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## G = A4×C3×C6order 216 = 23·33

### Direct product of C3×C6 and A4

Aliases: A4×C3×C6, C23⋊C33, C629C6, (C2×C62)⋊3C3, C22⋊(C32×C6), (C22×C6)⋊C32, (C2×C6)⋊2(C3×C6), SmallGroup(216,173)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — A4×C3×C6
 Chief series C1 — C22 — C2×C6 — C62 — C32×A4 — A4×C3×C6
 Lower central C22 — A4×C3×C6
 Upper central C1 — C3×C6

Generators and relations for A4×C3×C6
G = < a,b,c,d,e | a3=b6=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Subgroups: 316 in 136 conjugacy classes, 68 normal (10 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C32, C32, A4, C2×C6, C2×C6, C3×C6, C3×C6, C2×A4, C22×C6, C33, C3×A4, C62, C62, C32×C6, C6×A4, C2×C62, C32×A4, A4×C3×C6
Quotients: C1, C2, C3, C6, C32, A4, C3×C6, C2×A4, C33, C3×A4, C32×C6, C6×A4, C32×A4, A4×C3×C6

Smallest permutation representation of A4×C3×C6
On 54 points
Generators in S54
(1 15 11)(2 16 12)(3 17 7)(4 18 8)(5 13 9)(6 14 10)(19 33 29)(20 34 30)(21 35 25)(22 36 26)(23 31 27)(24 32 28)(37 51 47)(38 52 48)(39 53 43)(40 54 44)(41 49 45)(42 50 46)
(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)
(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 52)(50 53)(51 54)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)
(1 49 25)(2 50 26)(3 51 27)(4 52 28)(5 53 29)(6 54 30)(7 37 31)(8 38 32)(9 39 33)(10 40 34)(11 41 35)(12 42 36)(13 43 19)(14 44 20)(15 45 21)(16 46 22)(17 47 23)(18 48 24)

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

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

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

A4×C3×C6 is a maximal subgroup of   C6210Dic3

72 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Z 6A ··· 6H 6I ··· 6X 6Y ··· 6AP order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 3 3 1 ··· 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 A4 C2×A4 C3×A4 C6×A4 kernel A4×C3×C6 C32×A4 C6×A4 C2×C62 C3×A4 C62 C3×C6 C32 C6 C3 # reps 1 1 24 2 24 2 1 1 8 8

Matrix representation of A4×C3×C6 in GL4(𝔽7) generated by

 1 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 3 0 0 0 0 6 0 0 0 0 6 0 0 0 0 6
,
 1 0 0 0 0 1 0 0 0 4 6 0 0 2 0 6
,
 1 0 0 0 0 6 0 0 0 0 6 0 0 5 0 1
,
 2 0 0 0 0 2 6 0 0 0 5 4 0 0 6 0
G:=sub<GL(4,GF(7))| [1,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,1,4,2,0,0,6,0,0,0,0,6],[1,0,0,0,0,6,0,5,0,0,6,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,6,5,6,0,0,4,0] >;

A4×C3×C6 in GAP, Magma, Sage, TeX

A_4\times C_3\times C_6
% in TeX

G:=Group("A4xC3xC6");
// GroupNames label

G:=SmallGroup(216,173);
// by ID

G=gap.SmallGroup(216,173);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,2,1630,2927]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations

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