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## G = A4×C3.A4order 432 = 24·33

### Direct product of A4 and C3.A4

Aliases: A4×C3.A4, C3.1A42, (C22×A4)⋊C9, C241(C3×C9), (C3×A4).2A4, C223(C9×A4), C24⋊C91C3, (C23×C6).1C32, (A4×C2×C6).1C3, (C2×C6).1(C3×A4), C221(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

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