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## G = A4×C2×C6order 144 = 24·32

### Direct product of C2×C6 and A4

Aliases: A4×C2×C6, C22⋊C62, C242C32, C23⋊(C3×C6), (C22×C6)⋊2C6, (C23×C6)⋊1C3, (C2×C6)⋊3(C2×C6), SmallGroup(144,193)

Series: Derived Chief Lower central Upper central

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

Generators and relations for A4×C2×C6
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, C2×C6, C2×C6, C24, C3×C6, C2×A4, C22×C6, C22×C6, C3×A4, C62, C22×A4, C23×C6, C6×A4, A4×C2×C6
Quotients: C1, C2, C3, C22, C6, C32, A4, C2×C6, C3×C6, C2×A4, C3×A4, C62, C22×A4, C6×A4, A4×C2×C6

Smallest permutation representation of A4×C2×C6
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)]])

A4×C2×C6 is a maximal subgroup of   (C2×C6)⋊4S4  C24⋊He3  C24⋊3- 1+2

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C ··· 3H 6A ··· 6F 6G ··· 6N 6O ··· 6AF order 1 2 2 2 2 2 2 2 3 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 1 3 3 3 3 1 1 4 ··· 4 1 ··· 1 3 ··· 3 4 ··· 4

48 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×C2×C6 C6×A4 C22×A4 C23×C6 C2×A4 C22×C6 C2×C6 C6 C22 C2 # reps 1 3 6 2 18 6 1 3 2 6

Matrix representation of A4×C2×C6 in GL4(𝔽7) generated by

 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 0 0 0 3 1 0 0 0 0 6
,
 1 0 0 0 0 1 0 0 0 4 6 0 0 2 0 6
,
 4 0 0 0 0 4 5 0 0 0 3 1 0 0 5 0
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] >;

A4×C2×C6 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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