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## G = D10.11M4(2)  order 320 = 26·5

### 7th non-split extension by D10 of M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10.11M4(2)
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C23.2F5 — D10.11M4(2)
 Lower central C5 — C10 — D10.11M4(2)
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for D10.11M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd=a5c5 >

Subgroups: 762 in 202 conjugacy classes, 70 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C23, D5, D5, C10, C10, C2×C8, C22×C4, C22×C4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, C22×C8, C23×C4, C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×D5, C22×C10, C2×C22⋊C8, D5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D10⋊C8, C23.2F5, C2×D5⋊C8, D5×C22×C4, D10.11M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, F5, C22⋊C8, C2×C22⋊C4, C22×C8, C2×M4(2), C2×F5, C2×C22⋊C8, D5⋊C8, C22⋊F5, C22×F5, C2×D5⋊C8, D5⋊M4(2), C2×C22⋊F5, D10.11M4(2)

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 D4 M4(2) F5 C2×F5 C2×F5 C22⋊F5 D5⋊C8 D5⋊M4(2) kernel D10.11M4(2) D10⋊C8 C23.2F5 C2×D5⋊C8 D5×C22×C4 C2×C4×D5 C22×C20 C23×D5 C22×D5 C4×D5 D10 C22×C4 C2×C4 C23 C4 C22 C2 # reps 1 2 2 2 1 4 2 2 16 4 4 1 2 1 4 4 4

Matrix representation of D10.11M4(2) in GL8(𝔽41)

 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 0 0 0 0 0 0 7 7 0 0 0 0 0 0 34 40 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 40 34
,
 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 7 7 0 0 0 0 0 0 40 34 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 38 19 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 9 25 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 7 1 0 0
,
 1 0 0 0 0 0 0 0 37 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 4 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

`G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,7,34,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[38,0,0,0,0,0,0,0,19,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,25,32,0,0,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,37,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;`

D10.11M4(2) in GAP, Magma, Sage, TeX

`D_{10}._{11}M_4(2)`
`% in TeX`

`G:=Group("D10.11M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,1091);`
`// by ID`

`G=gap.SmallGroup(320,1091);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,184,136,6278,1595]);`
`// Polycyclic`

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

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