C5xC2^4:C6 - GroupNames

G = C₅×C₂⁴⋊C₆ order 480 = 2⁵·3·5

Direct product of C₅ and C₂⁴⋊C₆

direct product, metabelian, soluble, monomial

Aliases: C₅×C₂⁴⋊C₆, C₂⁴⋊₂C₃₀, C₂²≀C₂⋊C₁₅, C₂³⋊₁(C₅×A₄), C₂²⋊A₄⋊₂C₁₀, (C₂²×C₁₀)⋊₁A₄, (C₂³×C₁₀)⋊₁C₆, C₂².₂(C₁₀×A₄), (C₅×C₂²≀C₂)⋊C₃, (C₅×C₂²⋊A₄)⋊₁C₂, (C₂×C₁₀).₆(C₂×A₄), SmallGroup(480,656)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₅×C₂⁴⋊C₆

Chief series

C₁ — C₂² — C₂⁴ — C₂³×C₁₀ — C₅×C₂²⋊A₄ — C₅×C₂⁴⋊C₆

Lower central

C₂⁴ — C₅×C₂⁴⋊C₆

Upper central

C₁ — C₅

Generators and relations for C₅×C₂⁴⋊C₆
G = < a,b,c,d,e,f | a⁵=b²=c²=d²=e²=f⁶=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=ec=ce, cd=dc, fcf^-1=bcde, fef^-1=de=ed, fdf^-1=e >

Subgroups: 328 in 70 conjugacy classes, 14 normal (all characteristic)
C₁, C₂, C₃, C₄, C₂², C₂², C₅, C₆, C₂×C₄, D₄, C₂³, C₂³, C₁₀, A₄, C₁₅, C₂²⋊C₄, C₂×D₄, C₂⁴, C₂₀, C₂×C₁₀, C₂×C₁₀, C₂×A₄, C₃₀, C₂²≀C₂, C₂×C₂₀, C₅×D₄, C₂²×C₁₀, C₂²×C₁₀, C₂²⋊A₄, C₅×A₄, C₅×C₂²⋊C₄, D₄×C₁₀, C₂³×C₁₀, C₂⁴⋊C₆, C₁₀×A₄, C₅×C₂²≀C₂, C₅×C₂²⋊A₄, C₅×C₂⁴⋊C₆
Quotients: C₁, C₂, C₃, C₅, C₆, C₁₀, A₄, C₁₅, C₂×A₄, C₃₀, C₅×A₄, C₂⁴⋊C₆, C₁₀×A₄, C₅×C₂⁴⋊C₆

Smallest permutation representation of C₅×C₂⁴⋊C₆
►On 40 points

Generators in S₄₀

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

(1 35)(4 27)(5 13)(8 31)(9 18)(11 15)(20 22)(23 25)(29 33)(37 39)

(1 37)(4 23)(5 15)(8 33)(9 20)(11 13)(18 22)(25 27)(29 31)(35 39)

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

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

(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)

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

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

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

50 conjugacy classes

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

50 irreducible representations

dim	1	1	1	1	1	1	1	1	3	3	3	3	6	6
type	+	+							+	+			+
image	C₁	C₂	C₃	C₅	C₆	C₁₀	C₁₅	C₃₀	A₄	C₂×A₄	C₅×A₄	C₁₀×A₄	C₂⁴⋊C₆	C₅×C₂⁴⋊C₆
kernel	C₅×C₂⁴⋊C₆	C₅×C₂²⋊A₄	C₅×C₂²≀C₂	C₂⁴⋊C₆	C₂³×C₁₀	C₂²⋊A₄	C₂²≀C₂	C₂⁴	C₂²×C₁₀	C₂×C₁₀	C₂³	C₂²	C₅	C₁
# reps	1	1	2	4	2	4	8	8	1	1	4	4	2	8

Matrix representation of C₅×C₂⁴⋊C₆ ►in GL₆(𝔽₆₁)

20	0	0	0	0	0
0	20	0	0	0	0
0	0	20	0	0	0
0	0	0	20	0	0
0	0	0	0	20	0
0	0	0	0	0	20

0	1	60	0	0	0
1	0	60	0	0	0
0	0	60	0	0	0
0	0	8	1	0	0
0	0	8	0	1	0
0	0	8	0	0	1

60	0	0	0	0	0
60	0	1	0	0	0
60	1	0	0	0	0
8	0	0	1	0	0
8	0	0	0	1	0
8	0	0	0	0	1

0	1	60	0	0	0
1	0	60	0	0	0
0	0	60	0	0	0
53	53	0	60	60	60
0	0	8	0	0	1
0	0	8	0	1	0

60	0	0	0	0	0
60	0	1	0	0	0
60	1	0	0	0	0
8	0	0	0	1	0
8	0	0	1	0	0
0	53	53	60	60	60

0	0	0	60	0	1
53	53	53	59	60	60
0	0	0	60	1	0
31	31	31	8	0	0
31	32	31	8	0	0
31	31	32	8	0	0

G:=sub<GL(6,GF(61))| [20,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20],[0,1,0,0,0,0,1,0,0,0,0,0,60,60,60,8,8,8,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,60,60,8,8,8,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,53,0,0,1,0,0,53,0,0,60,60,60,0,8,8,0,0,0,60,0,0,0,0,0,60,0,1,0,0,0,60,1,0],[60,60,60,8,8,0,0,0,1,0,0,53,0,1,0,0,0,53,0,0,0,0,1,60,0,0,0,1,0,60,0,0,0,0,0,60],[0,53,0,31,31,31,0,53,0,31,32,31,0,53,0,31,31,32,60,59,60,8,8,8,0,60,1,0,0,0,1,60,0,0,0,0] >;

C₅×C₂⁴⋊C₆ in GAP, Magma, Sage, TeX

C_5\times C_2^4\rtimes C_6

% in TeX

G:=Group("C5xC2^4:C6");

// GroupNames label

G:=SmallGroup(480,656);

// by ID

G=gap.SmallGroup(480,656);

# by ID

G:=PCGroup([7,-2,-3,-5,-2,2,-2,2,4203,850,10504,1586,5052,8833]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^2=f^6=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,f*b*f^-1=e*c=c*e,c*d=d*c,f*c*f^-1=b*c*d*e,f*e*f^-1=d*e=e*d,f*d*f^-1=e>;

// generators/relations

G = C5×C24⋊C6 order 480 = 25·3·5

Direct product of C5 and C24⋊C6

G = C₅×C₂⁴⋊C₆ order 480 = 2⁵·3·5

Direct product of C₅ and C₂⁴⋊C₆