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G = D5⋊M5(2)  order 320 = 26·5

The semidirect product of D5 and M5(2) acting via M5(2)/C2×C8=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5⋊M5(2), D5⋊C164C2, C5⋊C162C22, (C4×D5).4C8, C52(C2×M5(2)), C8.33(C2×F5), (C2×C8).20F5, (C2×C40).24C4, C20.15(C2×C8), C40.40(C2×C4), (C8×D5).12C4, C8.F55C2, C20.C87C2, C4.10(D5⋊C8), D10.18(C2×C8), (C22×D5).9C8, C4.47(C22×F5), C10.10(C22×C8), C20.87(C22×C4), C52C8.37C23, C22.5(D5⋊C8), Dic5.18(C2×C8), (C2×Dic5).14C8, (C8×D5).63C22, (C2×C4×D5).39C4, (D5×C2×C8).31C2, C2.11(C2×D5⋊C8), (C2×C10).12(C2×C8), C52C8.53(C2×C4), (C4×D5).93(C2×C4), (C2×C4).134(C2×F5), (C2×C20).145(C2×C4), (C2×C52C8).338C22, SmallGroup(320,1053)

Series: Derived Chief Lower central Upper central

C1C10 — D5⋊M5(2)
C1C5C10C20C52C8C5⋊C16C20.C8 — D5⋊M5(2)
C5C10 — D5⋊M5(2)
C1C8C2×C8

Generators and relations for D5⋊M5(2)
 G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, cac-1=a3, ad=da, cbc-1=a2b, bd=db, dcd=c9 >

Subgroups: 250 in 90 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C16 [×4], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C16 [×2], M5(2) [×4], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×M5(2), C5⋊C16 [×4], C8×D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, D5⋊C16 [×2], C8.F5 [×2], C20.C8 [×2], D5×C2×C8, D5⋊M5(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, M5(2) [×2], C22×C8, C2×F5 [×3], C2×M5(2), D5⋊C8 [×2], C22×F5, C2×D5⋊C8, D5⋊M5(2)

Smallest permutation representation of D5⋊M5(2)
On 80 points
Generators in S80
(1 52 72 48 26)(2 33 53 27 73)(3 28 34 74 54)(4 75 29 55 35)(5 56 76 36 30)(6 37 57 31 77)(7 32 38 78 58)(8 79 17 59 39)(9 60 80 40 18)(10 41 61 19 65)(11 20 42 66 62)(12 67 21 63 43)(13 64 68 44 22)(14 45 49 23 69)(15 24 46 70 50)(16 71 25 51 47)
(1 26)(2 73)(3 54)(4 35)(5 30)(6 77)(7 58)(8 39)(9 18)(10 65)(11 62)(12 43)(13 22)(14 69)(15 50)(16 47)(19 41)(20 66)(23 45)(24 70)(27 33)(28 74)(31 37)(32 78)(36 56)(40 60)(44 64)(48 52)(51 71)(55 75)(59 79)(63 67)
(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)
(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)(33 41)(35 43)(37 45)(39 47)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F 5 8A8B8C8D8E8F8G8H8I8J8K8L10A10B10C16A···16P20A20B20C20D40A···40H
order122222444444588888888888810101016···162020202040···40
size1125510112551041111225555101044410···1044444···4

56 irreducible representations

dim111111111112444444
type++++++++
imageC1C2C2C2C2C4C4C4C8C8C8M5(2)F5C2×F5C2×F5D5⋊C8D5⋊C8D5⋊M5(2)
kernelD5⋊M5(2)D5⋊C16C8.F5C20.C8D5×C2×C8C8×D5C2×C40C2×C4×D5C4×D5C2×Dic5C22×D5D5C2×C8C8C2×C4C4C22C1
# reps122214228448121228

Matrix representation of D5⋊M5(2) in GL4(𝔽241) generated by

19019100
24024000
000190
005251
,
2405000
0100
0010
00240240
,
0010
0001
848400
3015700
,
1000
0100
002400
000240
G:=sub<GL(4,GF(241))| [190,240,0,0,191,240,0,0,0,0,0,52,0,0,190,51],[240,0,0,0,50,1,0,0,0,0,1,240,0,0,0,240],[0,0,84,30,0,0,84,157,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240] >;

D5⋊M5(2) in GAP, Magma, Sage, TeX

D_5\rtimes M_5(2)
% in TeX

G:=Group("D5:M5(2)");
// GroupNames label

G:=SmallGroup(320,1053);
// by ID

G=gap.SmallGroup(320,1053);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,80,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^9>;
// generators/relations

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