Copied to
clipboard

## G = C20.29M4(2)  order 320 = 26·5

### 4th non-split extension by C20 of M4(2) acting via M4(2)/C22=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C20.29M4(2)
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C20.C8 — C20.29M4(2)
 Lower central C5 — C10 — C2×C10 — C20.29M4(2)
 Upper central C1 — C4 — C2×C4 — C22×C4

Generators and relations for C20.29M4(2)
G = < a,b,c | a20=c2=1, b8=a10, bab-1=a3, ac=ca, cbc=a5b5 >

Subgroups: 162 in 58 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C16, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, M5(2), C2×M4(2), C52C8, C52C8, C2×C20, C2×C20, C22×C10, C23.C8, C5⋊C16, C2×C52C8, C4.Dic5, C22×C20, C20.C8, C2×C4.Dic5, C20.29M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C5⋊C8, C2×F5, C23.C8, C2×C5⋊C8, C22.F5, C22⋊F5, C23.2F5, C20.29M4(2)

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 type + + + + + - + - - + image C1 C2 C2 C4 C4 C8 C8 D4 M4(2) F5 C5⋊C8 C2×F5 C5⋊C8 C23.C8 C22.F5 C22⋊F5 C20.29M4(2) kernel C20.29M4(2) C20.C8 C2×C4.Dic5 C2×C5⋊2C8 C22×C20 C2×C20 C22×C10 C5⋊2C8 C20 C22×C4 C2×C4 C2×C4 C23 C5 C4 C4 C1 # reps 1 2 1 2 2 4 4 2 2 1 1 1 1 2 2 2 8

Matrix representation of C20.29M4(2) in GL4(𝔽241) generated by

 216 0 0 0 54 135 0 0 3 0 6 0 16 0 0 40
,
 62 0 239 0 224 0 79 240 154 1 179 0 127 0 197 0
,
 1 0 0 0 162 240 0 0 0 0 1 0 44 0 0 240
`G:=sub<GL(4,GF(241))| [216,54,3,16,0,135,0,0,0,0,6,0,0,0,0,40],[62,224,154,127,0,0,1,0,239,79,179,197,0,240,0,0],[1,162,0,44,0,240,0,0,0,0,1,0,0,0,0,240] >;`

C20.29M4(2) in GAP, Magma, Sage, TeX

`C_{20}._{29}M_4(2)`
`% in TeX`

`G:=Group("C20.29M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,250);`
`// by ID`

`G=gap.SmallGroup(320,250);`
`# by ID`

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

`G:=Group<a,b,c|a^20=c^2=1,b^8=a^10,b*a*b^-1=a^3,a*c=c*a,c*b*c=a^5*b^5>;`
`// generators/relations`

׿
×
𝔽