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## G = D10.3M4(2)  order 320 = 26·5

### 1st non-split extension by D10 of M4(2) acting via M4(2)/C8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D10.3M4(2)
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C2×C4×D5 — C2×C4×F5 — D10.3M4(2)
 Lower central C5 — C10 — D10.3M4(2)
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for D10.3M4(2)
G = < a,b,c,d | a10=b2=c8=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=dbd-1=a2b, dcd-1=a5c5 >

Subgroups: 418 in 118 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C2×C8, C2×C8, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C42, C22×C8, C52C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C2×F5, C2×F5, C22×D5, C22.7C42, C8×D5, C2×C52C8, C2×C40, D5⋊C8, C4×F5, C2×C5⋊C8, C2×C4×D5, C22×F5, D5×C2×C8, C2×D5⋊C8, C2×C4×F5, D10.3M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), F5, C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C2×F5, C22.7C42, C4×F5, C4⋊F5, C22⋊F5, C8×F5, C8⋊F5, D10.3Q8, D10.3M4(2)

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

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

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

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

56 conjugacy classes

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

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 4 type + + + + + - + + + image C1 C2 C2 C2 C4 C4 C4 C4 C8 D4 Q8 M4(2) F5 C2×F5 C4⋊F5 C22⋊F5 C4×F5 C8×F5 C8⋊F5 kernel D10.3M4(2) D5×C2×C8 C2×D5⋊C8 C2×C4×F5 C2×C5⋊2C8 C2×C40 C2×C5⋊C8 C22×F5 C2×F5 C4×D5 C4×D5 D10 C2×C8 C2×C4 C4 C4 C22 C2 C2 # reps 1 1 1 1 2 2 4 4 16 3 1 4 1 1 2 2 2 4 4

Matrix representation of D10.3M4(2) in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 40 0 0 1 1 1 1 0 0 40 0 0 0 0 0 0 40 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 0 0 1
,
 32 35 0 0 0 0 0 9 0 0 0 0 0 0 0 0 14 0 0 0 14 0 0 0 0 0 0 0 0 14 0 0 0 14 0 0
,
 9 0 0 0 0 0 14 32 0 0 0 0 0 0 0 0 9 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 9 0 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,40,0,0,0,1,0,0,0,0,40,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,40,1],[32,0,0,0,0,0,35,9,0,0,0,0,0,0,0,14,0,0,0,0,0,0,0,14,0,0,14,0,0,0,0,0,0,0,14,0],[9,14,0,0,0,0,0,32,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,9,0,0,9,0,0,0,0,0,0,0,9,0] >;`

D10.3M4(2) in GAP, Magma, Sage, TeX

`D_{10}._3M_4(2)`
`% in TeX`

`G:=Group("D10.3M4(2)");`
`// GroupNames label`

`G:=SmallGroup(320,230);`
`// by ID`

`G=gap.SmallGroup(320,230);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,136,6278,3156]);`
`// Polycyclic`

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

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