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## G = M4(2).13D10order 320 = 26·5

### 13rd non-split extension by M4(2) of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — M4(2).13D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — D4.9D10 — M4(2).13D10
 Lower central C5 — C10 — C2×C20 — M4(2).13D10
 Upper central C1 — C2 — C2×C4 — C8⋊C22

Generators and relations for M4(2).13D10
G = < a,b,c,d | a8=b2=c10=1, d2=a2, bab=a5, cac-1=a-1, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a6c-1 >

Subgroups: 350 in 104 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8, C52C8, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×C10, D4.3D4, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, D4.D5, C5⋊Q16, C5×M4(2), C5×D8, C5×SD16, C2×Dic10, D4×C10, C5×C4○D4, C20.53D4, C4.12D20, C20.D4, C2×D4.D5, D4.Dic5, D4.9D10, C5×C8⋊C22, M4(2).13D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C5⋊D4, C22×D5, D4.3D4, D4×D5, D42D5, C2×C5⋊D4, Dic5⋊D4, M4(2).13D10

Smallest permutation representation of M4(2).13D10
On 80 points
Generators in S80
```(1 51 56 6 33 18 13 38)(2 39 14 19 34 7 57 52)(3 53 58 8 35 20 15 40)(4 31 16 11 36 9 59 54)(5 55 60 10 37 12 17 32)(21 73 78 26 69 46 41 64)(22 65 42 47 70 27 79 74)(23 75 80 28 61 48 43 66)(24 67 44 49 62 29 71 76)(25 77 72 30 63 50 45 68)
(1 33)(3 35)(5 37)(7 39)(9 31)(11 54)(13 56)(15 58)(17 60)(19 52)(22 70)(24 62)(26 64)(28 66)(30 68)(42 79)(44 71)(46 73)(48 75)(50 77)
(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 29 56 76 33 67 13 49)(2 75 14 28 34 48 57 66)(3 27 58 74 35 65 15 47)(4 73 16 26 36 46 59 64)(5 25 60 72 37 63 17 45)(6 71 18 24 38 44 51 62)(7 23 52 80 39 61 19 43)(8 79 20 22 40 42 53 70)(9 21 54 78 31 69 11 41)(10 77 12 30 32 50 55 68)```

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

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

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

38 conjugacy classes

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

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 C5⋊D4 D4.3D4 D4×D5 D4⋊2D5 M4(2).13D10 kernel M4(2).13D10 C20.53D4 C4.12D20 C20.D4 C2×D4.D5 D4.Dic5 D4.9D10 C5×C8⋊C22 C5⋊2C8 C5×D4 C5×Q8 C8⋊C22 C2×C10 M4(2) C2×D4 C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 4 4 2 2 2 2

Matrix representation of M4(2).13D10 in GL6(𝔽41)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 1 1 0 0 1 2 0 0 0 0 0 40 0 0
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 38 7 0 0 0 0 7 38 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
,
 32 30 0 0 0 0 11 9 0 0 0 0 0 0 0 0 30 30 0 0 0 0 26 11 0 0 0 30 0 0 0 0 15 0 0 0

`G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,2,40,0,0,40,1,0,0,0,0,0,1,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[38,7,0,0,0,0,7,38,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[32,11,0,0,0,0,30,9,0,0,0,0,0,0,0,0,0,15,0,0,0,0,30,0,0,0,30,26,0,0,0,0,30,11,0,0] >;`

M4(2).13D10 in GAP, Magma, Sage, TeX

`M_4(2)._{13}D_{10}`
`% in TeX`

`G:=Group("M4(2).13D10");`
`// GroupNames label`

`G:=SmallGroup(320,827);`
`// by ID`

`G=gap.SmallGroup(320,827);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,219,1123,297,136,1684,851,438,102,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^2,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^6*c^-1>;`
`// generators/relations`

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