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## G = C40.9C23order 320 = 26·5

### 2nd non-split extension by C40 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C40.9C23
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C40.9C23
 Lower central C5 — C10 — C20 — C40.9C23
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for C40.9C23
G = < a,b,c,d | a40=b2=1, c2=d2=a20, bab=a19, ac=ca, dad-1=a21, bc=cb, bd=db, cd=dc >

Subgroups: 1054 in 262 conjugacy classes, 107 normal (21 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C2×C4○D4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, D8⋊C22, C40⋊C2, D40, Dic20, C2×C40, C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, D407C2, C8⋊D10, C8.D10, C10×M4(2), C2×C4○D20, C40.9C23
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, D20, C22×D5, D8⋊C22, C2×D20, C23×D5, C22×D20, C40.9C23

Smallest permutation representation of C40.9C23
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)
(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 55)(42 74)(43 53)(44 72)(45 51)(46 70)(47 49)(48 68)(50 66)(52 64)(54 62)(56 60)(57 79)(59 77)(61 75)(63 73)(65 71)(67 69)(76 80)
(1 58 21 78)(2 59 22 79)(3 60 23 80)(4 61 24 41)(5 62 25 42)(6 63 26 43)(7 64 27 44)(8 65 28 45)(9 66 29 46)(10 67 30 47)(11 68 31 48)(12 69 32 49)(13 70 33 50)(14 71 34 51)(15 72 35 52)(16 73 36 53)(17 74 37 54)(18 75 38 55)(19 76 39 56)(20 77 40 57)
(1 78 21 58)(2 59 22 79)(3 80 23 60)(4 61 24 41)(5 42 25 62)(6 63 26 43)(7 44 27 64)(8 65 28 45)(9 46 29 66)(10 67 30 47)(11 48 31 68)(12 69 32 49)(13 50 33 70)(14 71 34 51)(15 52 35 72)(16 73 36 53)(17 54 37 74)(18 75 38 55)(19 56 39 76)(20 77 40 57)```

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D20 D20 D8⋊C22 C40.9C23 kernel C40.9C23 D40⋊7C2 C8⋊D10 C8.D10 C10×M4(2) C2×C4○D20 C2×C20 C22×C10 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C5 C1 # reps 1 4 4 4 1 2 3 1 2 4 8 2 12 4 2 8

Matrix representation of C40.9C23 in GL4(𝔽41) generated by

 21 18 26 18 4 13 5 8 27 37 37 21 16 2 2 11
,
 34 40 23 27 7 7 35 27 0 0 25 12 0 0 30 16
,
 32 0 0 0 0 32 0 0 0 0 32 0 0 0 0 32
,
 9 0 0 2 0 9 18 1 0 0 32 0 0 0 0 32
`G:=sub<GL(4,GF(41))| [21,4,27,16,18,13,37,2,26,5,37,2,18,8,21,11],[34,7,0,0,40,7,0,0,23,35,25,30,27,27,12,16],[32,0,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[9,0,0,0,0,9,0,0,0,18,32,0,2,1,0,32] >;`

C40.9C23 in GAP, Magma, Sage, TeX

`C_{40}._9C_2^3`
`% in TeX`

`G:=Group("C40.9C2^3");`
`// GroupNames label`

`G:=SmallGroup(320,1420);`
`// by ID`

`G=gap.SmallGroup(320,1420);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,80,1684,102,12550]);`
`// Polycyclic`

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

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