C2xD5xS4 - GroupNames

G = C₂×D₅×S₄ order 480 = 2⁵·3·5

Direct product of C₂, D₅ and S₄

direct product, non-abelian, soluble, monomial

Aliases: C₂×D₅×S₄, C₅⋊S₄⋊C₂², C₂³⋊(S₃×D₅), C₁₀⋊₁(C₂×S₄), (C₅×S₄)⋊C₂², (C₅×A₄)⋊C₂³, C₅⋊₁(C₂²×S₄), (D₅×A₄)⋊C₂², (C₂²×D₅)⋊D₆, (C₂²×C₁₀)⋊D₆, (C₁₀×S₄)⋊₃C₂, (C₂×A₄)⋊₁D₁₀, (C₁₀×A₄)⋊C₂², A₄⋊₁(C₂²×D₅), (C₂³×D₅)⋊₃S₃, C₂²⋊(C₂×S₃×D₅), (C₂×C₅⋊S₄)⋊₅C₂, (C₂×D₅×A₄)⋊₃C₂, (C₂×C₁₀)⋊(C₂²×S₃), SmallGroup(480,1193)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₅×A₄ — C₂×D₅×S₄

Chief series

C₁ — C₂² — C₂×C₁₀ — C₅×A₄ — D₅×A₄ — D₅×S₄ — C₂×D₅×S₄

Lower central

C₅×A₄ — C₂×D₅×S₄

Upper central

C₁ — C₂

Generators and relations for C₂×D₅×S₄
G = < a,b,c,d,e,f,g | a²=b⁵=c²=d²=e²=f³=g²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, cbc=b^-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf^-1=gdg=de=ed, fef^-1=d, eg=ge, gfg=f^-1 >

Subgroups: 2128 in 262 conjugacy classes, 35 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₆, C₂×C₄, D₄, C₂³, C₂³, D₅, D₅, C₁₀, C₁₀, A₄, D₆, C₂×C₆, C₁₅, C₂²×C₄, C₂×D₄, C₂⁴, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, S₄, S₄, C₂×A₄, C₂×A₄, C₂²×S₃, C₅×S₃, C₃×D₅, D₁₅, C₃₀, C₂²×D₄, C₄×D₅, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, C₂×S₄, C₂×S₄, C₂²×A₄, S₃×D₅, C₅×A₄, C₆×D₅, S₃×C₁₀, D₃₀, C₂×C₄×D₅, C₂×D₂₀, D₄×D₅, C₂×C₅⋊D₄, D₄×C₁₀, C₂³×D₅, C₂³×D₅, C₂²×S₄, C₅×S₄, C₅⋊S₄, D₅×A₄, C₂×S₃×D₅, C₁₀×A₄, C₂×D₄×D₅, D₅×S₄, C₁₀×S₄, C₂×C₅⋊S₄, C₂×D₅×A₄, C₂×D₅×S₄
Quotients: C₁, C₂, C₂², S₃, C₂³, D₅, D₆, D₁₀, S₄, C₂²×S₃, C₂²×D₅, C₂×S₄, S₃×D₅, C₂²×S₄, C₂×S₃×D₅, D₅×S₄, C₂×D₅×S₄

Permutation representations of C₂×D₅×S₄
►On 30 points - transitive group 30T109

Generators in S₃₀

(1 13)(2 14)(3 15)(4 11)(5 12)(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)

(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)

(1 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 17)(18 20)(21 24)(22 23)(26 27)(28 30)

(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)

(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)

(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)

(6 21)(7 22)(8 23)(9 24)(10 25)(16 26)(17 27)(18 28)(19 29)(20 30)

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

G:=Group( (1,13)(2,14)(3,15)(4,11)(5,12)(6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,17)(18,20)(21,24)(22,23)(26,27)(28,30), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (6,21)(7,22)(8,23)(9,24)(10,25)(16,26)(17,27)(18,28)(19,29)(20,30) );

G=PermutationGroup([[(1,13),(2,14),(3,15),(4,11),(5,12),(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(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)], [(1,5),(2,4),(6,9),(7,8),(11,14),(12,13),(16,17),(18,20),(21,24),(22,23),(26,27),(28,30)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(6,21),(7,22),(8,23),(9,24),(10,25),(16,26),(17,27),(18,28),(19,29),(20,30)]])

G:=TransitiveGroup(30,109);

40 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	3	4A	4B	4C	4D	5A	5B	6A	6B	6C	10A	10B	10C	10D	10E	10F	10G	10H	10I	10J	15A	15B	20A	20B	20C	20D	30A	30B
order	1	2	2	2	2	2	2	2	2	2	2	2	3	4	4	4	4	5	5	6	6	6	10	10	10	10	10	10	10	10	10	10	15	15	20	20	20	20	30	30
size	1	1	3	3	5	5	6	6	15	15	30	30	8	6	6	30	30	2	2	8	40	40	2	2	6	6	6	6	12	12	12	12	16	16	12	12	12	12	16	16

40 irreducible representations

dim

type

image

C₁

C₂

S₃

D₅

D₆

D₁₀

S₄

C₂×S₄

S₃×D₅

C₂×S₃×D₅

D₅×S₄

C₂×D₅×S₄

kernel

C₂×D₅×S₄

D₅×S₄

C₁₀×S₄

C₂×C₅⋊S₄

C₂×D₅×A₄

C₂³×D₅

C₂×S₄

C₂²×D₅

C₂²×C₁₀

S₄

C₂×A₄

D₁₀

D₅

C₁₀

C₂³

C₂²

C₂

C₁

# reps

Matrix representation of C₂×D₅×S₄ ►in GL₅(𝔽₆₁)

60	0	0	0	0
0	60	0	0	0
0	0	60	0	0
0	0	0	60	0
0	0	0	0	60

0	1	0	0	0
60	17	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	60	0	0	0
60	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	60	0	0
0	0	0	60	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	60	0
0	0	0	0	60

1	0	0	0	0
0	1	0	0	0
0	0	0	0	1
0	0	1	0	0
0	0	0	1	0

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[0,60,0,0,0,1,17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,60,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₂×D₅×S₄ in GAP, Magma, Sage, TeX

C_2\times D_5\times S_4

% in TeX

G:=Group("C2xD5xS4");

// GroupNames label

G:=SmallGroup(480,1193);

// by ID

G=gap.SmallGroup(480,1193);

# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,234,3364,5052,1286,2953,2232]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^5=c^2=d^2=e^2=f^3=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=f^-1>;

// generators/relations

G = C2×D5×S4 order 480 = 25·3·5

Direct product of C2, D5 and S4

G = C₂×D₅×S₄ order 480 = 2⁵·3·5

Direct product of C₂, D₅ and S₄