C2^2xD5^2 - GroupNames

G = C₂²×D₅² order 400 = 2⁴·5²

Direct product of C₂², D₅ and D₅

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

Aliases: C₂²×D₅², C₅²⋊C₂⁴, C₁₀²⋊₅C₂², C₅⋊D₅⋊C₂³, (C₅×C₁₀)⋊C₂³, (C₅×D₅)⋊C₂³, (C₂×C₁₀)⋊₅D₁₀, C₅⋊₁(C₂³×D₅), C₁₀⋊₁(C₂²×D₅), (D₅×C₁₀)⋊₁₀C₂², (D₅×C₂×C₁₀)⋊₇C₂, (C₂×C₅⋊D₅)⋊₈C₂², (C₂²×C₅⋊D₅)⋊₆C₂, SmallGroup(400,218)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅² — C₂²×D₅²

Chief series

C₁ — C₅ — C₅² — C₅×D₅ — D₅² — C₂×D₅² — C₂²×D₅²

Lower central

C₅² — C₂²×D₅²

Upper central

C₁ — C₂²

Generators and relations for C₂²×D₅²
G = < a,b,c,d,e,f | a²=b²=c⁵=d²=e⁵=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1964 in 300 conjugacy classes, 104 normal (6 characteristic)
C₁, C₂, C₂, C₂², C₂², C₅, C₅, C₂³, D₅, D₅, C₁₀, C₁₀, C₂⁴, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₅², C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₅×D₅, C₅⋊D₅, C₅×C₁₀, C₂³×D₅, D₅², D₅×C₁₀, C₂×C₅⋊D₅, C₁₀², C₂×D₅², D₅×C₂×C₁₀, C₂²×C₅⋊D₅, C₂²×D₅²
Quotients: C₁, C₂, C₂², C₂³, D₅, C₂⁴, D₁₀, C₂²×D₅, C₂³×D₅, D₅², C₂×D₅², C₂²×D₅²

Smallest permutation representation of C₂²×D₅²
►On 40 points

Generators in S₄₀

(1 11)(2 12)(3 13)(4 14)(5 15)(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)

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

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

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

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

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

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

64 conjugacy classes

class	1	2A	2B	2C	2D	···	2K	2L	2M	2N	2O	5A	5B	5C	5D	5E	5F	5G	5H	10A	···	10L	10M	···	10X	10Y	···	10AN
order	1	2	2	2	2	···	2	2	2	2	2	5	5	5	5	5	5	5	5	10	···	10	10	···	10	10	···	10
size	1	1	1	1	5	···	5	25	25	25	25	2	2	2	2	4	4	4	4	2	···	2	4	···	4	10	···	10

64 irreducible representations

dim	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	D₅	D₁₀	D₁₀	D₅²	C₂×D₅²
kernel	C₂²×D₅²	C₂×D₅²	D₅×C₂×C₁₀	C₂²×C₅⋊D₅	C₂²×D₅	D₁₀	C₂×C₁₀	C₂²	C₂
# reps	1	12	2	1	4	24	4	4	12

Matrix representation of C₂²×D₅² ►in GL₆(𝔽₁₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	10	0	0	0
0	0	0	10	0	0
0	0	0	0	1	0
0	0	0	0	0	1

10	0	0	0	0	0
0	10	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

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

0	10	0	0	0	0
1	7	0	0	0	0
0	0	0	10	0	0
0	0	1	3	0	0
0	0	0	0	1	0
0	0	0	0	0	1

10	4	0	0	0	0
0	1	0	0	0	0
0	0	10	8	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(11))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,10,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,3,10],[0,1,0,0,0,0,10,7,0,0,0,0,0,0,0,1,0,0,0,0,10,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,0,0,0,0,0,4,1,0,0,0,0,0,0,10,0,0,0,0,0,8,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂²×D₅² in GAP, Magma, Sage, TeX

C_2^2\times D_5^2

% in TeX

G:=Group("C2^2xD5^2");

// GroupNames label

G:=SmallGroup(400,218);

// by ID

G=gap.SmallGroup(400,218);

# by ID

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

// Polycyclic

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

// generators/relations

G = C22×D52 order 400 = 24·52

Direct product of C22, D5 and D5

G = C₂²×D₅² order 400 = 2⁴·5²

Direct product of C₂², D₅ and D₅