C6xC2^4:C5 - GroupNames

G = C₆×C₂⁴⋊C₅ order 480 = 2⁵·3·5

Direct product of C₆ and C₂⁴⋊C₅

direct product, metabelian, soluble, monomial, A-group

Aliases: C₆×C₂⁴⋊C₅, C₂⁵⋊₂C₁₅, C₂⁴⋊₄C₃₀, (C₂⁴×C₆)⋊C₅, (C₂³×C₆)⋊₂C₁₀, SmallGroup(480,1204)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₆×C₂⁴⋊C₅

Chief series

C₁ — C₂⁴ — C₂⁴⋊C₅ — C₃×C₂⁴⋊C₅ — C₆×C₂⁴⋊C₅

Lower central

C₂⁴ — C₆×C₂⁴⋊C₅

Upper central

C₁ — C₆

Generators and relations for C₆×C₂⁴⋊C₅
G = < a,b,c,d,e,f | a⁶=b²=c²=d²=e²=f⁵=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=bcd, cd=dc, ce=ec, fcf^-1=cde, fdf^-1=de=ed, fef^-1=b >

Subgroups: 816 in 164 conjugacy classes, 12 normal (all characteristic)
C₁, C₂, C₂, C₃, C₂², C₅, C₆, C₆, C₂³, C₁₀, C₂×C₆, C₁₅, C₂⁴, C₂⁴, C₂²×C₆, C₃₀, C₂⁵, C₂³×C₆, C₂³×C₆, C₂⁴⋊C₅, C₂⁴×C₆, C₂×C₂⁴⋊C₅, C₃×C₂⁴⋊C₅, C₆×C₂⁴⋊C₅
Quotients: C₁, C₂, C₃, C₅, C₆, C₁₀, C₁₅, C₃₀, C₂⁴⋊C₅, C₂×C₂⁴⋊C₅, C₃×C₂⁴⋊C₅, C₆×C₂⁴⋊C₅

Permutation representations of C₆×C₂⁴⋊C₅
►On 30 points - transitive group 30T105

Generators in S₃₀

(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 4)(2 5)(3 6)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)

(1 4)(2 5)(3 6)(19 22)(20 23)(21 24)

(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)

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

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

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

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

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

G:=TransitiveGroup(30,105);

48 conjugacy classes

class	1	2A	2B	···	2G	3A	3B	5A	5B	5C	5D	6A	6B	6C	···	6N	10A	10B	10C	10D	15A	···	15H	30A	···	30H
order	1	2	2	···	2	3	3	5	5	5	5	6	6	6	···	6	10	10	10	10	15	···	15	30	···	30
size	1	1	5	···	5	1	1	16	16	16	16	1	1	5	···	5	16	16	16	16	16	···	16	16	···	16

48 irreducible representations

dim	1	1	1	1	1	1	1	1	5	5	5	5
type	+	+							+	+
image	C₁	C₂	C₃	C₅	C₆	C₁₀	C₁₅	C₃₀	C₂⁴⋊C₅	C₂×C₂⁴⋊C₅	C₃×C₂⁴⋊C₅	C₆×C₂⁴⋊C₅
kernel	C₆×C₂⁴⋊C₅	C₃×C₂⁴⋊C₅	C₂×C₂⁴⋊C₅	C₂⁴×C₆	C₂⁴⋊C₅	C₂³×C₆	C₂⁵	C₂⁴	C₆	C₃	C₂	C₁
# reps	1	1	2	4	2	4	8	8	3	3	6	6

Matrix representation of C₆×C₂⁴⋊C₅ ►in GL₅(𝔽₃₁)

26	0	0	0	0
0	26	0	0	0
0	0	26	0	0
0	0	0	26	0
0	0	0	0	26

30	0	0	0	0
0	30	0	0	0
0	0	30	0	0
0	0	0	30	0
0	0	0	0	1

30	0	0	0	0
0	30	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	30	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	30

30	0	0	0	0
0	30	0	0	0
0	0	30	0	0
0	0	0	1	0
0	0	0	0	30

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

G:=sub<GL(5,GF(31))| [26,0,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,0,26],[30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,1],[30,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,30],[30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,30],[0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C₆×C₂⁴⋊C₅ in GAP, Magma, Sage, TeX

C_6\times C_2^4\rtimes C_5

% in TeX

G:=Group("C6xC2^4:C5");

// GroupNames label

G:=SmallGroup(480,1204);

// by ID

G=gap.SmallGroup(480,1204);

# by ID

G:=PCGroup([7,-2,-3,-5,-2,2,2,2,1137,1593,2329,3695]);

// Polycyclic

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

// generators/relations

G = C6×C24⋊C5 order 480 = 25·3·5

Direct product of C6 and C24⋊C5

G = C₆×C₂⁴⋊C₅ order 480 = 2⁵·3·5

Direct product of C₆ and C₂⁴⋊C₅