C5xD5^2 - GroupNames

G = C₅×D₅² order 500 = 2²·5³

Direct product of C₅, D₅ and D₅

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

Aliases: C₅×D₅², C₅²⋊₆D₁₀, C₅³⋊₁C₂², (C₅×D₅)⋊C₁₀, C₅⋊D₅⋊₂C₁₀, C₅⋊₁(D₅×C₁₀), C₅²⋊₃(C₂×C₁₀), (D₅×C₅²)⋊₁C₂, (C₅×C₅⋊D₅)⋊₁C₂, SmallGroup(500,50)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅² — C₅×D₅²

Chief series

C₁ — C₅ — C₅² — C₅³ — D₅×C₅² — C₅×D₅²

Lower central

C₅² — C₅×D₅²

Upper central

C₁ — C₅

Generators and relations for C₅×D₅²
G = < a,b,c,d,e | a⁵=b⁵=c²=d⁵=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 344 in 72 conjugacy classes, 20 normal (8 characteristic)
C₁, C₂, C₂², C₅, C₅, C₅, D₅, D₅, C₁₀, D₁₀, C₂×C₁₀, C₅², C₅², C₅², C₅×D₅, C₅×D₅, C₅⋊D₅, C₅×C₁₀, D₅², D₅×C₁₀, C₅³, D₅×C₅², C₅×C₅⋊D₅, C₅×D₅²
Quotients: C₁, C₂, C₂², C₅, D₅, C₁₀, D₁₀, C₂×C₁₀, C₅×D₅, D₅², D₅×C₁₀, C₅×D₅²

Permutation representations of C₅×D₅²
►On 20 points - transitive group 20T124

Generators in S₂₀

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

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

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

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

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

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

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

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

G:=TransitiveGroup(20,124);

80 conjugacy classes

class	1	2A	2B	2C	5A	5B	5C	5D	5E	···	5X	5Y	···	5AR	10A	···	10H	10I	···	10AB	10AC	10AD	10AE	10AF
order	1	2	2	2	5	5	5	5	5	···	5	5	···	5	10	···	10	10	···	10	10	10	10	10
size	1	5	5	25	1	1	1	1	2	···	2	4	···	4	5	···	5	10	···	10	25	25	25	25

80 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+				+	+			+
image	C₁	C₂	C₂	C₅	C₁₀	C₁₀	D₅	D₁₀	C₅×D₅	D₅×C₁₀	D₅²	C₅×D₅²
kernel	C₅×D₅²	D₅×C₅²	C₅×C₅⋊D₅	D₅²	C₅×D₅	C₅⋊D₅	C₅×D₅	C₅²	D₅	C₅	C₅	C₁
# reps	1	2	1	4	8	4	4	4	16	16	4	16

Matrix representation of C₅×D₅² ►in GL₄(𝔽₁₁) generated by

9	0	0	0
0	9	0	0
0	0	1	0
0	0	0	1

1	0	0	0
0	1	0	0
0	0	3	10
0	0	1	0

1	0	0	0
0	1	0	0
0	0	3	10
0	0	8	8

0	10	0	0
1	3	0	0
0	0	1	0
0	0	0	1

1	3	0	0
0	10	0	0
0	0	1	0
0	0	0	1

G:=sub<GL(4,GF(11))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,1,0,0,10,0],[1,0,0,0,0,1,0,0,0,0,3,8,0,0,10,8],[0,1,0,0,10,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,3,10,0,0,0,0,1,0,0,0,0,1] >;

C₅×D₅² in GAP, Magma, Sage, TeX

C_5\times D_5^2

% in TeX

G:=Group("C5xD5^2");

// GroupNames label

G:=SmallGroup(500,50);

// by ID

G=gap.SmallGroup(500,50);

# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,808,10004]);

// Polycyclic

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

// generators/relations

G = C5×D52 order 500 = 22·53

Direct product of C5, D5 and D5

G = C₅×D₅² order 500 = 2²·5³

Direct product of C₅, D₅ and D₅