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G = A4xC2xC6order 144 = 24·32

Direct product of C2xC6 and A4

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

Aliases: A4xC2xC6, C22:C62, C24:2C32, C23:(C3xC6), (C22xC6):2C6, (C23xC6):1C3, (C2xC6):3(C2xC6), SmallGroup(144,193)

Series: Derived Chief Lower central Upper central

C1C22 — A4xC2xC6
C1C22C2xC6C3xA4C6xA4 — A4xC2xC6
C22 — A4xC2xC6
C1C2xC6

Generators and relations for A4xC2xC6
 G = < a,b,c,d,e | a2=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: 234 in 98 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C2, C3, C3, C22, C22, C6, C6, C23, C23, C32, A4, C2xC6, C2xC6, C24, C3xC6, C2xA4, C22xC6, C22xC6, C3xA4, C62, C22xA4, C23xC6, C6xA4, A4xC2xC6
Quotients: C1, C2, C3, C22, C6, C32, A4, C2xC6, C3xC6, C2xA4, C3xA4, C62, C22xA4, C6xA4, A4xC2xC6

Smallest permutation representation of A4xC2xC6
On 36 points
Generators in S36
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)
(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 4)(2 5)(3 6)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(13 16)(14 17)(15 18)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)
(1 15)(2 16)(3 17)(4 18)(5 13)(6 14)(7 10)(8 11)(9 12)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(31 34)(32 35)(33 36)
(1 32 22)(2 33 23)(3 34 24)(4 35 19)(5 36 20)(6 31 21)(7 28 18)(8 29 13)(9 30 14)(10 25 15)(11 26 16)(12 27 17)

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

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

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

A4xC2xC6 is a maximal subgroup of   (C2xC6):4S4  C24:He3  C24:3- 1+2

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C···3H6A···6F6G···6N6O···6AF
order12222222333···36···66···66···6
size11113333114···41···13···34···4

48 irreducible representations

dim1111113333
type++++
imageC1C2C3C3C6C6A4C2xA4C3xA4C6xA4
kernelA4xC2xC6C6xA4C22xA4C23xC6C2xA4C22xC6C2xC6C6C22C2
# reps13621861326

Matrix representation of A4xC2xC6 in GL4(F7) generated by

1000
0600
0060
0006
,
5000
0600
0060
0006
,
1000
0600
0310
0006
,
1000
0100
0460
0206
,
4000
0450
0031
0050
G:=sub<GL(4,GF(7))| [1,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[5,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,6,3,0,0,0,1,0,0,0,0,6],[1,0,0,0,0,1,4,2,0,0,6,0,0,0,0,6],[4,0,0,0,0,4,0,0,0,5,3,5,0,0,1,0] >;

A4xC2xC6 in GAP, Magma, Sage, TeX

A_4\times C_2\times C_6
% in TeX

G:=Group("A4xC2xC6");
// GroupNames label

G:=SmallGroup(144,193);
// by ID

G=gap.SmallGroup(144,193);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,2,556,989]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=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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