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G = C62.5Q8order 288 = 25·32

2nd non-split extension by C62 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C62.5Q8
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C3×C4.Dic3 — C62.5Q8
 Lower central C32 — C3×C6 — C3×C12 — C62.5Q8
 Upper central C1 — C4 — C2×C4

Generators and relations for C62.5Q8
G = < a,b,c,d | a6=b6=1, c4=b3, d2=c2, ab=ba, cac-1=a-1b3, dad-1=ab3, bc=cb, dbd-1=b-1, dcd-1=a3c3 >

Subgroups: 178 in 71 conjugacy classes, 32 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C8.C4, C3×C12, C62, C2×C3⋊C8, C4.Dic3, C3×M4(2), C3×C3⋊C8, C324C8, C6×C12, C12.53D4, C3×C4.Dic3, C2×C324C8, C62.5Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D6, C4⋊C4, Dic6, C4×S3, C3⋊D4, C8.C4, S32, Dic3⋊C4, C6.D6, D6⋊S3, C322Q8, C12.53D4, C62.C22, C62.5Q8

Smallest permutation representation of C62.5Q8
On 48 points
Generators in S48
```(1 40 25)(2 30 33 6 26 37)(3 34 27)(4 32 35 8 28 39)(5 36 29)(7 38 31)(9 42 24 13 46 20)(10 17 47)(11 44 18 15 48 22)(12 19 41)(14 21 43)(16 23 45)
(1 36 25 5 40 29)(2 37 26 6 33 30)(3 38 27 7 34 31)(4 39 28 8 35 32)(9 20 46 13 24 42)(10 21 47 14 17 43)(11 22 48 15 18 44)(12 23 41 16 19 45)
(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 11 3 13 5 15 7 9)(2 10 4 12 6 14 8 16)(17 28 19 30 21 32 23 26)(18 27 20 29 22 31 24 25)(33 47 35 41 37 43 39 45)(34 42 36 44 38 46 40 48)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 4C 6A 6B 6C ··· 6G 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E ··· 12J 24A ··· 24H order 1 2 2 3 3 3 4 4 4 6 6 6 ··· 6 8 8 8 8 8 8 8 8 12 12 12 12 12 ··· 12 24 ··· 24 size 1 1 2 2 2 4 1 1 2 2 2 4 ··· 4 12 12 12 12 18 18 18 18 2 2 2 2 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - + - + + - - image C1 C2 C2 C4 S3 D4 Q8 D6 C4×S3 C3⋊D4 Dic6 C8.C4 S32 C6.D6 D6⋊S3 C32⋊2Q8 C12.53D4 C62.5Q8 kernel C62.5Q8 C3×C4.Dic3 C2×C32⋊4C8 C32⋊4C8 C4.Dic3 C3×C12 C62 C2×C12 C12 C12 C2×C6 C32 C2×C4 C4 C4 C22 C3 C1 # reps 1 2 1 4 2 1 1 2 4 4 4 4 1 1 1 1 4 4

Matrix representation of C62.5Q8 in GL4(𝔽5) generated by

 1 3 0 0 4 3 0 0 0 0 2 3 0 0 4 4
,
 4 2 0 0 1 2 0 0 0 0 4 2 0 0 1 2
,
 0 0 3 0 0 0 0 3 1 0 0 0 0 1 0 0
,
 0 0 0 2 0 0 4 0 0 2 0 0 4 0 0 0
`G:=sub<GL(4,GF(5))| [1,4,0,0,3,3,0,0,0,0,2,4,0,0,3,4],[4,1,0,0,2,2,0,0,0,0,4,1,0,0,2,2],[0,0,1,0,0,0,0,1,3,0,0,0,0,3,0,0],[0,0,0,4,0,0,2,0,0,4,0,0,2,0,0,0] >;`

C62.5Q8 in GAP, Magma, Sage, TeX

`C_6^2._5Q_8`
`% in TeX`

`G:=Group("C6^2.5Q8");`
`// GroupNames label`

`G:=SmallGroup(288,226);`
`// by ID`

`G=gap.SmallGroup(288,226);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,36,100,675,346,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=1,c^4=b^3,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a*b^3,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=a^3*c^3>;`
`// generators/relations`

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