A4xC3:D4 - GroupNames

G = A₄×C₃⋊D₄ order 288 = 2⁵·3²

Direct product of A₄ and C₃⋊D₄

direct product, metabelian, soluble, monomial

Aliases: A₄×C₃⋊D₄, C₃⋊₂(D₄×A₄), D₆⋊₂(C₂×A₄), (C₃×A₄)⋊₉D₄, Dic₃⋊(C₂×A₄), C₂²⋊₃(S₃×A₄), (C₂³×C₆)⋊₄C₆, C₂⁴⋊₇(C₃×S₃), (S₃×C₂³)⋊₂C₆, (C₂²×A₄)⋊₃S₃, (Dic₃×A₄)⋊₄C₂, (C₂×A₄).₁₇D₆, (C₂²×Dic₃)⋊C₆, C₆.₁₁(C₂²×A₄), C₂³.₂₀(S₃×C₆), (C₆×A₄).₂₂C₂², (C₂×S₃×A₄)⋊₄C₂, (A₄×C₂×C₆)⋊₄C₂, (C₂×C₆)⋊₄(C₂×A₄), (C₂×C₆)⋊₂(C₃×D₄), C₂.₁₁(C₂×S₃×A₄), (C₂²×C₃⋊D₄)⋊C₃, C₂²⋊₂(C₃×C₃⋊D₄), (C₂²×C₆).₅(C₂×C₆), SmallGroup(288,928)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₆ — A₄×C₃⋊D₄

Chief series

C₁ — C₃ — C₂×C₆ — C₂²×C₆ — C₆×A₄ — C₂×S₃×A₄ — A₄×C₃⋊D₄

Lower central

C₂×C₆ — C₂²×C₆ — A₄×C₃⋊D₄

Upper central

C₁ — C₂ — C₂²

Generators and relations for A₄×C₃⋊D₄
G = < a,b,c,d,e,f | a²=b²=c³=d³=e⁴=f²=1, cac^-1=ab=ba, ad=da, ae=ea, af=fa, cbc^-1=a, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, ede^-1=fdf=d^-1, fef=e^-1 >

Subgroups: 698 in 142 conjugacy classes, 27 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₃², Dic₃, Dic₃, C₁₂, A₄, A₄, D₆, D₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂⁴, C₂⁴, C₃×S₃, C₃×C₆, C₂×Dic₃, C₃⋊D₄, C₃⋊D₄, C₃×D₄, C₂×A₄, C₂×A₄, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×D₄, C₃×Dic₃, C₃×A₄, S₃×C₆, C₆², C₄×A₄, C₂²×Dic₃, C₂×C₃⋊D₄, C₂²×A₄, C₂²×A₄, S₃×C₂³, C₂³×C₆, C₃×C₃⋊D₄, S₃×A₄, C₆×A₄, C₆×A₄, D₄×A₄, C₂²×C₃⋊D₄, Dic₃×A₄, C₂×S₃×A₄, A₄×C₂×C₆, A₄×C₃⋊D₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, A₄, D₆, C₂×C₆, C₃×S₃, C₃⋊D₄, C₃×D₄, C₂×A₄, S₃×C₆, C₂²×A₄, C₃×C₃⋊D₄, S₃×A₄, D₄×A₄, C₂×S₃×A₄, A₄×C₃⋊D₄

Smallest permutation representation of A₄×C₃⋊D₄
►On 36 points

Generators in S₃₆

(1 3)(2 4)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(33 35)(34 36)

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

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

(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)

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	3E	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	···	6Q	6R	6S	12A	12B
order	1	2	2	2	2	2	2	2	3	3	3	3	3	4	4	6	6	6	6	6	6	6	6	6	6	···	6	6	6	12	12
size	1	1	2	3	3	6	6	18	2	4	4	8	8	6	18	2	2	2	4	4	6	6	6	6	8	···	8	24	24	24	24

36 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	3	6	6	6	6
type	+	+	+	+					+	+	+						+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	D₄	D₆	C₃×S₃	C₃⋊D₄	C₃×D₄	S₃×C₆	C₃×C₃⋊D₄	A₄	C₂×A₄	C₂×A₄	C₂×A₄	S₃×A₄	D₄×A₄	C₂×S₃×A₄	A₄×C₃⋊D₄
kernel	A₄×C₃⋊D₄	Dic₃×A₄	C₂×S₃×A₄	A₄×C₂×C₆	C₂²×C₃⋊D₄	C₂²×Dic₃	S₃×C₂³	C₂³×C₆	C₂²×A₄	C₃×A₄	C₂×A₄	C₂⁴	A₄	C₂×C₆	C₂³	C₂²	C₃⋊D₄	Dic₃	D₆	C₂×C₆	C₂²	C₃	C₂	C₁
# reps	1	1	1	1	2	2	2	2	1	1	1	2	2	2	2	4	1	1	1	1	1	1	1	2

Matrix representation of A₄×C₃⋊D₄ ►in GL₅(𝔽₁₃)

1	0	0	0	0
0	1	0	0	0
0	0	0	12	1
0	0	0	12	0
0	0	1	12	0

1	0	0	0	0
0	1	0	0	0
0	0	12	0	0
0	0	12	0	1
0	0	12	1	0

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

9	0	0	0	0
0	3	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	12	0	0	0
1	0	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

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

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,12,12,12,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,12,12,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[9,0,0,0,0,0,3,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1] >;

A₄×C₃⋊D₄ in GAP, Magma, Sage, TeX

A_4\times C_3\rtimes D_4

% in TeX

G:=Group("A4xC3:D4");

// GroupNames label

G:=SmallGroup(288,928);

// by ID

G=gap.SmallGroup(288,928);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,197,648,271,9414]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^3=d^3=e^4=f^2=1,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c^-1=a,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,e*d*e^-1=f*d*f=d^-1,f*e*f=e^-1>;

// generators/relations

G = A4×C3⋊D4 order 288 = 25·32

Direct product of A4 and C3⋊D4

G = A₄×C₃⋊D₄ order 288 = 2⁵·3²

Direct product of A₄ and C₃⋊D₄