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## G = M4(2)⋊D10order 320 = 26·5

### 4th semidirect product of M4(2) and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — M4(2)⋊D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4⋊8D10 — M4(2)⋊D10
 Lower central C5 — C10 — C2×C20 — M4(2)⋊D10
 Upper central C1 — C2 — C2×C4 — C4≀C2

Generators and relations for M4(2)⋊D10
G = < a,b,c,d | a8=b2=c10=d2=1, bab=a5, cac-1=a-1b, dad=ab, cbc-1=a4b, bd=db, dcd=c-1 >

Subgroups: 782 in 152 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), M4(2), SD16, Q16, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4.D4, C4≀C2, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C52C8, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, D4.9D4, C40⋊C2, Dic20, C4.Dic5, D10⋊C4, D4.D5, C5⋊Q16, C4×C20, C5×M4(2), C2×Dic10, C2×D20, C2×D20, C4○D20, C4○D20, D4×D5, Q82D5, C5×C4○D4, D204C4, C20.46D4, C5×C4≀C2, C4.D20, C8.D10, D4.9D10, D48D10, M4(2)⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, D20, C22×D5, D4.9D4, C2×D20, D4×D5, C22⋊D20, M4(2)⋊D10

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4 D5 D10 D10 D10 D20 D20 D4.9D4 D4×D5 D4×D5 M4(2)⋊D10 kernel M4(2)⋊D10 D20⋊4C4 C20.46D4 C5×C4≀C2 C4.D20 C8.D10 D4.9D10 D4⋊8D10 Dic10 D20 C5×D4 C5×Q8 C22×D5 C4≀C2 C42 M4(2) C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 2 2 2 8

Matrix representation of M4(2)⋊D10 in GL4(𝔽41) generated by

 25 38 16 3 3 7 38 34 16 3 16 3 38 34 38 34
,
 0 0 39 28 0 0 13 2 2 13 0 0 28 39 0 0
,
 35 35 0 0 6 40 0 0 0 0 6 6 0 0 35 1
,
 35 35 0 0 40 6 0 0 0 0 6 6 0 0 1 35
`G:=sub<GL(4,GF(41))| [25,3,16,38,38,7,3,34,16,38,16,38,3,34,3,34],[0,0,2,28,0,0,13,39,39,13,0,0,28,2,0,0],[35,6,0,0,35,40,0,0,0,0,6,35,0,0,6,1],[35,40,0,0,35,6,0,0,0,0,6,1,0,0,6,35] >;`

M4(2)⋊D10 in GAP, Magma, Sage, TeX

`M_4(2)\rtimes D_{10}`
`% in TeX`

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

`G:=SmallGroup(320,452);`
`// by ID`

`G=gap.SmallGroup(320,452);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,58,1123,136,851,438,102,12550]);`
`// Polycyclic`

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

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