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G = Dic52order 400 = 24·52

Direct product of Dic5 and Dic5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: Dic52, C52:8C42, C102.1C22, C22.3D52, C5:3(C4xDic5), C52:6C4:3C4, (C5xDic5):8C4, (C2xC10).7D10, C10.25(C4xD5), C2.2(D5xDic5), (C2xDic5).6D5, (C10xDic5).7C2, C10.10(C2xDic5), C2.2(Dic5:2D5), (C5xC10).45(C2xC4), (C2xC52:6C4).1C2, SmallGroup(400,71)

Series: Derived Chief Lower central Upper central

C1C52 — Dic52
C1C5C52C5xC10C102C10xDic5 — Dic52
C52 — Dic52
C1C22

Generators and relations for Dic52
 G = < a,b,c,d | a10=c10=1, b2=a5, d2=c5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 316 in 76 conjugacy classes, 36 normal (8 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2xC4, C10, C10, C42, Dic5, Dic5, C20, C2xC10, C2xC10, C52, C2xDic5, C2xDic5, C2xC20, C5xC10, C5xC10, C4xDic5, C5xDic5, C52:6C4, C102, C10xDic5, C2xC52:6C4, Dic52
Quotients: C1, C2, C4, C22, C2xC4, D5, C42, Dic5, D10, C4xD5, C2xDic5, C4xDic5, D52, D5xDic5, Dic5:2D5, Dic52

Smallest permutation representation of Dic52
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 46 6 41)(2 45 7 50)(3 44 8 49)(4 43 9 48)(5 42 10 47)(11 54 16 59)(12 53 17 58)(13 52 18 57)(14 51 19 56)(15 60 20 55)(21 64 26 69)(22 63 27 68)(23 62 28 67)(24 61 29 66)(25 70 30 65)(31 74 36 79)(32 73 37 78)(33 72 38 77)(34 71 39 76)(35 80 40 75)
(1 17 7 13 3 19 9 15 5 11)(2 18 8 14 4 20 10 16 6 12)(21 33 25 37 29 31 23 35 27 39)(22 34 26 38 30 32 24 36 28 40)(41 53 45 57 49 51 43 55 47 59)(42 54 46 58 50 52 44 56 48 60)(61 79 67 75 63 71 69 77 65 73)(62 80 68 76 64 72 70 78 66 74)
(1 39 19 29)(2 40 20 30)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 34 14 24)(7 35 15 25)(8 36 16 26)(9 37 17 27)(10 38 18 28)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)

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

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

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

64 conjugacy classes

class 1 2A2B2C4A···4H4I4J4K4L5A5B5C5D5E5F5G5H10A···10L10M···10X20A···20P
order12224···444445555555510···1010···1020···20
size11115···525252525222244442···24···410···10

64 irreducible representations

dim111112222444
type++++-++-+
imageC1C2C2C4C4D5Dic5D10C4xD5D52D5xDic5Dic5:2D5
kernelDic52C10xDic5C2xC52:6C4C5xDic5C52:6C4C2xDic5Dic5C2xC10C10C22C2C2
# reps1218448416484

Matrix representation of Dic52 in GL4(F41) generated by

40000
04000
00040
00134
,
32000
03200
00040
00400
,
344000
1000
00400
00040
,
40000
7100
00320
00032
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,0,1,0,0,40,34],[32,0,0,0,0,32,0,0,0,0,0,40,0,0,40,0],[34,1,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[40,7,0,0,0,1,0,0,0,0,32,0,0,0,0,32] >;

Dic52 in GAP, Magma, Sage, TeX

{\rm Dic}_5^2
% in TeX

G:=Group("Dic5^2");
// GroupNames label

G:=SmallGroup(400,71);
// by ID

G=gap.SmallGroup(400,71);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,31,970,11525]);
// Polycyclic

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

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