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## G = D10⋊7M4(2)  order 320 = 26·5

### 1st semidirect product of D10 and M4(2) acting via M4(2)/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — D10⋊7M4(2)
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C4×D5 — D5×C22×C4 — D10⋊7M4(2)
 Lower central C5 — C2×C10 — D10⋊7M4(2)
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for D107M4(2)
G = < a,b,c,d | a10=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a3b, dbd=a8b, dcd=c5 >

Subgroups: 734 in 190 conjugacy classes, 61 normal (33 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C23, D5, C10, C10, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, C22⋊C8, C2×M4(2), C23×C4, C52C8, C40, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C24.4C4, C8⋊D5, C2×C52C8, C2×C40, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C23×D5, D101C8, C20.55D4, C5×C22⋊C8, C2×C8⋊D5, D5×C22×C4, D107M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, M4(2), C22×C4, C2×D4, D10, C2×C22⋊C4, C2×M4(2), C4×D5, C22×D5, C24.4C4, C8⋊D5, C2×C4×D5, D4×D5, D5×C22⋊C4, C2×C8⋊D5, D5×M4(2), D107M4(2)

Smallest permutation representation of D107M4(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)(11 20)(12 19)(13 18)(14 17)(15 16)(22 30)(23 29)(24 28)(25 27)(31 39)(32 38)(33 37)(34 36)(41 43)(44 50)(45 49)(46 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 70)(71 74)(72 73)(75 80)(76 79)(77 78)
(1 73 45 70 29 16 38 53)(2 72 46 69 30 15 39 52)(3 71 47 68 21 14 40 51)(4 80 48 67 22 13 31 60)(5 79 49 66 23 12 32 59)(6 78 50 65 24 11 33 58)(7 77 41 64 25 20 34 57)(8 76 42 63 26 19 35 56)(9 75 43 62 27 18 36 55)(10 74 44 61 28 17 37 54)
(1 24)(2 23)(3 22)(4 21)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 16)(12 15)(13 14)(17 20)(18 19)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 50)(39 49)(40 48)(51 60)(52 59)(53 58)(54 57)(55 56)(61 64)(62 63)(65 70)(66 69)(67 68)(71 80)(72 79)(73 78)(74 77)(75 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), (1,5)(2,4)(6,10)(7,9)(11,20)(12,19)(13,18)(14,17)(15,16)(22,30)(23,29)(24,28)(25,27)(31,39)(32,38)(33,37)(34,36)(41,43)(44,50)(45,49)(46,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,70)(71,74)(72,73)(75,80)(76,79)(77,78), (1,73,45,70,29,16,38,53)(2,72,46,69,30,15,39,52)(3,71,47,68,21,14,40,51)(4,80,48,67,22,13,31,60)(5,79,49,66,23,12,32,59)(6,78,50,65,24,11,33,58)(7,77,41,64,25,20,34,57)(8,76,42,63,26,19,35,56)(9,75,43,62,27,18,36,55)(10,74,44,61,28,17,37,54), (1,24)(2,23)(3,22)(4,21)(5,30)(6,29)(7,28)(8,27)(9,26)(10,25)(11,16)(12,15)(13,14)(17,20)(18,19)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,60)(52,59)(53,58)(54,57)(55,56)(61,64)(62,63)(65,70)(66,69)(67,68)(71,80)(72,79)(73,78)(74,77)(75,76)>;`

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

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

68 conjugacy classes

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

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D5 M4(2) M4(2) D10 D10 C4×D5 C4×D5 C8⋊D5 D4×D5 D5×M4(2) kernel D10⋊7M4(2) D10⋊1C8 C20.55D4 C5×C22⋊C8 C2×C8⋊D5 D5×C22×C4 C2×C4×D5 C22×Dic5 C23×D5 C4×D5 C22⋊C8 D10 C2×C10 C2×C8 C22×C4 C2×C4 C23 C22 C4 C2 # reps 1 2 1 1 2 1 4 2 2 4 2 4 4 4 2 4 4 16 4 4

Matrix representation of D107M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 7 7 0 0 0 0 34 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 32 40 0 0 0 0 0 0 7 7 0 0 0 0 40 34 0 0 0 0 0 0 1 0 0 0 0 0 25 40
,
 32 39 0 0 0 0 36 9 0 0 0 0 0 0 32 0 0 0 0 0 22 9 0 0 0 0 0 0 9 37 0 0 0 0 20 32
,
 1 0 0 0 0 0 32 40 0 0 0 0 0 0 1 0 0 0 0 0 34 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,1,25,0,0,0,0,0,40],[32,36,0,0,0,0,39,9,0,0,0,0,0,0,32,22,0,0,0,0,0,9,0,0,0,0,0,0,9,20,0,0,0,0,37,32],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;`

D107M4(2) in GAP, Magma, Sage, TeX

`D_{10}\rtimes_7M_4(2)`
`% in TeX`

`G:=Group("D10:7M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,353);`
`// by ID`

`G=gap.SmallGroup(320,353);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,58,136,12550]);`
`// Polycyclic`

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

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