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G = A4xC3xC6order 216 = 23·33

Direct product of C3xC6 and A4

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

Aliases: A4xC3xC6, C23:C33, C62:9C6, (C2xC62):3C3, C22:(C32xC6), (C22xC6):C32, (C2xC6):2(C3xC6), SmallGroup(216,173)

Series: Derived Chief Lower central Upper central

C1C22 — A4xC3xC6
C1C22C2xC6C62C32xA4 — A4xC3xC6
C22 — A4xC3xC6
C1C3xC6

Generators and relations for A4xC3xC6
 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, C2xC6, C2xC6, C3xC6, C3xC6, C2xA4, C22xC6, C33, C3xA4, C62, C62, C32xC6, C6xA4, C2xC62, C32xA4, A4xC3xC6
Quotients: C1, C2, C3, C6, C32, A4, C3xC6, C2xA4, C33, C3xA4, C32xC6, C6xA4, C32xA4, A4xC3xC6

Smallest permutation representation of A4xC3xC6
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)]])

A4xC3xC6 is a maximal subgroup of   C62:10Dic3

72 conjugacy classes

class 1 2A2B2C3A···3H3I···3Z6A···6H6I···6X6Y···6AP
order12223···33···36···66···66···6
size11331···14···41···13···34···4

72 irreducible representations

dim1111113333
type++++
imageC1C2C3C3C6C6A4C2xA4C3xA4C6xA4
kernelA4xC3xC6C32xA4C6xA4C2xC62C3xA4C62C3xC6C32C6C3
# reps112422421188

Matrix representation of A4xC3xC6 in GL4(F7) generated by

1000
0400
0040
0004
,
3000
0600
0060
0006
,
1000
0100
0460
0206
,
1000
0600
0060
0501
,
2000
0260
0054
0060
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] >;

A4xC3xC6 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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