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G = M4(2).19D10order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).19D10, (C4×D5).91D4, C4.147(D4×D5), C20.90(C2×D4), C4.D46D5, C23.7(C4×D5), C20.D44C2, (D5×M4(2))⋊6C2, (C2×C20).2C23, (C2×D4).122D10, C4.12D205C2, (D4×C10).12C22, (C22×Dic5).4C4, C4.Dic5.1C22, D10.21(C22⋊C4), (C5×M4(2)).9C22, Dic5.54(C22⋊C4), (C2×Dic10).48C22, C53(M4(2).8C22), (C2×C5⋊D4).2C4, (C2×C4×D5).6C22, C22.15(C2×C4×D5), (C5×C4.D4)⋊6C2, C2.14(D5×C22⋊C4), (C2×C4).2(C22×D5), (C2×D42D5).2C2, C10.54(C2×C22⋊C4), (C22×C10).7(C2×C4), (C2×Dic5).2(C2×C4), (C22×D5).20(C2×C4), (C2×C10).110(C22×C4), SmallGroup(320,372)

Series: Derived Chief Lower central Upper central

C1C2×C10 — M4(2).19D10
C1C5C10C20C2×C20C2×C4×D5C2×D42D5 — M4(2).19D10
C5C10C2×C10 — M4(2).19D10
C1C2C2×C4C4.D4

Generators and relations for M4(2).19D10
 G = < a,b,c,d | a8=b2=c10=1, d2=a4, bab=a5, cac-1=ab, dad-1=a5b, bc=cb, bd=db, dcd-1=a4c-1 >

Subgroups: 558 in 150 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×4], C2×C4, C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×2], C23, D5 [×2], C10, C10 [×3], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10, C2×C10, C2×C10 [×4], C4.D4, C4.D4, C4.10D4 [×2], C2×M4(2) [×2], C2×C4○D4, C52C8 [×2], C40 [×2], Dic10 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], M4(2).8C22, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5 [×2], C5×M4(2) [×2], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C4.12D20 [×2], C20.D4, C5×C4.D4, D5×M4(2) [×2], C2×D42D5, M4(2).19D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], C22×D5, M4(2).8C22, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, M4(2).19D10

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H10A10B10C10D10E10F10G10H20A20B20C20D40A···40H
order12222224444444558888888810101010101010102020202040···40
size1124410102255102020224444202020202244888844448···8

44 irreducible representations

dim1111111122222448
type+++++++++++-
imageC1C2C2C2C2C2C4C4D4D5D10D10C4×D5M4(2).8C22D4×D5M4(2).19D10
kernelM4(2).19D10C4.12D20C20.D4C5×C4.D4D5×M4(2)C2×D42D5C22×Dic5C2×C5⋊D4C4×D5C4.D4M4(2)C2×D4C23C5C4C1
# reps1211214442428242

Matrix representation of M4(2).19D10 in GL8(𝔽41)

90000000
032000000
00900000
000320000
00004062539
0000732340
000012202919
00001210910
,
400000000
040000000
004000000
000400000
00001000
000014000
00000010
0000202540
,
06060000
60600000
035010000
350100000
0000150210
00001214219
000030260
00009332927
,
0350350000
3503500000
040060000
400600000
0000260200
000029272032
000010150
00009311214

G:=sub<GL(8,GF(41))| [9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,7,12,12,0,0,0,0,6,3,20,10,0,0,0,0,25,23,29,9,0,0,0,0,39,40,19,10],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,1,0,2,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,25,0,0,0,0,0,0,0,40],[0,6,0,35,0,0,0,0,6,0,35,0,0,0,0,0,0,6,0,1,0,0,0,0,6,0,1,0,0,0,0,0,0,0,0,0,15,12,3,9,0,0,0,0,0,14,0,33,0,0,0,0,21,21,26,29,0,0,0,0,0,9,0,27],[0,35,0,40,0,0,0,0,35,0,40,0,0,0,0,0,0,35,0,6,0,0,0,0,35,0,6,0,0,0,0,0,0,0,0,0,26,29,1,9,0,0,0,0,0,27,0,31,0,0,0,0,20,20,15,12,0,0,0,0,0,32,0,14] >;

M4(2).19D10 in GAP, Magma, Sage, TeX

M_4(2)._{19}D_{10}
% in TeX

G:=Group("M4(2).19D10");
// GroupNames label

G:=SmallGroup(320,372);
// by ID

G=gap.SmallGroup(320,372);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,219,58,570,136,438,12550]);
// Polycyclic

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

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