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G = C5xD5.D5order 500 = 22·53

Direct product of C5 and D5.D5

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

Aliases: C5xD5.D5, C53:2C4, C52:5C20, C52:11F5, C52:4Dic5, D5.(C5xD5), C5:(C5xDic5), C5:3(C5xF5), (C5xD5).1D5, (C5xD5).3C10, (D5xC52).2C2, SmallGroup(500,42)

Series: Derived Chief Lower central Upper central

C1C52 — C5xD5.D5
C1C5C52C5xD5D5xC52 — C5xD5.D5
C52 — C5xD5.D5
C1C5

Generators and relations for C5xD5.D5
 G = < a,b,c,d,e | a5=b5=c2=d5=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b2, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 184 in 44 conjugacy classes, 14 normal (all characteristic)
Quotients: C1, C2, C4, C5, D5, C10, Dic5, C20, F5, C5xD5, C5xDic5, C5xF5, D5.D5, C5xD5.D5
5C2
2C5
2C5
4C5
4C5
4C5
4C5
4C5
4C5
25C4
5C10
5C10
10C10
10C10
2C52
2C52
4C52
4C52
4C52
4C52
4C52
4C52
5F5
5Dic5
25C20
2C5xD5
2C5xD5
5C5xC10
5C5xF5
5C5xDic5

Permutation representations of C5xD5.D5
On 20 points - transitive group 20T127
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 9 7 10 8)(11 12 13 14 15)(16 20 19 18 17)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 19 7 14)(2 20 8 15)(3 16 9 11)(4 17 10 12)(5 18 6 13)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,9,7,10,8)(11,12,13,14,15)(16,20,19,18,17), (1,9)(2,10)(3,6)(4,7)(5,8)(11,17)(12,18)(13,19)(14,20)(15,16), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,19,7,14)(2,20,8,15)(3,16,9,11)(4,17,10,12)(5,18,6,13)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,9,7,10,8)(11,12,13,14,15)(16,20,19,18,17), (1,9)(2,10)(3,6)(4,7)(5,8)(11,17)(12,18)(13,19)(14,20)(15,16), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,19,7,14)(2,20,8,15)(3,16,9,11)(4,17,10,12)(5,18,6,13) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,3,5,2,4),(6,9,7,10,8),(11,12,13,14,15),(16,20,19,18,17)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,17),(12,18),(13,19),(14,20),(15,16)], [(1,3,5,2,4),(6,8,10,7,9),(11,14,12,15,13),(16,19,17,20,18)], [(1,19,7,14),(2,20,8,15),(3,16,9,11),(4,17,10,12),(5,18,6,13)]])

G:=TransitiveGroup(20,127);

65 conjugacy classes

class 1  2 4A4B5A5B5C5D5E···5N5O···5AM10A10B10C10D10E···10N20A···20H
order124455555···55···51010101010···1020···20
size15252511112···24···4555510···1025···25

65 irreducible representations

dim11111122224444
type+++-+
imageC1C2C4C5C10C20D5Dic5C5xD5C5xDic5F5C5xF5D5.D5C5xD5.D5
kernelC5xD5.D5D5xC52C53D5.D5C5xD5C52C5xD5C52D5C5C52C5C5C1
# reps112448228814416

Matrix representation of C5xD5.D5 in GL4(F41) generated by

18000
01800
00180
00018
,
10000
03700
00160
2151618
,
03700
10000
3921833
39262523
,
10000
01000
00370
1724037
,
00370
5363720
01000
301105
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[10,0,0,2,0,37,0,15,0,0,16,16,0,0,0,18],[0,10,39,39,37,0,2,26,0,0,18,25,0,0,33,23],[10,0,0,17,0,10,0,24,0,0,37,0,0,0,0,37],[0,5,0,30,0,36,10,11,37,37,0,0,0,20,0,5] >;

C5xD5.D5 in GAP, Magma, Sage, TeX

C_5\times D_5.D_5
% in TeX

G:=Group("C5xD5.D5");
// GroupNames label

G:=SmallGroup(500,42);
// by ID

G=gap.SmallGroup(500,42);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,1603,7504,1014]);
// Polycyclic

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

Export

Subgroup lattice of C5xD5.D5 in TeX

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