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G = M4(2).22D10order 320 = 26·5

5th non-split extension by M4(2) of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — M4(2).22D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4○D20 — D4.8D10 — M4(2).22D10
 Lower central C5 — C10 — C20 — M4(2).22D10
 Upper central C1 — C4 — C2×C4 — C4≀C2

Generators and relations for M4(2).22D10
G = < a,b,c,d | a8=b2=c10=1, d2=a6b, bab=a5, cac-1=dad-1=a-1b, cbc-1=a4b, bd=db, dcd-1=a6bc-1 >

Subgroups: 326 in 104 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C42, C2×C8, M4(2), M4(2), D8, SD16, Q16, C4○D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C8⋊C4, C4≀C2, C4≀C2, C8.C4, C8○D4, C4○D8, C52C8, C52C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C8.26D4, C8×D5, C8⋊D5, C2×C52C8, C2×C52C8, C4.Dic5, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4×C20, C5×M4(2), C4○D20, C5×C4○D4, C42.D5, D204C4, C20.53D4, C5×C4≀C2, D20.2C4, D4.Dic5, D4.8D10, M4(2).22D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, D10, C4×D4, C4×D5, C22×D5, C8.26D4, C2×C4×D5, D4×D5, D42D5, Dic54D4, M4(2).22D10

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

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 D5 C4○D4 D10 D10 D10 C4×D5 C4×D5 C8.26D4 D4×D5 D4⋊2D5 M4(2).22D10 kernel M4(2).22D10 C42.D5 D20⋊4C4 C20.53D4 C5×C4≀C2 D20.2C4 D4.Dic5 D4.8D10 D4⋊D5 D4.D5 Q8⋊D5 C5⋊Q16 C5⋊2C8 C4≀C2 C2×C10 C42 M4(2) C4○D4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 2 2 2 8

Matrix representation of M4(2).22D10 in GL6(𝔽41)

 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 17 0 0 0 0 27 40 0 0 0 0 0 0 40 24 0 0 0 0 14 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 17 1 0 0 0 0 0 0 1 0 0 0 0 0 24 40
,
 10 0 0 0 0 0 0 37 0 0 0 0 0 0 40 24 0 0 0 0 0 1 0 0 0 0 0 0 9 30 0 0 0 0 11 32
,
 0 4 0 0 0 0 31 0 0 0 0 0 0 0 0 0 40 24 0 0 0 0 0 1 0 0 9 30 0 0 0 0 11 32 0 0

`G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,27,0,0,0,0,17,40,0,0,0,0,0,0,40,14,0,0,0,0,24,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,17,0,0,0,0,0,1,0,0,0,0,0,0,1,24,0,0,0,0,0,40],[10,0,0,0,0,0,0,37,0,0,0,0,0,0,40,0,0,0,0,0,24,1,0,0,0,0,0,0,9,11,0,0,0,0,30,32],[0,31,0,0,0,0,4,0,0,0,0,0,0,0,0,0,9,11,0,0,0,0,30,32,0,0,40,0,0,0,0,0,24,1,0,0] >;`

M4(2).22D10 in GAP, Magma, Sage, TeX

`M_4(2)._{22}D_{10}`
`% in TeX`

`G:=Group("M4(2).22D10");`
`// GroupNames label`

`G:=SmallGroup(320,450);`
`// by ID`

`G=gap.SmallGroup(320,450);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,219,58,136,1684,851,438,102,12550]);`
`// Polycyclic`

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

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