C5xC2^2:S4 - GroupNames

G = C₅×C₂²⋊S₄ order 480 = 2⁵·3·5

Direct product of C₅ and C₂²⋊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

Generators and relations for C₅×C₂²⋊S₄
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 >

Subgroups: 500 in 112 conjugacy classes, 14 normal (8 characteristic)
C₁, C₂, C₃, C₄, C₂², C₂², C₅, S₃, C₂×C₄, D₄, C₂³, C₁₀, A₄, C₁₅, C₂²⋊C₄, C₂×D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, S₄, C₅×S₃, C₂²≀C₂, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²⋊A₄, C₅×A₄, C₅×C₂²⋊C₄, D₄×C₁₀, C₂³×C₁₀, C₂²⋊S₄, C₅×S₄, C₅×C₂²≀C₂, C₅×C₂²⋊A₄, C₅×C₂²⋊S₄
Quotients: C₁, C₂, C₅, S₃, C₁₀, S₄, C₅×S₃, C₂²⋊S₄, C₅×S₄, C₅×C₂²⋊S₄

Smallest permutation representation of C₅×C₂²⋊S₄
►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 30)(2 26)(3 27)(4 28)(5 29)(6 11)(7 12)(8 13)(9 14)(10 15)(16 36)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 31)(25 32)

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

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

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

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

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

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

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

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

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

Matrix representation of C₅×C₂²⋊S₄ ►in 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] >;

C₅×C₂²⋊S₄ 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