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## G = C40.23D4order 320 = 26·5

### 23rd non-split extension by C40 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C40.23D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D20.3C4 — C40.23D4
 Lower central C5 — C10 — C2×C20 — C40.23D4
 Upper central C1 — C2 — C2×C4 — C2×D8

Generators and relations for C40.23D4
G = < a,b,c | a40=c2=1, b4=a20, bab-1=a-1, cac=a9, cbc=a20b3 >

Subgroups: 398 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C22×C10, D4.4D4, C8×D5, C8⋊D5, C4.Dic5, C4.Dic5, D4⋊D5, D4.D5, C2×C40, C5×D8, C4○D20, D4×C10, C40.6C4, C20.D4, D20.3C4, D4.D10, C10×D8, C40.23D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4.4D4, D4×D5, D42D5, C2×C5⋊D4, C202D4, C40.23D4

Smallest permutation representation of C40.23D4
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 41 11 71 21 61 31 51)(2 80 12 70 22 60 32 50)(3 79 13 69 23 59 33 49)(4 78 14 68 24 58 34 48)(5 77 15 67 25 57 35 47)(6 76 16 66 26 56 36 46)(7 75 17 65 27 55 37 45)(8 74 18 64 28 54 38 44)(9 73 19 63 29 53 39 43)(10 72 20 62 30 52 40 42)
(1 66)(2 75)(3 44)(4 53)(5 62)(6 71)(7 80)(8 49)(9 58)(10 67)(11 76)(12 45)(13 54)(14 63)(15 72)(16 41)(17 50)(18 59)(19 68)(20 77)(21 46)(22 55)(23 64)(24 73)(25 42)(26 51)(27 60)(28 69)(29 78)(30 47)(31 56)(32 65)(33 74)(34 43)(35 52)(36 61)(37 70)(38 79)(39 48)(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), (1,41,11,71,21,61,31,51)(2,80,12,70,22,60,32,50)(3,79,13,69,23,59,33,49)(4,78,14,68,24,58,34,48)(5,77,15,67,25,57,35,47)(6,76,16,66,26,56,36,46)(7,75,17,65,27,55,37,45)(8,74,18,64,28,54,38,44)(9,73,19,63,29,53,39,43)(10,72,20,62,30,52,40,42), (1,66)(2,75)(3,44)(4,53)(5,62)(6,71)(7,80)(8,49)(9,58)(10,67)(11,76)(12,45)(13,54)(14,63)(15,72)(16,41)(17,50)(18,59)(19,68)(20,77)(21,46)(22,55)(23,64)(24,73)(25,42)(26,51)(27,60)(28,69)(29,78)(30,47)(31,56)(32,65)(33,74)(34,43)(35,52)(36,61)(37,70)(38,79)(39,48)(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), (1,41,11,71,21,61,31,51)(2,80,12,70,22,60,32,50)(3,79,13,69,23,59,33,49)(4,78,14,68,24,58,34,48)(5,77,15,67,25,57,35,47)(6,76,16,66,26,56,36,46)(7,75,17,65,27,55,37,45)(8,74,18,64,28,54,38,44)(9,73,19,63,29,53,39,43)(10,72,20,62,30,52,40,42), (1,66)(2,75)(3,44)(4,53)(5,62)(6,71)(7,80)(8,49)(9,58)(10,67)(11,76)(12,45)(13,54)(14,63)(15,72)(16,41)(17,50)(18,59)(19,68)(20,77)(21,46)(22,55)(23,64)(24,73)(25,42)(26,51)(27,60)(28,69)(29,78)(30,47)(31,56)(32,65)(33,74)(34,43)(35,52)(36,61)(37,70)(38,79)(39,48)(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)], [(1,41,11,71,21,61,31,51),(2,80,12,70,22,60,32,50),(3,79,13,69,23,59,33,49),(4,78,14,68,24,58,34,48),(5,77,15,67,25,57,35,47),(6,76,16,66,26,56,36,46),(7,75,17,65,27,55,37,45),(8,74,18,64,28,54,38,44),(9,73,19,63,29,53,39,43),(10,72,20,62,30,52,40,42)], [(1,66),(2,75),(3,44),(4,53),(5,62),(6,71),(7,80),(8,49),(9,58),(10,67),(11,76),(12,45),(13,54),(14,63),(15,72),(16,41),(17,50),(18,59),(19,68),(20,77),(21,46),(22,55),(23,64),(24,73),(25,42),(26,51),(27,60),(28,69),(29,78),(30,47),(31,56),(32,65),(33,74),(34,43),(35,52),(36,61),(37,70),(38,79),(39,48),(40,57)]])`

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 8A 8B 8C 8D 8E 8F 8G 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 4 4 4 5 5 8 8 8 8 8 8 8 10 ··· 10 10 ··· 10 20 20 20 20 40 ··· 40 size 1 1 2 8 8 20 2 2 20 2 2 2 2 4 20 20 40 40 2 ··· 2 8 ··· 8 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 C5⋊D4 D4.4D4 D4×D5 D4⋊2D5 C40.23D4 kernel C40.23D4 C40.6C4 C20.D4 D20.3C4 D4.D10 C10×D8 C40 Dic10 D20 C2×D8 C2×C10 C2×C8 C2×D4 C8 C5 C4 C22 C1 # reps 1 1 2 1 2 1 2 1 1 2 2 2 4 8 2 2 2 8

Matrix representation of C40.23D4 in GL6(𝔽41)

 16 0 0 0 0 0 0 18 0 0 0 0 0 0 0 40 0 0 0 0 1 24 0 0 0 0 0 0 0 40 0 0 0 0 1 24
,
 0 15 0 0 0 0 30 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 17 40 0 0 0 0 1 24 0 0
,
 0 15 0 0 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

`G:=sub<GL(6,GF(41))| [16,0,0,0,0,0,0,18,0,0,0,0,0,0,0,1,0,0,0,0,40,24,0,0,0,0,0,0,0,1,0,0,0,0,40,24],[0,30,0,0,0,0,15,0,0,0,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,40,0,0,0,0,40,0,0,0],[0,11,0,0,0,0,15,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C40.23D4 in GAP, Magma, Sage, TeX

`C_{40}._{23}D_4`
`% in TeX`

`G:=Group("C40.23D4");`
`// GroupNames label`

`G:=SmallGroup(320,787);`
`// by ID`

`G=gap.SmallGroup(320,787);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,1123,297,136,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=c^2=1,b^4=a^20,b*a*b^-1=a^-1,c*a*c=a^9,c*b*c=a^20*b^3>;`
`// generators/relations`

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