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G = C40.6Q8order 320 = 26·5

6th non-split extension by C40 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C40.6Q8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C20.4C8 — C40.6Q8
 Lower central C5 — C10 — C20 — C40 — C40.6Q8
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for C40.6Q8
G = < a,b,c | a40=1, b4=a20, c2=a5b2, bab-1=a31, cac-1=a29, cbc-1=a15b3 >

Smallest permutation representation of C40.6Q8
On 80 points
Generators in S80
```(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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 65 11 55 21 45 31 75)(2 56 12 46 22 76 32 66)(3 47 13 77 23 67 33 57)(4 78 14 68 24 58 34 48)(5 69 15 59 25 49 35 79)(6 60 16 50 26 80 36 70)(7 51 17 41 27 71 37 61)(8 42 18 72 28 62 38 52)(9 73 19 63 29 53 39 43)(10 64 20 54 30 44 40 74)
(1 75 16 70 31 65 6 60 21 55 36 50 11 45 26 80)(2 64 17 59 32 54 7 49 22 44 37 79 12 74 27 69)(3 53 18 48 33 43 8 78 23 73 38 68 13 63 28 58)(4 42 19 77 34 72 9 67 24 62 39 57 14 52 29 47)(5 71 20 66 35 61 10 56 25 51 40 46 15 41 30 76)```

`G:=sub<Sym(80)| (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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,65,11,55,21,45,31,75)(2,56,12,46,22,76,32,66)(3,47,13,77,23,67,33,57)(4,78,14,68,24,58,34,48)(5,69,15,59,25,49,35,79)(6,60,16,50,26,80,36,70)(7,51,17,41,27,71,37,61)(8,42,18,72,28,62,38,52)(9,73,19,63,29,53,39,43)(10,64,20,54,30,44,40,74), (1,75,16,70,31,65,6,60,21,55,36,50,11,45,26,80)(2,64,17,59,32,54,7,49,22,44,37,79,12,74,27,69)(3,53,18,48,33,43,8,78,23,73,38,68,13,63,28,58)(4,42,19,77,34,72,9,67,24,62,39,57,14,52,29,47)(5,71,20,66,35,61,10,56,25,51,40,46,15,41,30,76)>;`

`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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,65,11,55,21,45,31,75)(2,56,12,46,22,76,32,66)(3,47,13,77,23,67,33,57)(4,78,14,68,24,58,34,48)(5,69,15,59,25,49,35,79)(6,60,16,50,26,80,36,70)(7,51,17,41,27,71,37,61)(8,42,18,72,28,62,38,52)(9,73,19,63,29,53,39,43)(10,64,20,54,30,44,40,74), (1,75,16,70,31,65,6,60,21,55,36,50,11,45,26,80)(2,64,17,59,32,54,7,49,22,44,37,79,12,74,27,69)(3,53,18,48,33,43,8,78,23,73,38,68,13,63,28,58)(4,42,19,77,34,72,9,67,24,62,39,57,14,52,29,47)(5,71,20,66,35,61,10,56,25,51,40,46,15,41,30,76) );`

`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),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,65,11,55,21,45,31,75),(2,56,12,46,22,76,32,66),(3,47,13,77,23,67,33,57),(4,78,14,68,24,58,34,48),(5,69,15,59,25,49,35,79),(6,60,16,50,26,80,36,70),(7,51,17,41,27,71,37,61),(8,42,18,72,28,62,38,52),(9,73,19,63,29,53,39,43),(10,64,20,54,30,44,40,74)], [(1,75,16,70,31,65,6,60,21,55,36,50,11,45,26,80),(2,64,17,59,32,54,7,49,22,44,37,79,12,74,27,69),(3,53,18,48,33,43,8,78,23,73,38,68,13,63,28,58),(4,42,19,77,34,72,9,67,24,62,39,57,14,52,29,47),(5,71,20,66,35,61,10,56,25,51,40,46,15,41,30,76)]])`

44 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 10A 10B 10C 10D 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I ··· 40P order 1 2 2 4 4 4 4 5 5 8 8 8 8 8 10 10 10 10 16 16 16 16 20 20 20 20 20 20 40 ··· 40 40 ··· 40 size 1 1 2 2 2 40 40 2 2 2 2 4 8 8 2 2 4 4 20 20 20 20 2 2 2 2 4 4 4 ··· 4 8 ··· 8

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + - + + + - - + image C1 C2 C2 C2 C4 Q8 D4 D5 SD16 SD16 D10 Dic10 C4×D5 C5⋊D4 C8.Q8 D4.D5 Q8⋊D5 C40.6Q8 kernel C40.6Q8 C20.4C8 C40⋊6C4 C5×C8.C4 C5⋊2C16 C40 C2×C20 C8.C4 C20 C2×C10 C2×C8 C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 2 2 2 8

Matrix representation of C40.6Q8 in GL4(𝔽241) generated by

 37 68 0 0 173 116 0 0 61 22 75 116 219 122 125 68
,
 56 77 239 0 164 149 0 239 70 223 185 164 18 42 77 92
,
 104 108 91 144 2 137 232 150 218 225 153 167 216 23 164 88
`G:=sub<GL(4,GF(241))| [37,173,61,219,68,116,22,122,0,0,75,125,0,0,116,68],[56,164,70,18,77,149,223,42,239,0,185,77,0,239,164,92],[104,2,218,216,108,137,225,23,91,232,153,164,144,150,167,88] >;`

C40.6Q8 in GAP, Magma, Sage, TeX

`C_{40}._6Q_8`
`% in TeX`

`G:=Group("C40.6Q8");`
`// GroupNames label`

`G:=SmallGroup(320,52);`
`// by ID`

`G=gap.SmallGroup(320,52);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,365,36,758,184,346,80,851,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=1,b^4=a^20,c^2=a^5*b^2,b*a*b^-1=a^31,c*a*c^-1=a^29,c*b*c^-1=a^15*b^3>;`
`// generators/relations`

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