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G = D5xC5:D4order 400 = 24·52

Direct product of D5 and C5:D4

direct product, metabelian, supersoluble, monomial

Aliases: D5xC5:D4, D10:6D10, Dic5:1D10, C102:1C22, C5:5(D4xD5), C22:1D52, (C5xD5):2D4, C52:6(C2xD4), (C2xC10):1D10, C5:D20:5C2, (D5xDic5):3C2, (C22xD5):3D5, C52:2D4:4C2, C52:7D4:2C2, C52:6C4:C22, (D5xC10):3C22, (C5xC10).17C23, (C5xDic5):1C22, C10.17(C22xD5), (C2xD52):3C2, (D5xC2xC10):3C2, C5:2(C2xC5:D4), C2.17(C2xD52), (C5xC5:D4):3C2, (C2xC5:D5):2C22, SmallGroup(400,179)

Series: Derived Chief Lower central Upper central

C1C5xC10 — D5xC5:D4
C1C5C52C5xC10D5xC10C2xD52 — D5xC5:D4
C52C5xC10 — D5xC5:D4
C1C2C22

Generators and relations for D5xC5:D4
 G = < a,b,c,d,e | a5=b2=c5=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 836 in 124 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2xC4, D4, C23, D5, D5, C10, C10, C2xD4, Dic5, Dic5, C20, D10, D10, C2xC10, C2xC10, C52, C4xD5, D20, C2xDic5, C5:D4, C5:D4, C5xD4, C22xD5, C22xD5, C22xC10, C5xD5, C5xD5, C5:D5, C5xC10, C5xC10, D4xD5, C2xC5:D4, C5xDic5, C52:6C4, D52, D5xC10, D5xC10, C2xC5:D5, C102, D5xDic5, C52:2D4, C5:D20, C5xC5:D4, C52:7D4, C2xD52, D5xC2xC10, D5xC5:D4
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C5:D4, C22xD5, D4xD5, C2xC5:D4, D52, C2xD52, D5xC5:D4

Smallest permutation representation of D5xC5:D4
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 30)(2 29)(3 28)(4 27)(5 26)(6 23)(7 22)(8 21)(9 25)(10 24)(11 38)(12 37)(13 36)(14 40)(15 39)(16 33)(17 32)(18 31)(19 35)(20 34)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)(21 24 22 25 23)(26 29 27 30 28)(31 33 35 32 34)(36 38 40 37 39)
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)

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

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

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

52 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B5C5D5E5F5G5H10A···10H10I···10V10W···10AD10AE10AF20A20B
order12222222445555555510···1010···1010···1010102020
size112551010501050222244442···24···410···1020202020

52 irreducible representations

dim1111111122222224444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D5D5D10D10D10C5:D4D4xD5D52C2xD52D5xC5:D4
kernelD5xC5:D4D5xDic5C52:2D4C5:D20C5xC5:D4C52:7D4C2xD52D5xC2xC10C5xD5C5:D4C22xD5Dic5D10C2xC10D5C5C22C2C1
# reps1111111122226482448

Matrix representation of D5xC5:D4 in GL4(F41) generated by

1000
0100
00040
00134
,
1000
0100
00134
00040
,
16000
01800
0010
0001
,
02500
18000
0010
0001
,
02500
23000
0010
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,34],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,34,40],[16,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[0,18,0,0,25,0,0,0,0,0,1,0,0,0,0,1],[0,23,0,0,25,0,0,0,0,0,1,0,0,0,0,1] >;

D5xC5:D4 in GAP, Magma, Sage, TeX

D_5\times C_5\rtimes D_4
% in TeX

G:=Group("D5xC5:D4");
// GroupNames label

G:=SmallGroup(400,179);
// by ID

G=gap.SmallGroup(400,179);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,116,970,11525]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^5=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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