Copied to
clipboard

## G = C8.21D20order 320 = 26·5

### 7th non-split extension by C8 of D20 acting via D20/D10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C8.21D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D20.3C4 — C8.21D20
 Lower central C5 — C10 — C2×C20 — C8.21D20
 Upper central C1 — C2 — C2×C4 — C8.C4

Generators and relations for C8.21D20
G = < a,b,c | a8=c2=1, b20=a4, bab-1=cac=a-1, cbc=a4b19 >

Subgroups: 606 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), M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C52C8, C40, C40, Dic10, C4×D5, D20, D20, C5⋊D4, C2×C20, C22×D5, D4.4D4, C8×D5, C8⋊D5, C40⋊C2, D40, C4.Dic5, C2×C40, C5×M4(2), C2×D20, C4○D20, C20.46D4, C5×C8.C4, D20.3C4, C2×D40, C8⋊D10, C8.21D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C22×D5, D4.4D4, C2×D20, D4×D5, Q82D5, C4⋊D20, C8.21D20

Smallest permutation representation of C8.21D20
On 80 points
Generators in S80
```(1 51 11 61 21 71 31 41)(2 42 32 72 22 62 12 52)(3 53 13 63 23 73 33 43)(4 44 34 74 24 64 14 54)(5 55 15 65 25 75 35 45)(6 46 36 76 26 66 16 56)(7 57 17 67 27 77 37 47)(8 48 38 78 28 68 18 58)(9 59 19 69 29 79 39 49)(10 50 40 80 30 70 20 60)
(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 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 73)(34 72)(35 71)(36 70)(37 69)(38 68)(39 67)(40 66)```

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

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

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

44 conjugacy classes

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

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 D20 D4.4D4 D4×D5 Q8⋊2D5 C8.21D20 kernel C8.21D20 C20.46D4 C5×C8.C4 D20.3C4 C2×D40 C8⋊D10 C40 Dic10 D20 C8.C4 C2×C10 C2×C8 M4(2) C8 C5 C4 C22 C1 # reps 1 2 1 1 1 2 2 1 1 2 2 2 4 8 2 2 2 8

Matrix representation of C8.21D20 in GL4(𝔽41) generated by

 12 33 0 0 8 5 0 0 0 0 5 8 0 0 33 12
,
 0 0 0 1 0 0 40 6 13 2 0 0 39 25 0 0
,
 8 5 0 0 12 33 0 0 0 0 33 12 0 0 5 8
`G:=sub<GL(4,GF(41))| [12,8,0,0,33,5,0,0,0,0,5,33,0,0,8,12],[0,0,13,39,0,0,2,25,0,40,0,0,1,6,0,0],[8,12,0,0,5,33,0,0,0,0,33,5,0,0,12,8] >;`

C8.21D20 in GAP, Magma, Sage, TeX

`C_8._{21}D_{20}`
`% in TeX`

`G:=Group("C8.21D20");`
`// GroupNames label`

`G:=SmallGroup(320,524);`
`// by ID`

`G=gap.SmallGroup(320,524);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,219,226,1123,136,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=c^2=1,b^20=a^4,b*a*b^-1=c*a*c=a^-1,c*b*c=a^4*b^19>;`
`// generators/relations`

׿
×
𝔽