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## G = C40.Q8order 320 = 26·5

### 1st non-split extension by C40 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — C40.Q8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C40⋊6C4 — C40.Q8
 Lower central C5 — C10 — C20 — C40 — C40.Q8
 Upper central C1 — C2 — C2×C4 — C2×C8 — M5(2)

Generators and relations for C40.Q8
G = < a,b,c | a40=1, b4=a10, c2=a15b2, bab-1=a21, cac-1=a19, cbc-1=a20b3 >

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

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

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

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

56 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 40J 40K 40L 80A ··· 80P 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 40 40 80 ··· 80 size 1 1 2 2 2 40 40 2 2 2 2 4 40 40 2 2 4 4 4 4 4 4 2 2 2 2 4 4 2 ··· 2 4 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + - + + - + - + image C1 C2 C2 C2 C4 Q8 D4 D5 SD16 SD16 Dic5 D10 Dic10 D20 C40⋊C2 C40⋊C2 C8.Q8 C40.Q8 kernel C40.Q8 C40⋊6C4 C40.6C4 C5×M5(2) C80 C40 C2×C20 M5(2) C20 C2×C10 C16 C2×C8 C8 C2×C4 C4 C22 C5 C1 # reps 1 1 1 1 4 1 1 2 2 2 4 2 4 4 8 8 2 8

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

 200 225 19 61 125 184 184 4 0 0 166 125 0 0 116 173
,
 117 44 82 0 78 161 8 8 204 171 202 78 107 118 163 2
,
 190 50 46 121 189 51 165 119 0 0 173 204 0 0 125 68
`G:=sub<GL(4,GF(241))| [200,125,0,0,225,184,0,0,19,184,166,116,61,4,125,173],[117,78,204,107,44,161,171,118,82,8,202,163,0,8,78,2],[190,189,0,0,50,51,0,0,46,165,173,125,121,119,204,68] >;`

C40.Q8 in GAP, Magma, Sage, TeX

`C_{40}.Q_8`
`% in TeX`

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

`G:=SmallGroup(320,71);`
`// by ID`

`G=gap.SmallGroup(320,71);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,387,675,80,1684,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=1,b^4=a^10,c^2=a^15*b^2,b*a*b^-1=a^21,c*a*c^-1=a^19,c*b*c^-1=a^20*b^3>;`
`// generators/relations`

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