C6xC2^2:A4

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

Aliases: C₆×C₂²⋊A₄, C₂⁵⋊₃C₃², (C₂⁴×C₆)⋊₂C₃, (C₂³×C₆)⋊₈C₆, C₂²⋊₂(C₆×A₄), C₂⁴⋊₅(C₃×C₆), C₂³⋊₄(C₃×A₄), (C₂²×C₆)⋊₃A₄, (C₂×C₆)⋊₂(C₂×A₄), SmallGroup(288,1042)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — C₆×C₂²⋊A₄

Chief series

C₁ — C₂² — C₂⁴ — C₂³×C₆ — C₃×C₂²⋊A₄ — C₆×C₂²⋊A₄

Lower central

C₂⁴ — C₆×C₂²⋊A₄

Upper central

C₁ — C₆

Subgroups: 1044 in 324 conjugacy classes, 36 normal (10 characteristic)
C₁, C₂, C₂^[×10], C₃, C₃^[×3], C₂²^[×5], C₂²^[×50], C₆, C₆^[×13], C₂³^[×5], C₂³^[×50], C₃², A₄^[×15], C₂×C₆^[×5], C₂×C₆^[×50], C₂⁴, C₂⁴^[×10], C₃×C₆, C₂×A₄^[×15], C₂²×C₆^[×5], C₂²×C₆^[×50], C₂⁵, C₃×A₄^[×5], C₂²⋊A₄^[×3], C₂³×C₆, C₂³×C₆^[×10], C₆×A₄^[×5], C₂×C₂²⋊A₄^[×3], C₂⁴×C₆, C₃×C₂²⋊A₄, C₆×C₂²⋊A₄

Quotients:
C₁, C₂, C₃^[×4], C₆^[×4], C₃², A₄^[×5], C₃×C₆, C₂×A₄^[×5], C₃×A₄^[×5], C₂²⋊A₄, C₆×A₄^[×5], C₂×C₂²⋊A₄, C₃×C₂²⋊A₄, C₆×C₂²⋊A₄

Generators and relations
G = < a,b,c,d,e,f | a⁶=b²=c²=d²=e²=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf^-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf^-1=b, fdf^-1=de=ed, fef^-1=d >

Smallest permutation representation
►On 36 points

Generators in S₃₆

(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)

(7 10)(8 11)(9 12)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)

(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 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 35)(8 36)(9 31)(10 32)(11 33)(12 34)

(7 35)(8 36)(9 31)(10 32)(11 33)(12 34)(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₇)

3	0	0	0	0	0
0	3	0	0	0	0
0	0	3	0	0	0
0	0	0	6	0	0
0	0	0	0	6	0
0	0	0	0	0	6

1	0	0	0	0	0
0	6	0	0	0	0
3	0	6	0	0	0
0	0	0	1	0	0
0	0	0	0	6	0
0	0	0	0	0	6

6	0	0	0	0	0
0	6	0	0	0	0
4	2	1	0	0	0
0	0	0	6	0	0
0	0	0	0	6	0
0	0	0	0	0	1

6	0	0	0	0	0
0	1	0	0	0	0
0	5	6	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	6	0	0	0	0
3	0	6	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	2	0	0	0	0
6	3	3	0	0	0
0	0	4	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	0	1	0	0

G:=sub<GL(6,GF(7))| [3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[1,0,3,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,0,6],[6,0,4,0,0,0,0,6,2,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,1],[6,0,0,0,0,0,0,1,5,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,3,0,0,0,0,6,0,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,6,0,0,0,0,2,3,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;

48 conjugacy classes

class	1	2A	2B	···	2K	3A	3B	3C	···	3H	6A	6B	6C	···	6V	6W	···	6AB
order	1	2	2	···	2	3	3	3	···	3	6	6	6	···	6	6	···	6
size	1	1	3	···	3	1	1	16	···	16	1	1	3	···	3	16	···	16

48 irreducible representations

dim	1	1	1	1	1	1	3	3	3	3
type	+	+					+	+
image	C₁	C₂	C₃	C₃	C₆	C₆	A₄	C₂×A₄	C₃×A₄	C₆×A₄
kernel	C₆×C₂²⋊A₄	C₃×C₂²⋊A₄	C₂×C₂²⋊A₄	C₂⁴×C₆	C₂²⋊A₄	C₂³×C₆	C₂²×C₆	C₂×C₆	C₂³	C₂²
# reps	1	1	6	2	6	2	5	5	10	10

In GAP, Magma, Sage, TeX

C_6\times C_2^2\rtimes A_4

% in TeX

G:=Group("C6xC2^2:A4");

// GroupNames label

G:=SmallGroup(288,1042);

// by ID

G=gap.SmallGroup(288,1042);

# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,514,956,3036,5305]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=b^2=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,f*c*f^-1=b,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;

// generators/relations

G = C₆×C₂²⋊A₄ order 288 = 2⁵·3²

Direct product of C₆ and C₂²⋊A₄

G = C6×C22⋊A4 order 288 = 25·32

Direct product of C6 and C22⋊A4

G = C₆×C₂²⋊A₄ order 288 = 2⁵·3²

Direct product of C₆ and C₂²⋊A₄