C5xC2^3:A4

direct product, non-abelian, soluble, monomial

Aliases: C₅×C₂³⋊A₄, 2₊⁽¹⁺⁴⁾⋊₂C₁₅, Q₈⋊₂(C₅×A₄), (C₅×Q₈)⋊₂A₄, C₂³⋊₂(C₅×A₄), (C₂²×C₁₀)⋊₂A₄, C₁₀.₂(C₂²⋊A₄), (C₅×2₊⁽¹⁺⁴⁾)⋊₂C₃, C₂.₂(C₅×C₂²⋊A₄), SmallGroup(480,1134)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — 2₊⁽¹⁺⁴⁾ — C₅×C₂³⋊A₄

Chief series

C₁ — C₂ — C₂³ — 2₊⁽¹⁺⁴⁾ — C₅×2₊⁽¹⁺⁴⁾ — C₅×C₂³⋊A₄

Lower central

2₊⁽¹⁺⁴⁾ — C₅×C₂³⋊A₄

Upper central

C₁ — C₁₀

Subgroups: 350 in 92 conjugacy classes, 18 normal (8 characteristic)
C₁, C₂, C₂^[×3], C₃, C₄^[×2], C₂²^[×7], C₅, C₆, C₂×C₄^[×3], D₄^[×6], Q₈^[×2], C₂³^[×3], C₂³, C₁₀, C₁₀^[×3], A₄^[×3], C₁₅, C₂×D₄^[×3], C₄○D₄^[×2], C₂₀^[×2], C₂×C₁₀^[×7], SL₂(𝔽₃)^[×2], C₂×A₄^[×3], C₃₀, 2₊⁽¹⁺⁴⁾, C₂×C₂₀^[×3], C₅×D₄^[×6], C₅×Q₈^[×2], C₂²×C₁₀^[×3], C₂²×C₁₀, C₅×A₄^[×3], D₄×C₁₀^[×3], C₅×C₄○D₄^[×2], C₂³⋊A₄, C₅×SL₂(𝔽₃)^[×2], C₁₀×A₄^[×3], C₅×2₊⁽¹⁺⁴⁾, C₅×C₂³⋊A₄

Quotients:
C₁, C₃, C₅, A₄^[×5], C₁₅, C₂²⋊A₄, C₅×A₄^[×5], C₂³⋊A₄, C₅×C₂²⋊A₄, C₅×C₂³⋊A₄

Generators and relations
G = < a,b,c,d,e,f,g | a⁵=b²=c²=d²=e²=f²=g³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, gbg^-1=bc=cb, fbf=bd=db, be=eb, ece=cd=dc, cf=fc, gcg^-1=b, de=ed, df=fd, dg=gd, geg^-1=ef=fe, gfg^-1=e >

Smallest permutation representation
►On 40 points

Generators in S₄₀

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

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

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

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

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

(6 24 28)(7 25 29)(8 21 30)(9 22 26)(10 23 27)(16 38 33)(17 39 34)(18 40 35)(19 36 31)(20 37 32)

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

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

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

Matrix representation ►G ⊆ GL₇(𝔽₆₁)

58	0	0	0	0	0	0
0	58	0	0	0	0	0
0	0	58	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

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

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

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

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

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

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

G:=sub<GL(7,GF(61))| [58,0,0,0,0,0,0,0,58,0,0,0,0,0,0,0,58,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,60,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,60,0,0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

55 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	4A	4B	5A	5B	5C	5D	6A	6B	10A	10B	10C	10D	10E	···	10P	15A	···	15H	20A	···	20H	30A	···	30H
order	1	2	2	2	2	3	3	4	4	5	5	5	5	6	6	10	10	10	10	10	···	10	15	···	15	20	···	20	30	···	30
size	1	1	6	6	6	16	16	6	6	1	1	1	1	16	16	1	1	1	1	6	···	6	16	···	16	6	···	6	16	···	16

55 irreducible representations

dim	1	1	1	1	3	3	3	3	4	4	4
type	+				+	+			+
image	C₁	C₃	C₅	C₁₅	A₄	A₄	C₅×A₄	C₅×A₄	C₂³⋊A₄	C₂³⋊A₄	C₅×C₂³⋊A₄
kernel	C₅×C₂³⋊A₄	C₅×2₊⁽¹⁺⁴⁾	C₂³⋊A₄	2₊⁽¹⁺⁴⁾	C₅×Q₈	C₂²×C₁₀	Q₈	C₂³	C₅	C₅	C₁
# reps	1	2	4	8	2	3	8	12	1	2	12

In GAP, Magma, Sage, TeX

C_5\times C_2^3\rtimes A_4

% in TeX

G:=Group("C5xC2^3:A4");

// GroupNames label

G:=SmallGroup(480,1134);

// by ID

G=gap.SmallGroup(480,1134);

# by ID

G:=PCGroup([7,-3,-5,-2,2,-2,2,-2,632,1263,4204,375,7565,1027]);

// Polycyclic

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

// generators/relations

G = C₅×C₂³⋊A₄ order 480 = 2⁵·3·5

Direct product of C₅ and C₂³⋊A₄

G = C5×C23⋊A4 order 480 = 25·3·5

Direct product of C5 and C23⋊A4

G = C₅×C₂³⋊A₄ order 480 = 2⁵·3·5

Direct product of C₅ and C₂³⋊A₄