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G = D5xM5(2)  order 320 = 26·5

Direct product of D5 and M5(2)

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

Aliases: D5xM5(2), C16:6D10, C80:7C22, C40.66C23, (D5xC16):7C2, (C8xD5).9C4, C80:C2:5C2, (C4xD5).1C8, C8.35(C4xD5), C4.15(C8xD5), C5:5(C2xM5(2)), C20.36(C2xC8), C40.79(C2xC4), C22.7(C8xD5), D10.11(C2xC8), (C2xC8).272D10, (C5xM5(2)):5C2, (C2xDic5).7C8, C20.4C8:13C2, (C22xD5).5C8, C8.60(C22xD5), C5:2C16:11C22, C10.39(C22xC8), Dic5.13(C2xC8), (C8xD5).47C22, (C2xC40).230C22, C20.190(C22xC4), C2.16(D5xC2xC8), (C2xC4xD5).12C4, (D5xC2xC8).28C2, C4.105(C2xC4xD5), (C2xC5:2C8).11C4, (C2xC10).20(C2xC8), C5:2C8.44(C2xC4), (C4xD5).82(C2xC4), (C2xC4).148(C4xD5), (C2xC20).245(C2xC4), SmallGroup(320,533)

Series: Derived Chief Lower central Upper central

C1C10 — D5xM5(2)
C1C5C10C20C40C8xD5D5xC2xC8 — D5xM5(2)
C5C10 — D5xM5(2)
C1C8M5(2)

Generators and relations for D5xM5(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, C4, C4, C22, C22, C5, C8, C8, C2xC4, C2xC4, C23, D5, D5, C10, C10, C16, C16, C2xC8, C2xC8, C22xC4, Dic5, C20, D10, D10, C2xC10, C2xC16, M5(2), M5(2), C22xC8, C5:2C8, C40, C4xD5, C2xDic5, C2xC20, C22xD5, C2xM5(2), C5:2C16, C80, C8xD5, C2xC5:2C8, C2xC40, C2xC4xD5, D5xC16, C80:C2, C20.4C8, C5xM5(2), D5xC2xC8, D5xM5(2)
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, D5, C2xC8, C22xC4, D10, M5(2), C22xC8, C4xD5, C22xD5, C2xM5(2), C8xD5, C2xC4xD5, D5xC2xC8, D5xM5(2)

Smallest permutation representation of D5xM5(2)
On 80 points
Generators in S80
(1 20 64 74 37)(2 21 49 75 38)(3 22 50 76 39)(4 23 51 77 40)(5 24 52 78 41)(6 25 53 79 42)(7 26 54 80 43)(8 27 55 65 44)(9 28 56 66 45)(10 29 57 67 46)(11 30 58 68 47)(12 31 59 69 48)(13 32 60 70 33)(14 17 61 71 34)(15 18 62 72 35)(16 19 63 73 36)
(1 45)(2 46)(3 47)(4 48)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 79)(18 80)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 71)(26 72)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(49 57)(50 58)(51 59)(52 60)(53 61)(54 62)(55 63)(56 64)
(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)(34 42)(36 44)(38 46)(40 48)(49 57)(51 59)(53 61)(55 63)(65 73)(67 75)(69 77)(71 79)

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

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

G=PermutationGroup([[(1,20,64,74,37),(2,21,49,75,38),(3,22,50,76,39),(4,23,51,77,40),(5,24,52,78,41),(6,25,53,79,42),(7,26,54,80,43),(8,27,55,65,44),(9,28,56,66,45),(10,29,57,67,46),(11,30,58,68,47),(12,31,59,69,48),(13,32,60,70,33),(14,17,61,71,34),(15,18,62,72,35),(16,19,63,73,36)], [(1,45),(2,46),(3,47),(4,48),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,79),(18,80),(19,65),(20,66),(21,67),(22,68),(23,69),(24,70),(25,71),(26,72),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(49,57),(50,58),(51,59),(52,60),(53,61),(54,62),(55,63),(56,64)], [(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),(34,42),(36,44),(38,46),(40,48),(49,57),(51,59),(53,61),(55,63),(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)C4xD5C4xD5C8xD5C8xD5D5xM5(2)
kernelD5xM5(2)D5xC16C80:C2C20.4C8C5xM5(2)D5xC2xC8C8xD5C2xC5:2C8C2xC4xD5C4xD5C2xDic5C22xD5M5(2)C16C2xC8D5C8C2xC4C4C22C1
# reps122111422844242844888

Matrix representation of D5xM5(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] >;

D5xM5(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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