C3xA4:D4 - GroupNames

G = C₃×A₄⋊D₄ order 288 = 2⁵·3²

Direct product of C₃ and A₄⋊D₄

direct product, non-abelian, soluble, monomial

Aliases: C₃×A₄⋊D₄, A₄⋊C₄⋊C₆, (C₂×C₆)⋊₃S₄, (C₂×S₄)⋊₂C₆, (C₆×S₄)⋊₅C₂, (C₃×A₄)⋊₇D₄, A₄⋊₂(C₃×D₄), C₂.₁₁(C₆×S₄), C₆.₄₉(C₂×S₄), C₂⁴⋊₃(C₃×S₃), (C₂³×C₆)⋊₁S₃, C₂²⋊₃(C₃×S₄), (C₂²×A₄)⋊₄C₆, C₂³.₅(S₃×C₆), (C₂²×C₆).₁₂D₆, (C₆×A₄).₁₂C₂², (A₄×C₂×C₆)⋊₂C₂, (C₃×A₄⋊C₄)⋊₄C₂, C₂²⋊(C₃×C₃⋊D₄), (C₂×C₆)⋊₃(C₃⋊D₄), (C₂×A₄).₅(C₂×C₆), SmallGroup(288,906)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂² — C₂×A₄ — C₃×A₄⋊D₄

Chief series

C₁ — C₂² — A₄ — C₂×A₄ — C₆×A₄ — C₆×S₄ — C₃×A₄⋊D₄

Lower central

A₄ — C₂×A₄ — C₃×A₄⋊D₄

Upper central

C₁ — C₆ — C₂×C₆

Generators and relations for C₃×A₄⋊D₄
G = < a,b,c,d,e,f | a³=b²=c²=d³=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd^-1=ebe^-1=fbf=bc=cb, dcd^-1=b, ce=ec, cf=fc, ede^-1=fdf=d^-1, fef=e^-1 >

Subgroups: 518 in 134 conjugacy classes, 24 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, C₂³, C₂³, C₃², Dic₃, C₁₂, A₄, A₄, D₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂×D₄, C₂⁴, C₃×S₃, C₃×C₆, C₃⋊D₄, C₂×C₁₂, C₃×D₄, S₄, C₂×A₄, C₂×A₄, C₂²×C₆, C₂²×C₆, C₂²≀C₂, C₃×Dic₃, C₃×A₄, S₃×C₆, C₆², C₃×C₂²⋊C₄, A₄⋊C₄, C₆×D₄, C₂×S₄, C₂²×A₄, C₂²×A₄, C₂³×C₆, C₃×C₃⋊D₄, C₃×S₄, C₆×A₄, C₆×A₄, C₃×C₂²≀C₂, A₄⋊D₄, C₃×A₄⋊C₄, C₆×S₄, A₄×C₂×C₆, C₃×A₄⋊D₄
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, D₄, D₆, C₂×C₆, C₃×S₃, C₃⋊D₄, C₃×D₄, S₄, S₃×C₆, C₂×S₄, C₃×C₃⋊D₄, C₃×S₄, A₄⋊D₄, C₆×S₄, C₃×A₄⋊D₄

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

Generators in S₃₆

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

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

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

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

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

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

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

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

42 conjugacy classes

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

42 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
type	+	+	+	+					+	+	+						+	+			+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	D₄	D₆	C₃×S₃	C₃×D₄	C₃⋊D₄	S₃×C₆	C₃×C₃⋊D₄	S₄	C₂×S₄	C₃×S₄	C₆×S₄	A₄⋊D₄	C₃×A₄⋊D₄
kernel	C₃×A₄⋊D₄	C₃×A₄⋊C₄	C₆×S₄	A₄×C₂×C₆	A₄⋊D₄	A₄⋊C₄	C₂×S₄	C₂²×A₄	C₂³×C₆	C₃×A₄	C₂²×C₆	C₂⁴	A₄	C₂×C₆	C₂³	C₂²	C₂×C₆	C₆	C₂²	C₂	C₃	C₁
# reps	1	1	1	1	2	2	2	2	1	1	1	2	2	2	2	4	2	2	4	4	1	2

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

1	0	0	0	0
0	1	0	0	0
0	0	9	0	0
0	0	0	9	0
0	0	0	0	9

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

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

3	7	0	0	0
0	9	0	0	0
0	0	1	0	11
0	0	0	0	12
0	0	0	1	12

3	12	0	0	0
10	10	0	0	0
0	0	12	0	0
0	0	0	0	12
0	0	0	12	0

3	7	0	0	0
10	10	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,12,0,0,0,12,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0,0,0,12,0,0,0,0,0,12],[3,0,0,0,0,7,9,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,11,12,12],[3,10,0,0,0,12,10,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0],[3,10,0,0,0,7,10,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C₃×A₄⋊D₄ in GAP, Magma, Sage, TeX

C_3\times A_4\rtimes D_4

% in TeX

G:=Group("C3xA4:D4");

// GroupNames label

G:=SmallGroup(288,906);

// by ID

G=gap.SmallGroup(288,906);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,197,1684,6053,285,3534,475]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^3=e^4=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=e*b*e^-1=f*b*f=b*c=c*b,d*c*d^-1=b,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 = C3×A4⋊D4 order 288 = 25·32

Direct product of C3 and A4⋊D4

G = C₃×A₄⋊D₄ order 288 = 2⁵·3²

Direct product of C₃ and A₄⋊D₄