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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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D10.3M4(2), (C2×F5)⋊C8, (C2×C8)⋊3F5, (C2×C40)⋊1C4, D5.(C4⋊C8), C2.5(C8×F5), C10.4(C4×C8), D10.6(C2×C8), C4.28(C4⋊F5), D5.(C22⋊C8), C20.28(C4⋊C4), (C4×D5).29Q8, (C4×D5).116D4, C2.3(C8⋊F5), C10.4(C8⋊C4), D10.21(C4⋊C4), (C22×F5).2C4, C22.16(C4×F5), (C2×C10).11C42, C4.37(C22⋊F5), C20.35(C22⋊C4), Dic5.23(C4⋊C4), D10.30(C22⋊C4), C2.1(D10.3Q8), C51(C22.7C42), C10.7(C2.C42), Dic5.31(C22⋊C4), (C2×C5⋊C8)⋊3C4, (C2×C4×F5).7C2, (D5×C2×C8).11C2, (C2×C52C8)⋊21C4, (C2×D5⋊C8).7C2, (C2×C4).157(C2×F5), (C2×C20).163(C2×C4), (C2×C4×D5).407C22, (C22×D5).85(C2×C4), (C2×Dic5).122(C2×C4), SmallGroup(320,230)

Series: Derived Chief Lower central Upper central

C1C10 — D10.3M4(2)
C1C5C10D10C4×D5C2×C4×D5C2×C4×F5 — D10.3M4(2)
C5C10 — D10.3M4(2)
C1C2×C4C2×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 [×3], C2 [×4], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×4], C2×C4, C2×C4 [×13], C23, D5 [×4], C10 [×3], C42 [×2], C2×C8, C2×C8 [×7], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2×C42, C22×C8 [×2], C52C8, C40, C5⋊C8 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C2×F5 [×4], C22×D5, C22.7C42, C8×D5 [×2], C2×C52C8, C2×C40, D5⋊C8 [×2], C4×F5 [×2], C2×C5⋊C8 [×2], C2×C4×D5, C22×F5 [×2], D5×C2×C8, C2×D5⋊C8, C2×C4×F5, D10.3M4(2)
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], F5, C2.C42, C4×C8, C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], 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 22)(2 21)(3 30)(4 29)(5 28)(6 27)(7 26)(8 25)(9 24)(10 23)(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 23 53 50 11)(2 67 32 71 24 60 49 14)(3 64 31 74 25 57 48 17)(4 61 40 77 26 54 47 20)(5 68 39 80 27 51 46 13)(6 65 38 73 28 58 45 16)(7 62 37 76 29 55 44 19)(8 69 36 79 30 52 43 12)(9 66 35 72 21 59 42 15)(10 63 34 75 22 56 41 18)
(1 38 23 45)(2 35 22 48)(3 32 21 41)(4 39 30 44)(5 36 29 47)(6 33 28 50)(7 40 27 43)(8 37 26 46)(9 34 25 49)(10 31 24 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), (1,22)(2,21)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(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), (1,70,33,78,23,53,50,11)(2,67,32,71,24,60,49,14)(3,64,31,74,25,57,48,17)(4,61,40,77,26,54,47,20)(5,68,39,80,27,51,46,13)(6,65,38,73,28,58,45,16)(7,62,37,76,29,55,44,19)(8,69,36,79,30,52,43,12)(9,66,35,72,21,59,42,15)(10,63,34,75,22,56,41,18), (1,38,23,45)(2,35,22,48)(3,32,21,41)(4,39,30,44)(5,36,29,47)(6,33,28,50)(7,40,27,43)(8,37,26,46)(9,34,25,49)(10,31,24,42)(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), (1,22)(2,21)(3,30)(4,29)(5,28)(6,27)(7,26)(8,25)(9,24)(10,23)(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), (1,70,33,78,23,53,50,11)(2,67,32,71,24,60,49,14)(3,64,31,74,25,57,48,17)(4,61,40,77,26,54,47,20)(5,68,39,80,27,51,46,13)(6,65,38,73,28,58,45,16)(7,62,37,76,29,55,44,19)(8,69,36,79,30,52,43,12)(9,66,35,72,21,59,42,15)(10,63,34,75,22,56,41,18), (1,38,23,45)(2,35,22,48)(3,32,21,41)(4,39,30,44)(5,36,29,47)(6,33,28,50)(7,40,27,43)(8,37,26,46)(9,34,25,49)(10,31,24,42)(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)], [(1,22),(2,21),(3,30),(4,29),(5,28),(6,27),(7,26),(8,25),(9,24),(10,23),(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)], [(1,70,33,78,23,53,50,11),(2,67,32,71,24,60,49,14),(3,64,31,74,25,57,48,17),(4,61,40,77,26,54,47,20),(5,68,39,80,27,51,46,13),(6,65,38,73,28,58,45,16),(7,62,37,76,29,55,44,19),(8,69,36,79,30,52,43,12),(9,66,35,72,21,59,42,15),(10,63,34,75,22,56,41,18)], [(1,38,23,45),(2,35,22,48),(3,32,21,41),(4,39,30,44),(5,36,29,47),(6,33,28,50),(7,40,27,43),(8,37,26,46),(9,34,25,49),(10,31,24,42),(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 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I···4P 5 8A8B8C8D8E···8P10A10B10C20A20B20C20D40A···40H
order12222222444444444···4588888···81010102020202040···40
size111155551111555510···104222210···1044444444···4

56 irreducible representations

dim1111111112224444444
type+++++-+++
imageC1C2C2C2C4C4C4C4C8D4Q8M4(2)F5C2×F5C4⋊F5C22⋊F5C4×F5C8×F5C8⋊F5
kernelD10.3M4(2)D5×C2×C8C2×D5⋊C8C2×C4×F5C2×C52C8C2×C40C2×C5⋊C8C22×F5C2×F5C4×D5C4×D5D10C2×C8C2×C4C4C4C22C2C2
# reps11112244163141122244

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

4000000
0400000
0000040
001111
0040000
0004000
,
100000
010000
000100
001000
0040404040
000001
,
32350000
090000
0000140
0014000
0000014
0001400
,
900000
14320000
000090
009000
000009
000900

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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