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## G = C40.9Q8order 320 = 26·5

### 9th non-split extension by C40 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C40.9Q8
 Chief series C1 — C5 — C10 — C20 — C40 — C2×C40 — C8×Dic5 — C40.9Q8
 Lower central C5 — C10 — C20 — C40.9Q8
 Upper central C1 — C8 — C2×C8 — M5(2)

Generators and relations for C40.9Q8
G = < a,b,c | a40=1, b4=a30, c2=a35b2, bab-1=a21, cac-1=a9, cbc-1=a35b3 >

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

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

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

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

68 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 4E 4F 4G 5A 5B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 10A 10B 10C 10D 16A 16B 16C 16D 16E 16F 16G 16H 20A 20B 20C 20D 20E 20F 40A ··· 40H 40I 40J 40K 40L 80A ··· 80P order 1 2 2 4 4 4 4 4 4 4 5 5 8 8 8 8 8 8 8 8 8 8 10 10 10 10 16 16 16 16 16 16 16 16 20 20 20 20 20 20 40 ··· 40 40 40 40 40 80 ··· 80 size 1 1 2 1 1 2 10 10 10 10 2 2 1 1 1 1 2 2 10 10 10 10 2 2 4 4 4 4 4 4 20 20 20 20 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + - + + - image C1 C2 C2 C2 C4 C4 C8 D4 Q8 D5 M4(2) D10 Dic10 C5⋊D4 C4×D5 C8.C8 C8×D5 C8⋊D5 C40.9Q8 kernel C40.9Q8 C20.4C8 C8×Dic5 C5×M5(2) C2×C5⋊2C8 C4×Dic5 C5⋊2C8 C40 C40 M5(2) C2×C10 C2×C8 C8 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 2 2 8 1 1 2 2 2 4 4 4 8 8 8 8

Matrix representation of C40.9Q8 in GL4(𝔽241) generated by

 211 36 0 0 0 30 0 0 0 0 0 52 0 0 190 190
,
 211 201 0 0 191 30 0 0 0 0 177 0 0 0 0 177
,
 30 170 0 0 0 233 0 0 0 0 190 189 0 0 50 51
`G:=sub<GL(4,GF(241))| [211,0,0,0,36,30,0,0,0,0,0,190,0,0,52,190],[211,191,0,0,201,30,0,0,0,0,177,0,0,0,0,177],[30,0,0,0,170,233,0,0,0,0,190,50,0,0,189,51] >;`

C40.9Q8 in GAP, Magma, Sage, TeX

`C_{40}._9Q_8`
`% in TeX`

`G:=Group("C40.9Q8");`
`// GroupNames label`

`G:=SmallGroup(320,69);`
`// by ID`

`G=gap.SmallGroup(320,69);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,141,36,100,570,136,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=1,b^4=a^30,c^2=a^35*b^2,b*a*b^-1=a^21,c*a*c^-1=a^9,c*b*c^-1=a^35*b^3>;`
`// generators/relations`

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