Dic3xC3.A4 - GroupNames

G = Dic₃×C₃.A₄ order 432 = 2⁴·3³

Direct product of Dic₃ and C₃.A₄

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

Aliases: Dic₃×C₃.A₄, C₆².₆C₁₂, (C₂×C₆)⋊C₃₆, C₆.₁₈(S₃×A₄), (C₂²×C₆).C₁₈, (C₂×C₆²).₅C₆, C₃².₂(C₄×A₄), C₂³.₂(S₃×C₉), (C₂²×Dic₃)⋊C₉, C₃.₄(Dic₃×A₄), (C₃×Dic₃).₂A₄, C₂²⋊₂(C₉×Dic₃), C₃⋊(C₄×C₃.A₄), (Dic₃×C₂×C₆).C₃, (C₃×C₃.A₄)⋊₁C₄, C₂.₁(S₃×C₃.A₄), C₆.₃(C₂×C₃.A₄), (C₃×C₆).₁₃(C₂×A₄), (C₂×C₃.A₄).₃S₃, (C₆×C₃.A₄).₁C₂, (C₂×C₆).₈(C₃×Dic₃), (C₂²×C₆).₁₈(C₃×S₃), SmallGroup(432,271)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — Dic₃×C₃.A₄

Chief series

C₁ — C₃ — C₂×C₆ — C₆² — C₂×C₆² — C₆×C₃.A₄ — Dic₃×C₃.A₄

Lower central

C₂×C₆ — Dic₃×C₃.A₄

Upper central

C₁ — C₆

Generators and relations for Dic₃×C₃.A₄
G = < a,b,c,d,e,f | a⁶=c³=d²=e²=1, b²=a³, f³=c, bab^-1=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf^-1=de=ed, fef^-1=d >

Subgroups: 244 in 76 conjugacy classes, 25 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₆, C₆, C₂×C₄, C₂³, C₉, C₃², Dic₃, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂²×C₄, C₁₈, C₃×C₆, C₃×C₆, C₂×Dic₃, C₂×C₁₂, C₂²×C₆, C₂²×C₆, C₃×C₉, C₃₆, C₃.A₄, C₃.A₄, C₃×Dic₃, C₃×Dic₃, C₆², C₆², C₂²×Dic₃, C₂²×C₁₂, C₃×C₁₈, C₂×C₃.A₄, C₂×C₃.A₄, C₆×Dic₃, C₂×C₆², C₉×Dic₃, C₃×C₃.A₄, C₄×C₃.A₄, Dic₃×C₂×C₆, C₆×C₃.A₄, Dic₃×C₃.A₄
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, C₉, Dic₃, C₁₂, A₄, C₁₈, C₃×S₃, C₂×A₄, C₃₆, C₃.A₄, C₃×Dic₃, C₄×A₄, S₃×C₉, C₂×C₃.A₄, S₃×A₄, C₉×Dic₃, C₄×C₃.A₄, Dic₃×A₄, S₃×C₃.A₄, Dic₃×C₃.A₄

Smallest permutation representation of Dic₃×C₃.A₄
►On 36 points

Generators in S₃₆

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

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

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

(1 19)(2 20)(4 22)(5 23)(7 25)(8 26)(10 31)(11 32)(13 34)(14 35)(16 28)(17 29)

(2 20)(3 21)(5 23)(6 24)(8 26)(9 27)(11 32)(12 33)(14 35)(15 36)(17 29)(18 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 31 32 33 34 35 36)

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

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

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

72 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	···	6O	9A	···	9F	9G	···	9L	12A	12B	12C	12D	12E	12F	12G	12H	18A	···	18F	18G	···	18L	36A	···	36L
order	1	2	2	2	3	3	3	3	3	4	4	4	4	6	6	6	6	6	6	6	6	6	6	···	6	9	···	9	9	···	9	12	12	12	12	12	12	12	12	18	···	18	18	···	18	36	···	36
size	1	1	3	3	1	1	2	2	2	3	3	9	9	1	1	2	2	2	3	3	3	3	6	···	6	4	···	4	8	···	8	3	3	3	3	9	9	9	9	4	···	4	8	···	8	12	···	12

72 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	3	3	3	6	6	6	6
type	+	+								+	-					+	+					+	-
image	C₁	C₂	C₃	C₄	C₆	C₉	C₁₂	C₁₈	C₃₆	S₃	Dic₃	C₃×S₃	C₃×Dic₃	S₃×C₉	C₉×Dic₃	A₄	C₂×A₄	C₃.A₄	C₄×A₄	C₂×C₃.A₄	C₄×C₃.A₄	S₃×A₄	Dic₃×A₄	S₃×C₃.A₄	Dic₃×C₃.A₄
kernel	Dic₃×C₃.A₄	C₆×C₃.A₄	Dic₃×C₂×C₆	C₃×C₃.A₄	C₂×C₆²	C₂²×Dic₃	C₆²	C₂²×C₆	C₂×C₆	C₂×C₃.A₄	C₃.A₄	C₂²×C₆	C₂×C₆	C₂³	C₂²	C₃×Dic₃	C₃×C₆	Dic₃	C₃²	C₆	C₃	C₆	C₃	C₂	C₁
# reps	1	1	2	2	2	6	4	6	12	1	1	2	2	6	6	1	1	2	2	2	4	1	1	2	2

Matrix representation of Dic₃×C₃.A₄ ►in GL₅(𝔽₃₇)

11	5	0	0	0
0	27	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

3	19	0	0	0
17	34	0	0	0
0	0	36	0	0
0	0	0	36	0
0	0	0	0	36

1	0	0	0	0
0	1	0	0	0
0	0	26	0	0
0	0	0	26	0
0	0	0	0	26

1	0	0	0	0
0	1	0	0	0
0	0	36	0	0
0	0	0	1	12
0	0	0	0	36

1	0	0	0	0
0	1	0	0	0
0	0	36	0	12
0	0	0	36	25
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	0	16	0
0	0	21	21	0
0	0	0	15	16

G:=sub<GL(5,GF(37))| [11,0,0,0,0,5,27,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,17,0,0,0,19,34,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,0,26],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,1,0,0,0,0,12,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,12,25,1],[1,0,0,0,0,0,1,0,0,0,0,0,0,21,0,0,0,16,21,15,0,0,0,0,16] >;

Dic₃×C₃.A₄ in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3.A_4

% in TeX

G:=Group("Dic3xC3.A4");

// GroupNames label

G:=SmallGroup(432,271);

// by ID

G=gap.SmallGroup(432,271);

# by ID

G:=PCGroup([7,-2,-3,-2,-3,-2,2,-3,42,92,1901,768,14118]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=c^3=d^2=e^2=1,b^2=a^3,f^3=c,b*a*b^-1=a^-1,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,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 = Dic3×C3.A4 order 432 = 24·33

Direct product of Dic3 and C3.A4

G = Dic₃×C₃.A₄ order 432 = 2⁴·3³

Direct product of Dic₃ and C₃.A₄