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## G = C3×A42order 432 = 24·33

### Direct product of C3, A4 and A4

Aliases: C3×A42, C24⋊C33, C22⋊A4⋊C32, (C23×C6)⋊C32, (C22×A4)⋊C32, C221(C32×A4), (A4×C2×C6)⋊3C3, (C2×C6)⋊1(C3×A4), (C3×C22⋊A4)⋊3C3, SmallGroup(432,750)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C3×A42
 Chief series C1 — C22 — C24 — C22×A4 — A42 — C3×A42
 Lower central C24 — C3×A42
 Upper central C1 — C3

Generators and relations for C3×A42
G = < a,b,c,d,e,f,g | a3=b2=c2=d3=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, dbd-1=bc=cb, be=eb, bf=fb, bg=gb, dcd-1=b, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >

Subgroups: 880 in 158 conjugacy classes, 42 normal (5 characteristic)
C1, C2, C3, C3, C22, C22, C6, C23, C32, A4, A4, C2×C6, C2×C6, C24, C3×C6, C2×A4, C22×C6, C33, C3×A4, C3×A4, C62, C22×A4, C22⋊A4, C23×C6, C6×A4, C32×A4, A42, A4×C2×C6, C3×C22⋊A4, C3×A42
Quotients: C1, C3, C32, A4, C33, C3×A4, C32×A4, A42, C3×A42

Smallest permutation representation of C3×A42
On 36 points
Generators in S36
(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 15)(2 13)(3 14)(4 8)(5 9)(6 7)(10 18)(11 16)(12 17)(19 29)(20 30)(21 28)(22 27)(23 25)(24 26)(31 36)(32 34)(33 35)
(1 10)(2 11)(3 12)(4 32)(5 33)(6 31)(7 36)(8 34)(9 35)(13 16)(14 17)(15 18)(19 24)(20 22)(21 23)(25 28)(26 29)(27 30)
(1 5 26)(2 6 27)(3 4 25)(7 30 16)(8 28 17)(9 29 18)(10 35 24)(11 36 22)(12 34 23)(13 31 20)(14 32 21)(15 33 19)
(1 10)(2 11)(3 12)(4 34)(5 35)(6 36)(7 31)(8 32)(9 33)(13 16)(14 17)(15 18)(19 29)(20 30)(21 28)(22 27)(23 25)(24 26)
(1 15)(2 13)(3 14)(4 32)(5 33)(6 31)(7 36)(8 34)(9 35)(10 18)(11 16)(12 17)(19 26)(20 27)(21 25)(22 30)(23 28)(24 29)
(1 6 25)(2 4 26)(3 5 27)(7 23 15)(8 24 13)(9 22 14)(10 31 28)(11 32 29)(12 33 30)(16 34 19)(17 35 20)(18 36 21)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 3O ··· 3Z 6A 6B 6C 6D 6E 6F 6G ··· 6R order 1 2 2 2 3 3 3 ··· 3 3 ··· 3 6 6 6 6 6 6 6 ··· 6 size 1 3 3 9 1 1 4 ··· 4 16 ··· 16 3 3 3 3 9 9 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 3 3 3 9 9 type + + + image C1 C3 C3 C3 A4 C3×A4 C3×A4 A42 C3×A42 kernel C3×A42 A42 A4×C2×C6 C3×C22⋊A4 C3×A4 A4 C2×C6 C3 C1 # reps 1 18 4 4 2 12 4 1 2

Matrix representation of C3×A42 in GL6(𝔽7)

 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 6 6 0 0 0 0 0 1 0 0 0 0 1 0
,
 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 6 6 6 0 0 0 1 0 0
,
 2 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 6 6 6
,
 0 0 1 0 0 0 6 6 6 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 6 6 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 4 0 0 0 0 0 0 0 4 0 0 0 3 3 3 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(7))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,0,0,0,0,0,6,0,1,0,0,0,6,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,6,1,0,0,0,0,6,0,0,0,0,1,6,0],[2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,6,0,0,0,0,0,6,0,0,0,0,1,6],[0,6,1,0,0,0,0,6,0,0,0,0,1,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,6,0,0,0,1,0,6,0,0,0,0,0,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,3,0,0,0,0,0,3,0,0,0,0,4,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C3×A42 in GAP, Magma, Sage, TeX

C_3\times A_4^2
% in TeX

G:=Group("C3xA4^2");
// GroupNames label

G:=SmallGroup(432,750);
// by ID

G=gap.SmallGroup(432,750);
# by ID

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

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