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## G = C23.9D20order 320 = 26·5

### 2nd non-split extension by C23 of D20 acting via D20/C10=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C23.9D20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C22×C20 — C23.21D10 — C23.9D20
 Lower central C5 — C20 — C23.9D20
 Upper central C1 — C4 — C2×M4(2)

Generators and relations for C23.9D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=abc, ab=ba, dad-1=eae-1=ac=ca, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=bd19 >

Subgroups: 310 in 94 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C42⋊C2, C2×M4(2), C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C4.9C42, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C5×M4(2), C22×C20, C23.21D10, C10×M4(2), C23.9D20
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C4.9C42, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42, C23.9D20

Smallest permutation representation of C23.9D20
On 80 points
Generators in S80
```(1 52)(2 73)(3 54)(4 75)(5 56)(6 77)(7 58)(8 79)(9 60)(10 41)(11 62)(12 43)(13 64)(14 45)(15 66)(16 47)(17 68)(18 49)(19 70)(20 51)(21 72)(22 53)(23 74)(24 55)(25 76)(26 57)(27 78)(28 59)(29 80)(30 61)(31 42)(32 63)(33 44)(34 65)(35 46)(36 67)(37 48)(38 69)(39 50)(40 71)
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 41)(31 42)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 56 21 76)(2 4)(3 54 23 74)(5 52 25 72)(6 40)(7 50 27 70)(8 38)(9 48 29 68)(10 36)(11 46 31 66)(12 34)(13 44 33 64)(14 32)(15 42 35 62)(16 30)(17 80 37 60)(18 28)(19 78 39 58)(20 26)(22 24)(41 47)(43 45)(49 79)(51 77)(53 75)(55 73)(57 71)(59 69)(61 67)(63 65)```

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F ··· 4M 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 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 1 1 2 2 2 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 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + - + + - + - + image C1 C2 C2 C4 C4 D4 Q8 D4 D5 Dic5 D10 Dic10 C4×D5 C5⋊D4 D20 C4.9C42 C23.9D20 kernel C23.9D20 C23.21D10 C10×M4(2) C4×Dic5 C2×C40 C2×C20 C2×C20 C22×C10 C2×M4(2) C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C5 C1 # reps 1 2 1 8 4 2 1 1 2 4 2 4 8 8 4 2 8

Matrix representation of C23.9D20 in GL4(𝔽41) generated by

 24 40 26 1 1 17 40 33 0 0 17 1 0 0 40 24
,
 24 40 0 0 1 17 0 0 0 0 24 40 0 0 1 17
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 34 29 17 21 12 32 20 0 13 19 7 12 22 23 29 9
,
 1 17 4 31 24 40 18 37 0 0 9 30 0 0 11 32
`G:=sub<GL(4,GF(41))| [24,1,0,0,40,17,0,0,26,40,17,40,1,33,1,24],[24,1,0,0,40,17,0,0,0,0,24,1,0,0,40,17],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[34,12,13,22,29,32,19,23,17,20,7,29,21,0,12,9],[1,24,0,0,17,40,0,0,4,18,9,11,31,37,30,32] >;`

C23.9D20 in GAP, Magma, Sage, TeX

`C_2^3._9D_{20}`
`% in TeX`

`G:=Group("C2^3.9D20");`
`// GroupNames label`

`G:=SmallGroup(320,115);`
`// by ID`

`G=gap.SmallGroup(320,115);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,1123,136,102,12550]);`
`// Polycyclic`

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

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