Copied to
clipboard

G = C5×C5⋊D4order 200 = 23·52

Direct product of C5 and C5⋊D4

direct product, metabelian, supersoluble, monomial

Aliases: C5×C5⋊D4, C10C2, C526D4, Dic5⋊C10, D102C10, C1022C2, C10.21D10, C52(C5×D4), C22⋊(C5×D5), (C2×C10)⋊1D5, (C2×C10)⋊2C10, (D5×C10)⋊4C2, C2.5(D5×C10), C10.5(C2×C10), (C5×Dic5)⋊4C2, (C5×C10).10C22, SmallGroup(200,31)

Series: Derived Chief Lower central Upper central

C1C10 — C5×C5⋊D4
C1C5C10C5×C10D5×C10 — C5×C5⋊D4
C5C10 — C5×C5⋊D4
C1C10C2×C10

Generators and relations for C5×C5⋊D4
 G = < a,b,c,d | a5=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

2C2
10C2
2C5
2C5
5C4
5C22
2D5
2C10
2C10
2C10
2C10
2C10
2C10
2C10
2C10
10C10
5D4
2C2×C10
2C2×C10
5C20
5C2×C10
2C5×C10
2C5×D5
5C5×D4

Permutation representations of C5×C5⋊D4
On 20 points - transitive group 20T53
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 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 17 7 12)(2 18 8 13)(3 19 9 14)(4 20 10 15)(5 16 6 11)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 16)(7 17)(8 18)(9 19)(10 20)

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

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

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

G:=TransitiveGroup(20,53);

C5×C5⋊D4 is a maximal subgroup of   Dic5.D10  D10.4D10  D10⋊D10  C5×D4×D5

65 conjugacy classes

class 1 2A2B2C 4 5A5B5C5D5E···5N10A10B10C10D10E···10AL10AM10AN10AO10AP20A20B20C20D
order1222455555···51010101010···101010101020202020
size112101011112···211112···21010101010101010

65 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C5C10C10C10D4D5D10C5⋊D4C5×D4C5×D5D5×C10C5×C5⋊D4
kernelC5×C5⋊D4C5×Dic5D5×C10C102C5⋊D4Dic5D10C2×C10C52C2×C10C10C5C5C22C2C1
# reps11114444122448816

Matrix representation of C5×C5⋊D4 in GL2(𝔽11) generated by

90
09
,
81
210
,
24
79
,
11
010
G:=sub<GL(2,GF(11))| [9,0,0,9],[8,2,1,10],[2,7,4,9],[1,0,1,10] >;

C5×C5⋊D4 in GAP, Magma, Sage, TeX

C_5\times C_5\rtimes D_4
% in TeX

G:=Group("C5xC5:D4");
// GroupNames label

G:=SmallGroup(200,31);
// by ID

G=gap.SmallGroup(200,31);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-5,221,4004]);
// Polycyclic

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

Export

Subgroup lattice of C5×C5⋊D4 in TeX

׿
×
𝔽