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## G = C20.10M4(2)  order 320 = 26·5

### 4th non-split extension by C20 of M4(2) acting via M4(2)/C4=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.10M4(2)
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C20.C8 — C20.10M4(2)
 Lower central C5 — C10 — C2×C10 — C20.10M4(2)
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for C20.10M4(2)
G = < a,b,c | a20=c2=1, b8=a10, bab-1=a7, cac=a9, cbc=a5b5 >

Subgroups: 226 in 58 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, C2×C10, M5(2), C2×M4(2), C52C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C23.C8, C5⋊C16, C8⋊D5, C2×C52C8, C2×C40, C2×C4×D5, C20.C8, C2×C8⋊D5, C20.10M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C2×F5, C23.C8, D5⋊C8, C4.F5, C22⋊F5, D10⋊C8, C20.10M4(2)

Smallest permutation representation of C20.10M4(2)
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 70 27 55 6 65 32 50 11 80 37 45 16 75 22 60)(2 73 36 42 7 68 21 57 12 63 26 52 17 78 31 47)(3 76 25 49 8 71 30 44 13 66 35 59 18 61 40 54)(4 79 34 56 9 74 39 51 14 69 24 46 19 64 29 41)(5 62 23 43 10 77 28 58 15 72 33 53 20 67 38 48)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 23)(22 32)(24 30)(25 39)(26 28)(27 37)(29 35)(31 33)(34 40)(36 38)(41 49)(42 58)(43 47)(44 56)(46 54)(48 52)(51 59)(53 57)(61 79)(62 68)(63 77)(64 66)(65 75)(67 73)(69 71)(70 80)(72 78)(74 76)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5 8A 8B 8C 8D 8E 8F 10A 10B 10C 16A ··· 16H 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 4 4 4 4 5 8 8 8 8 8 8 10 10 10 16 ··· 16 20 20 20 20 40 ··· 40 size 1 1 2 20 1 1 2 20 4 4 4 10 10 10 10 4 4 4 20 ··· 20 4 4 4 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C4 C4 C8 C8 D4 M4(2) F5 C2×F5 C23.C8 C4.F5 C22⋊F5 D5⋊C8 C20.10M4(2) kernel C20.10M4(2) C20.C8 C2×C8⋊D5 C2×C40 C2×C4×D5 C2×Dic5 C22×D5 C5⋊2C8 C20 C2×C8 C2×C4 C5 C4 C4 C22 C1 # reps 1 2 1 2 2 4 4 2 2 1 1 2 2 2 2 8

Matrix representation of C20.10M4(2) in GL4(𝔽241) generated by

 46 64 0 0 223 64 0 0 0 0 0 110 0 0 195 46
,
 0 0 189 1 0 0 189 52 25 90 0 0 50 216 0 0
,
 189 1 0 0 189 52 0 0 0 0 189 1 0 0 189 52
`G:=sub<GL(4,GF(241))| [46,223,0,0,64,64,0,0,0,0,0,195,0,0,110,46],[0,0,25,50,0,0,90,216,189,189,0,0,1,52,0,0],[189,189,0,0,1,52,0,0,0,0,189,189,0,0,1,52] >;`

C20.10M4(2) in GAP, Magma, Sage, TeX

`C_{20}._{10}M_4(2)`
`% in TeX`

`G:=Group("C20.10M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,229);`
`// by ID`

`G=gap.SmallGroup(320,229);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,100,1123,102,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b,c|a^20=c^2=1,b^8=a^10,b*a*b^-1=a^7,c*a*c=a^9,c*b*c=a^5*b^5>;`
`// generators/relations`

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