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## G = M4(2).15D10order 320 = 26·5

### 15th 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).15D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — D4⋊D10 — M4(2).15D10
 Lower central C5 — C10 — C2×C20 — M4(2).15D10
 Upper central C1 — C2 — C2×C4 — C8.C22

Generators and relations for M4(2).15D10
G = < a,b,c,d | a8=b2=c10=1, d2=a2, bab=a5, cac-1=a3, dad-1=a3b, cbc-1=dbd-1=a4b, dcd-1=a2c-1 >

Subgroups: 414 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, D5, C10, C10, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C52C8, C52C8, C40, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, D4.3D4, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, D4⋊D5, Q8⋊D5, C5×M4(2), C5×SD16, C5×Q16, C2×D20, Q8×C10, C5×C4○D4, C20.53D4, C20.46D4, C20.10D4, C2×Q8⋊D5, D4.Dic5, D4⋊D10, C5×C8.C22, M4(2).15D10
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).15D10

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

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

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

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

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 10F 20A 20B 20C 20D 20E ··· 20J 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 40 2 2 4 8 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).15D10 kernel M4(2).15D10 C20.53D4 C20.46D4 C20.10D4 C2×Q8⋊D5 D4.Dic5 D4⋊D10 C5×C8.C22 C5⋊2C8 C5×D4 C5×Q8 C8.C22 C2×C10 M4(2) C2×Q8 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).15D10 in GL8(𝔽41)

 0 0 17 40 0 0 0 0 0 0 1 24 0 0 0 0 24 1 0 0 0 0 0 0 40 17 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 26 0 0 0 0 0 11 30 0 0 0 0 0 0 26 30 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 0 0 7 7 0 0 0 0 0 0 34 40 0 0 0 0 7 7 0 0 0 0 0 0 34 40 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
,
 0 0 34 34 0 0 0 0 0 0 1 7 0 0 0 0 7 7 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 0 0 0 1 39 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 0 0 1 40 0 0

`G:=sub<GL(8,GF(41))| [0,0,24,40,0,0,0,0,0,0,1,17,0,0,0,0,17,1,0,0,0,0,0,0,40,24,0,0,0,0,0,0,0,0,0,0,0,0,11,26,0,0,0,0,0,0,30,30,0,0,0,0,0,26,0,0,0,0,0,0,11,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,34,1,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,39,40,0,0] >;`

M4(2).15D10 in GAP, Magma, Sage, TeX

`M_4(2)._{15}D_{10}`
`% in TeX`

`G:=Group("M4(2).15D10");`
`// GroupNames label`

`G:=SmallGroup(320,830);`
`// by ID`

`G=gap.SmallGroup(320,830);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,219,184,1123,297,136,1684,851,438,102,12550]);`
`// 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^3,d*a*d^-1=a^3*b,c*b*c^-1=d*b*d^-1=a^4*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

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