A4xC3.A4 - GroupNames

G = A₄×C₃.A₄ order 432 = 2⁴·3³

Direct product of A₄ and C₃.A₄

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

Aliases: A₄×C₃.A₄, C₃.₁A₄², (C₂²×A₄)⋊C₉, C₂⁴⋊₁(C₃×C₉), (C₃×A₄).₂A₄, C₂²⋊₃(C₉×A₄), C₂⁴⋊C₉⋊₁C₃, (C₂³×C₆).₁C₃², (A₄×C₂×C₆).₁C₃, (C₂×C₆).₁(C₃×A₄), C₂²⋊₁(C₃×C₃.A₄), (C₂²×C₃.A₄)⋊₁C₃, SmallGroup(432,524)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁴ — A₄×C₃.A₄

Chief series

C₁ — C₂² — C₂⁴ — C₂³×C₆ — A₄×C₂×C₆ — A₄×C₃.A₄

Lower central

C₂⁴ — A₄×C₃.A₄

Upper central

C₁ — C₃

Generators and relations for A₄×C₃.A₄
G = < a,b,c,d,e,f,g | a²=b²=c³=d³=e²=f²=1, g³=d, cac^-1=ab=ba, ad=da, ae=ea, af=fa, ag=ga, cbc^-1=a, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg^-1=ef=fe, gfg^-1=e >

Subgroups: 358 in 74 conjugacy classes, 21 normal (13 characteristic)
C₁, C₂, C₃, C₃, C₂², C₂², C₆, C₂³, C₉, C₃², A₄, C₂×C₆, C₂×C₆, C₂⁴, C₁₈, C₃×C₆, C₂×A₄, C₂²×C₆, C₃×C₉, C₃.A₄, C₃.A₄, C₂×C₁₈, C₃×A₄, C₆², C₂²×A₄, C₂³×C₆, C₂×C₃.A₄, C₆×A₄, C₉×A₄, C₃×C₃.A₄, C₂²×C₃.A₄, C₂⁴⋊C₉, A₄×C₂×C₆, A₄×C₃.A₄
Quotients: C₁, C₃, C₉, C₃², A₄, C₃×C₉, C₃.A₄, C₃×A₄, C₉×A₄, C₃×C₃.A₄, A₄², A₄×C₃.A₄

Smallest permutation representation of A₄×C₃.A₄
►On 54 points

Generators in S₅₄

(1 25)(2 26)(3 27)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 52)(11 53)(12 54)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)

(10 52)(11 53)(12 54)(13 46)(14 47)(15 48)(16 49)(17 50)(18 51)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 37)(35 38)(36 39)

(1 49 31)(2 50 32)(3 51 33)(4 52 34)(5 53 35)(6 54 36)(7 46 28)(8 47 29)(9 48 30)(10 37 19)(11 38 20)(12 39 21)(13 40 22)(14 41 23)(15 42 24)(16 43 25)(17 44 26)(18 45 27)

(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)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)

(2 26)(3 27)(5 20)(6 21)(8 23)(9 24)(11 53)(12 54)(14 47)(15 48)(17 50)(18 51)(29 41)(30 42)(32 44)(33 45)(35 38)(36 39)

(1 25)(3 27)(4 19)(6 21)(7 22)(9 24)(10 52)(12 54)(13 46)(15 48)(16 49)(18 51)(28 40)(30 42)(31 43)(33 45)(34 37)(36 39)

(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)(37 38 39 40 41 42 43 44 45)(46 47 48 49 50 51 52 53 54)

G:=sub<Sym(54)| (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,52)(11,53)(12,54)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51), (10,52)(11,53)(12,54)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (1,49,31)(2,50,32)(3,51,33)(4,52,34)(5,53,35)(6,54,36)(7,46,28)(8,47,29)(9,48,30)(10,37,19)(11,38,20)(12,39,21)(13,40,22)(14,41,23)(15,42,24)(16,43,25)(17,44,26)(18,45,27), (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)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,26)(3,27)(5,20)(6,21)(8,23)(9,24)(11,53)(12,54)(14,47)(15,48)(17,50)(18,51)(29,41)(30,42)(32,44)(33,45)(35,38)(36,39), (1,25)(3,27)(4,19)(6,21)(7,22)(9,24)(10,52)(12,54)(13,46)(15,48)(16,49)(18,51)(28,40)(30,42)(31,43)(33,45)(34,37)(36,39), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54)>;

