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,8,0,0,0,0,0,0,3,1] >;`

C62.65C23 in GAP, Magma, Sage, TeX

`C_6^2._{65}C_2^3`
`% in TeX`

`G:=Group("C6^2.65C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,543);`
`// by ID`

`G=gap.SmallGroup(288,543);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,422,135,58,1356,9414]);`
`// Polycyclic`

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

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