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G = Dic5.D10order 400 = 24·52

3rd non-split extension by Dic5 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial

Aliases: D10.3D10, Dic5.3D10, C102.7C22, C5:D4:3D5, C22.1D52, C5:4(C4oD20), C5:D20:2C2, (D5xDic5):5C2, (C2xDic5):3D5, (C2xC10).3D10, C52:6(C4oD4), C52:7D4:1C2, C5:3(D4:2D5), C52:2Q8:4C2, (C10xDic5):6C2, Dic5:2D5:2C2, (C5xC10).11C23, (D5xC10).3C22, C10.11(C22xD5), C52:6C4.3C22, (C5xDic5).21C22, C2.12(C2xD52), (C5xC5:D4):1C2, (C2xC5:D5).2C22, SmallGroup(400,173)

Series: Derived Chief Lower central Upper central

C1C5xC10 — Dic5.D10
C1C5C52C5xC10D5xC10D5xDic5 — Dic5.D10
C52C5xC10 — Dic5.D10
C1C2C22

Generators and relations for Dic5.D10
 G = < a,b,c,d | a10=c10=1, b2=d2=a5, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 596 in 96 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2xC4, D4, Q8, D5, C10, C10, C4oD4, Dic5, Dic5, C20, D10, D10, C2xC10, C2xC10, C52, Dic10, C4xD5, D20, C2xDic5, C2xDic5, C5:D4, C5:D4, C2xC20, C5xD4, C5xD5, C5:D5, C5xC10, C5xC10, C4oD20, D4:2D5, C5xDic5, C52:6C4, D5xC10, C2xC5:D5, C102, D5xDic5, Dic5:2D5, C5:D20, C52:2Q8, C10xDic5, C5xC5:D4, C52:7D4, Dic5.D10
Quotients: C1, C2, C22, C23, D5, C4oD4, D10, C22xD5, C4oD20, D4:2D5, D52, C2xD52, Dic5.D10

Smallest permutation representation of Dic5.D10
On 40 points
Generators in S40
(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)
(1 26 6 21)(2 25 7 30)(3 24 8 29)(4 23 9 28)(5 22 10 27)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)
(1 29 5 25 9 21 3 27 7 23)(2 28 6 24 10 30 4 26 8 22)(11 31 17 35 13 39 19 33 15 37)(12 40 18 34 14 38 20 32 16 36)
(1 34 6 39)(2 33 7 38)(3 32 8 37)(4 31 9 36)(5 40 10 35)(11 21 16 26)(12 30 17 25)(13 29 18 24)(14 28 19 23)(15 27 20 22)

G:=sub<Sym(40)| (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), (1,26,6,21)(2,25,7,30)(3,24,8,29)(4,23,9,28)(5,22,10,27)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35), (1,29,5,25,9,21,3,27,7,23)(2,28,6,24,10,30,4,26,8,22)(11,31,17,35,13,39,19,33,15,37)(12,40,18,34,14,38,20,32,16,36), (1,34,6,39)(2,33,7,38)(3,32,8,37)(4,31,9,36)(5,40,10,35)(11,21,16,26)(12,30,17,25)(13,29,18,24)(14,28,19,23)(15,27,20,22)>;

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), (1,26,6,21)(2,25,7,30)(3,24,8,29)(4,23,9,28)(5,22,10,27)(11,34,16,39)(12,33,17,38)(13,32,18,37)(14,31,19,36)(15,40,20,35), (1,29,5,25,9,21,3,27,7,23)(2,28,6,24,10,30,4,26,8,22)(11,31,17,35,13,39,19,33,15,37)(12,40,18,34,14,38,20,32,16,36), (1,34,6,39)(2,33,7,38)(3,32,8,37)(4,31,9,36)(5,40,10,35)(11,21,16,26)(12,30,17,25)(13,29,18,24)(14,28,19,23)(15,27,20,22) );

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)], [(1,26,6,21),(2,25,7,30),(3,24,8,29),(4,23,9,28),(5,22,10,27),(11,34,16,39),(12,33,17,38),(13,32,18,37),(14,31,19,36),(15,40,20,35)], [(1,29,5,25,9,21,3,27,7,23),(2,28,6,24,10,30,4,26,8,22),(11,31,17,35,13,39,19,33,15,37),(12,40,18,34,14,38,20,32,16,36)], [(1,34,6,39),(2,33,7,38),(3,32,8,37),(4,31,9,36),(5,40,10,35),(11,21,16,26),(12,30,17,25),(13,29,18,24),(14,28,19,23),(15,27,20,22)]])

52 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E5A5B5C5D5E5F5G5H10A···10H10I···10V10W10X20A···20H20I20J
order12222444445555555510···1010···10101020···202020
size112105055101050222244442···24···4202010···102020

52 irreducible representations

dim1111111122222224444
type+++++++++++++-++
imageC1C2C2C2C2C2C2C2D5D5C4oD4D10D10D10C4oD20D4:2D5D52C2xD52Dic5.D10
kernelDic5.D10D5xDic5Dic5:2D5C5:D20C52:2Q8C10xDic5C5xC5:D4C52:7D4C2xDic5C5:D4C52Dic5D10C2xC10C5C5C22C2C1
# reps1111111122262482448

Matrix representation of Dic5.D10 in GL6(F41)

4000000
0400000
001000
000100
00003440
000010
,
12230000
24290000
0040000
0004000
000010
00003440
,
12230000
33290000
00403400
007700
000010
00003440
,
15390000
31260000
001700
0004000
000010
00003440

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[12,24,0,0,0,0,23,29,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[12,33,0,0,0,0,23,29,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[15,31,0,0,0,0,39,26,0,0,0,0,0,0,1,0,0,0,0,0,7,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40] >;

Dic5.D10 in GAP, Magma, Sage, TeX

{\rm Dic}_5.D_{10}
% in TeX

G:=Group("Dic5.D10");
// GroupNames label

G:=SmallGroup(400,173);
// by ID

G=gap.SmallGroup(400,173);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,970,11525]);
// Polycyclic

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

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