G:=Group( (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,52)(11,53)(12,54)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51), (10,52)(11,53)(12,54)(13,46)(14,47)(15,48)(16,49)(17,50)(18,51)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,37)(35,38)(36,39), (1,49,31)(2,50,32)(3,51,33)(4,52,34)(5,53,35)(6,54,36)(7,46,28)(8,47,29)(9,48,30)(10,37,19)(11,38,20)(12,39,21)(13,40,22)(14,41,23)(15,42,24)(16,43,25)(17,44,26)(18,45,27), (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)(37,40,43)(38,41,44)(39,42,45)(46,49,52)(47,50,53)(48,51,54), (2,26)(3,27)(5,20)(6,21)(8,23)(9,24)(11,53)(12,54)(14,47)(15,48)(17,50)(18,51)(29,41)(30,42)(32,44)(33,45)(35,38)(36,39), (1,25)(3,27)(4,19)(6,21)(7,22)(9,24)(10,52)(12,54)(13,46)(15,48)(16,49)(18,51)(28,40)(30,42)(31,43)(33,45)(34,37)(36,39), (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)(37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54) );

G=PermutationGroup([[(1,25),(2,26),(3,27),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,52),(11,53),(12,54),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51)], [(10,52),(11,53),(12,54),(13,46),(14,47),(15,48),(16,49),(17,50),(18,51),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,37),(35,38),(36,39)], [(1,49,31),(2,50,32),(3,51,33),(4,52,34),(5,53,35),(6,54,36),(7,46,28),(8,47,29),(9,48,30),(10,37,19),(11,38,20),(12,39,21),(13,40,22),(14,41,23),(15,42,24),(16,43,25),(17,44,26),(18,45,27)], [(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),(37,40,43),(38,41,44),(39,42,45),(46,49,52),(47,50,53),(48,51,54)], [(2,26),(3,27),(5,20),(6,21),(8,23),(9,24),(11,53),(12,54),(14,47),(15,48),(17,50),(18,51),(29,41),(30,42),(32,44),(33,45),(35,38),(36,39)], [(1,25),(3,27),(4,19),(6,21),(7,22),(9,24),(10,52),(12,54),(13,46),(15,48),(16,49),(18,51),(28,40),(30,42),(31,43),(33,45),(34,37),(36,39)], [(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),(37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54)]])

48 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	6A	6B	6C	6D	6E	6F	6G	···	6L	9A	···	9F	9G	···	9R	18A	···	18F
order	1	2	2	2	3	3	3	···	3	6	6	6	6	6	6	6	···	6	9	···	9	9	···	9	18	···	18
size	1	3	3	9	1	1	4	···	4	3	3	3	3	9	9	12	···	12	4	···	4	16	···	16	12	···	12

48 irreducible representations

dim	1	1	1	1	1	3	3	3	3	3	9	9
type	+					+	+				+
image	C₁	C₃	C₃	C₃	C₉	A₄	A₄	C₃.A₄	C₃×A₄	C₉×A₄	A₄²	A₄×C₃.A₄
kernel	A₄×C₃.A₄	C₂²×C₃.A₄	C₂⁴⋊C₉	A₄×C₂×C₆	C₂²×A₄	C₃.A₄	C₃×A₄	A₄	C₂×C₆	C₂²	C₃	C₁
# reps	1	2	4	2	18	1	1	6	4	6	1	2

Matrix representation of A₄×C₃.A₄ ►in GL₆(𝔽₁₉)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	18	18	18
0	0	0	1	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	1	0	0
0	0	0	18	18	18

11	0	0	0	0	0
0	11	0	0	0	0
0	0	11	0	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	18	18	18

7	0	0	0	0	0
0	7	0	0	0	0
0	0	7	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	18	1	0	0	0
0	18	0	0	0	0
1	18	0	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

18	0	0	0	0	0
18	0	1	0	0	0
18	1	0	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,1,0,0,0,0,18,0,0,0,0,1,18,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,1,0,18,0,0,0,0,0,18],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,18,0,0,0,0,0,18,0,0,0,0,1,18],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,1,0,0,0,18,18,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,18,18,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,0,9,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

A₄×C₃.A₄ in GAP, Magma, Sage, TeX

A_4\times C_3.A_4

% in TeX

G:=Group("A4xC3.A4");

// GroupNames label

G:=SmallGroup(432,524);

// by ID

G=gap.SmallGroup(432,524);

# by ID

G:=PCGroup([7,-3,-3,-3,-2,2,-2,2,50,766,326,13613,5298]);

// Polycyclic

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

// generators/relations

G = A4×C3.A4 order 432 = 24·33

Direct product of A4 and C3.A4

G = A₄×C₃.A₄ order 432 = 2⁴·3³

Direct product of A₄ and C₃.A₄