C5xC2^2:S4

direct product, non-abelian, soluble, monomial

Aliases: C₅×C₂²⋊S₄, C₂²⋊(C₅×S₄), (C₂×C₁₀)⋊₁S₄, C₂⁴⋊₃(C₅×S₃), C₂²⋊A₄⋊₃C₁₀, (C₂³×C₁₀)⋊₃S₃, (C₅×C₂²⋊A₄)⋊₅C₂, SmallGroup(480,1200)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₂²⋊A₄ — C₅×C₂²⋊S₄

Chief series

C₁ — C₂² — C₂⁴ — C₂²⋊A₄ — C₅×C₂²⋊A₄ — C₅×C₂²⋊S₄

Lower central

C₂²⋊A₄ — C₅×C₂²⋊S₄

Upper central

C₁ — C₅

Subgroups: 500 in 112 conjugacy classes, 14 normal (8 characteristic)
C₁, C₂^[×5], C₃, C₄^[×3], C₂²^[×3], C₂²^[×11], C₅, S₃, C₂×C₄^[×3], D₄^[×6], C₂³^[×5], C₁₀^[×5], A₄^[×4], C₁₅, C₂²⋊C₄^[×3], C₂×D₄^[×3], C₂⁴, C₂₀^[×3], C₂×C₁₀^[×3], C₂×C₁₀^[×11], S₄^[×3], C₅×S₃, C₂²≀C₂, C₂×C₂₀^[×3], C₅×D₄^[×6], C₂²×C₁₀^[×5], C₂²⋊A₄, C₅×A₄^[×4], C₅×C₂²⋊C₄^[×3], D₄×C₁₀^[×3], C₂³×C₁₀, C₂²⋊S₄, C₅×S₄^[×3], C₅×C₂²≀C₂, C₅×C₂²⋊A₄, C₅×C₂²⋊S₄

Quotients:
C₁, C₂, C₅, S₃, C₁₀, S₄^[×3], C₅×S₃, C₂²⋊S₄, C₅×S₄^[×3], C₅×C₂²⋊S₄

Generators and relations
G = < a,b,c,d,e,f,g | a⁵=b²=c²=d²=e²=f³=g²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fcf^-1=bc=cb, bd=db, be=eb, fbf^-1=gbg=c, cd=dc, ce=ec, gcg=b, fdf^-1=gdg=de=ed, fef^-1=d, eg=ge, gfg=f^-1 >

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

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

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

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

(6 14 20)(7 15 16)(8 11 17)(9 12 18)(10 13 19)(21 35 27)(22 31 28)(23 32 29)(24 33 30)(25 34 26)

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₆₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	34	0	0
0	0	0	0	34	0
0	0	0	0	0	34

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

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

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

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

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

0	1	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	60	0	0
0	0	0	0	0	60

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,0,0,0,0,0,0,34,0,0,0,0,0,0,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,1,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,0,1,0,0,0,60,1,0],[60,60,60,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,1,60,0],[0,1,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,0,0,0,1,60,0,0,0,1,0,60,0,0,0,0,0,60],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,60] >;

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	3	4A	4B	4C	5A	5B	5C	5D	10A	···	10L	10M	10N	10O	10P	10Q	10R	10S	10T	15A	15B	15C	15D	20A	···	20L
order	1	2	2	2	2	2	3	4	4	4	5	5	5	5	10	···	10	10	10	10	10	10	10	10	10	15	15	15	15	20	···	20
size	1	3	3	3	6	12	32	12	12	12	1	1	1	1	3	···	3	6	6	6	6	12	12	12	12	32	32	32	32	12	···	12

50 irreducible representations

dim	1	1	1	1	2	2	3	3	6	6
type	+	+			+		+		+
image	C₁	C₂	C₅	C₁₀	S₃	C₅×S₃	S₄	C₅×S₄	C₂²⋊S₄	C₅×C₂²⋊S₄
kernel	C₅×C₂²⋊S₄	C₅×C₂²⋊A₄	C₂²⋊S₄	C₂²⋊A₄	C₂³×C₁₀	C₂⁴	C₂×C₁₀	C₂²	C₅	C₁
# reps	1	1	4	4	1	4	6	24	1	4

In GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes S_4

% in TeX

G:=Group("C5xC2^2:S4");

// GroupNames label

G:=SmallGroup(480,1200);

// by ID

G=gap.SmallGroup(480,1200);

# by ID

G:=PCGroup([7,-2,-5,-3,-2,2,-2,2,422,1683,185,1054,333,10085,1531,5886,608]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^5=b^2=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,f*c*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=g*b*g=c,c*d=d*c,c*e=e*c,g*c*g=b,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 = C₅×C₂²⋊S₄ order 480 = 2⁵·3·5

Direct product of C₅ and C₂²⋊S₄

G = C5×C22⋊S4 order 480 = 25·3·5

Direct product of C5 and C22⋊S4

G = C₅×C₂²⋊S₄ order 480 = 2⁵·3·5

Direct product of C₅ and C₂²⋊S₄