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G = D5×M5(2)  order 320 = 26·5

Direct product of D5 and M5(2)

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×M5(2), C166D10, C807C22, C40.66C23, (D5×C16)⋊7C2, (C8×D5).9C4, C80⋊C25C2, (C4×D5).1C8, C8.35(C4×D5), C4.15(C8×D5), C55(C2×M5(2)), C20.36(C2×C8), C40.79(C2×C4), C22.7(C8×D5), D10.11(C2×C8), (C2×C8).272D10, (C5×M5(2))⋊5C2, (C2×Dic5).7C8, C20.4C813C2, (C22×D5).5C8, C8.60(C22×D5), C52C1611C22, C10.39(C22×C8), Dic5.13(C2×C8), (C8×D5).47C22, (C2×C40).230C22, C20.190(C22×C4), C2.16(D5×C2×C8), (C2×C4×D5).12C4, (D5×C2×C8).28C2, C4.105(C2×C4×D5), (C2×C52C8).11C4, (C2×C10).20(C2×C8), C52C8.44(C2×C4), (C4×D5).82(C2×C4), (C2×C4).148(C4×D5), (C2×C20).245(C2×C4), SmallGroup(320,533)

Series: Derived Chief Lower central Upper central

C1C10 — D5×M5(2)
C1C5C10C20C40C8×D5D5×C2×C8 — D5×M5(2)
C5C10 — D5×M5(2)
C1C8M5(2)

Generators and relations for D5×M5(2)
 G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c9 >

Subgroups: 238 in 90 conjugacy classes, 53 normal (41 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 [×2], C16 [×2], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C16 [×2], M5(2), M5(2) [×3], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×M5(2), C52C16 [×2], C80 [×2], C8×D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, D5×C16 [×2], C80⋊C2 [×2], C20.4C8, C5×M5(2), D5×C2×C8, D5×M5(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, D5, C2×C8 [×6], C22×C4, D10 [×3], M5(2) [×2], C22×C8, C4×D5 [×2], C22×D5, C2×M5(2), C8×D5 [×2], C2×C4×D5, D5×C2×C8, D5×M5(2)

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

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D8E8F8G8H8I8J8K8L10A10B10C10D16A···16H16I···16P20A20B20C20D20E20F40A···40H40I40J40K40L80A···80P
order122222444444558888888888881010101016···1616···1620202020202040···404040404080···80
size11255101125510221111225555101022442···210···102222442···244444···4

80 irreducible representations

dim111111111111222222224
type+++++++++
imageC1C2C2C2C2C2C4C4C4C8C8C8D5D10D10M5(2)C4×D5C4×D5C8×D5C8×D5D5×M5(2)
kernelD5×M5(2)D5×C16C80⋊C2C20.4C8C5×M5(2)D5×C2×C8C8×D5C2×C52C8C2×C4×D5C4×D5C2×Dic5C22×D5M5(2)C16C2×C8D5C8C2×C4C4C22C1
# reps122111422844242844888

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

1000
0100
0001
00240189
,
240000
024000
0001
0010
,
15400
11124000
0010
0001
,
1000
11624000
0010
0001
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,189],[240,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0],[1,111,0,0,54,240,0,0,0,0,1,0,0,0,0,1],[1,116,0,0,0,240,0,0,0,0,1,0,0,0,0,1] >;

D5×M5(2) in GAP, Magma, Sage, TeX

D_5\times M_5(2)
% in TeX

G:=Group("D5xM5(2)");
// GroupNames label

G:=SmallGroup(320,533);
// by ID

G=gap.SmallGroup(320,533);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,58,80,102,12550]);
// Polycyclic

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

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