C3xD4xA4 - GroupNames

G = C₃×D₄×A₄ order 288 = 2⁵·3²

Direct product of C₃, D₄ and A₄

direct product, metabelian, soluble, monomial

Aliases: C₃×D₄×A₄, C₂³.₇C₆², C₄⋊(C₆×A₄), (C₄×A₄)⋊₃C₆, C₁₂⋊₃(C₂×A₄), (C₁₂×A₄)⋊₇C₂, C₂²⋊₃(C₆×A₄), (C₂³×C₆)⋊₂C₆, C₂⁴⋊₃(C₃×C₆), (C₂²×D₄)⋊C₃², C₂²⋊(D₄×C₃²), (C₂²×A₄)⋊₃C₆, (C₂²×C₁₂)⋊₃C₆, C₆.₂₁(C₂²×A₄), (C₆×A₄).₂₄C₂², (D₄×C₂×C₆)⋊C₃, (A₄×C₂×C₆)⋊₁C₂, C₂.₂(A₄×C₂×C₆), (C₂×C₆)⋊₃(C₂×A₄), (C₂×C₆)⋊₆(C₃×D₄), (C₂²×C₄)⋊(C₃×C₆), (C₂×A₄).₇(C₂×C₆), (C₂²×C₆).₄₀(C₂×C₆), SmallGroup(288,980)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₃×D₄×A₄

Chief series

C₁ — C₂² — C₂³ — C₂²×C₆ — C₆×A₄ — A₄×C₂×C₆ — C₃×D₄×A₄

Lower central

C₂² — C₂³ — C₃×D₄×A₄

Upper central

C₁ — C₆ — C₃×D₄

Generators and relations for C₃×D₄×A₄
G = < a,b,c,d,e,f | a³=b⁴=c²=d²=e²=f³=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b^-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf^-1=de=ed, fef^-1=d >

Subgroups: 516 in 164 conjugacy classes, 48 normal (20 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², C₂², C₆, C₆, C₂×C₄, D₄, D₄, C₂³, C₂³, C₃², C₁₂, C₁₂, A₄, C₂×C₆, C₂×C₆, C₂×C₆, C₂²×C₄, C₂×D₄, C₂⁴, C₃×C₆, C₂×C₁₂, C₃×D₄, C₃×D₄, C₂×A₄, C₂×A₄, C₂²×C₆, C₂²×C₆, C₂²×D₄, C₃×C₁₂, C₃×A₄, C₆², C₄×A₄, C₂²×C₁₂, C₆×D₄, C₂²×A₄, C₂³×C₆, D₄×C₃², C₆×A₄, C₆×A₄, D₄×A₄, D₄×C₂×C₆, C₁₂×A₄, A₄×C₂×C₆, C₃×D₄×A₄
Quotients: C₁, C₂, C₃, C₂², C₆, D₄, C₃², A₄, C₂×C₆, C₃×C₆, C₃×D₄, C₂×A₄, C₃×A₄, C₆², C₂²×A₄, D₄×C₃², C₆×A₄, D₄×A₄, A₄×C₂×C₆, C₃×D₄×A₄

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

Generators in S₃₆

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

(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)

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

(5 7)(6 8)(9 11)(10 12)(17 19)(18 20)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)

(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)

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

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	···	3H	4A	4B	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	6K	···	6P	6Q	6R	6S	6T	6U	···	6AF	12A	12B	12C	12D	12E	···	12J
order	1	2	2	2	2	2	2	2	3	3	3	···	3	4	4	6	6	6	6	6	6	6	6	6	6	6	···	6	6	6	6	6	6	···	6	12	12	12	12	12	···	12
size	1	1	2	2	3	3	6	6	1	1	4	···	4	2	6	1	1	2	2	2	2	3	3	3	3	4	···	4	6	6	6	6	8	···	8	2	2	6	6	8	···	8

60 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	3	3	3	3	3	3	6	6
type	+	+	+							+			+	+	+				+
image	C₁	C₂	C₂	C₃	C₃	C₆	C₆	C₆	C₆	D₄	C₃×D₄	C₃×D₄	A₄	C₂×A₄	C₂×A₄	C₃×A₄	C₆×A₄	C₆×A₄	D₄×A₄	C₃×D₄×A₄
kernel	C₃×D₄×A₄	C₁₂×A₄	A₄×C₂×C₆	D₄×A₄	D₄×C₂×C₆	C₄×A₄	C₂²×C₁₂	C₂²×A₄	C₂³×C₆	C₃×A₄	A₄	C₂×C₆	C₃×D₄	C₁₂	C₂×C₆	D₄	C₄	C₂²	C₃	C₁
# reps	1	1	2	6	2	6	2	12	4	1	6	2	1	1	2	2	2	4	1	2

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

3	0	0	0	0
0	3	0	0	0
0	0	9	0	0
0	0	0	9	0
0	0	0	0	9

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

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

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	12	12	12
0	0	1	0	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

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

C₃×D₄×A₄ in GAP, Magma, Sage, TeX

C_3\times D_4\times A_4

% in TeX

G:=Group("C3xD4xA4");

// GroupNames label

G:=SmallGroup(288,980);

// by ID

G=gap.SmallGroup(288,980);

# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,2,533,1531,2666]);

// Polycyclic

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

// generators/relations

G = C3×D4×A4 order 288 = 25·32

Direct product of C3, D4 and A4

G = C₃×D₄×A₄ order 288 = 2⁵·3²

Direct product of C₃, D₄ and A₄