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## G = M4(2).31D10order 320 = 26·5

### 4th non-split extension by M4(2) of D10 acting via D10/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — M4(2).31D10
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C2×C4○D20 — M4(2).31D10
 Lower central C5 — C10 — C2×C10 — M4(2).31D10
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for M4(2).31D10
G = < a,b,c,d | a8=b2=1, c10=d2=a4, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c9 >

Subgroups: 574 in 150 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4.D4, C4.10D4, C2×M4(2), C2×M4(2), C2×C4○D4, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, M4(2).8C22, C2×C52C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C20.46D4, C4.12D20, C2×C4.Dic5, C10×M4(2), C2×C4○D20, M4(2).31D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, M4(2).8C22, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, M4(2).31D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 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 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 2 2 2 20 20 1 1 2 2 2 20 20 2 2 4 4 4 4 20 20 20 20 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D5 D10 D10 C4×D5 D20 C5⋊D4 C4×D5 M4(2).8C22 M4(2).31D10 kernel M4(2).31D10 C20.46D4 C4.12D20 C2×C4.Dic5 C10×M4(2) C2×C4○D20 C2×C4×D5 C2×C5⋊D4 C2×C20 C2×M4(2) M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C5 C1 # reps 1 2 2 1 1 1 4 4 4 2 4 2 4 8 8 4 2 8

Matrix representation of M4(2).31D10 in GL4(𝔽41) generated by

 0 0 9 0 0 0 0 9 18 35 0 0 6 23 0 0
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 28 28 0 0 13 32 0 0 0 0 28 28 0 0 13 32
,
 28 28 0 0 32 13 0 0 0 0 28 28 0 0 32 13
`G:=sub<GL(4,GF(41))| [0,0,18,6,0,0,35,23,9,0,0,0,0,9,0,0],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[28,13,0,0,28,32,0,0,0,0,28,13,0,0,28,32],[28,32,0,0,28,13,0,0,0,0,28,32,0,0,28,13] >;`

M4(2).31D10 in GAP, Magma, Sage, TeX

`M_4(2)._{31}D_{10}`
`% in TeX`

`G:=Group("M4(2).31D10");`
`// GroupNames label`

`G:=SmallGroup(320,759);`
`// by ID`

`G=gap.SmallGroup(320,759);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,58,1123,136,438,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^8=b^2=1,c^10=d^2=a^4,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^9>;`
`// generators/relations`

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