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G = C4×C3.A4order 144 = 24·32

Direct product of C4 and C3.A4

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

Aliases: C4×C3.A4, C22⋊C36, C12.1A4, C23.C18, C3.(C4×A4), (C22×C4)⋊C9, (C2×C6).C12, C6.4(C2×A4), (C22×C12).C3, (C22×C6).2C6, C2.1(C2×C3.A4), (C2×C3.A4).2C2, SmallGroup(144,34)

Series: Derived Chief Lower central Upper central

C1C22 — C4×C3.A4
C1C22C2×C6C22×C6C2×C3.A4 — C4×C3.A4
C22 — C4×C3.A4
C1C12

Generators and relations for C4×C3.A4
 G = < a,b,c,d,e | a4=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

3C2
3C2
3C4
3C22
3C22
3C6
3C6
4C9
3C2×C4
3C2×C4
3C2×C6
3C12
3C2×C6
4C18
3C2×C12
3C2×C12
4C36

Smallest permutation representation of C4×C3.A4
On 36 points
Generators in S36
(1 16 21 31)(2 17 22 32)(3 18 23 33)(4 10 24 34)(5 11 25 35)(6 12 26 36)(7 13 27 28)(8 14 19 29)(9 15 20 30)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(2 22)(3 23)(5 25)(6 26)(8 19)(9 20)(11 35)(12 36)(14 29)(15 30)(17 32)(18 33)
(1 21)(3 23)(4 24)(6 26)(7 27)(9 20)(10 34)(12 36)(13 28)(15 30)(16 31)(18 33)
(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)

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

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

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

C4×C3.A4 is a maximal subgroup of   C12.S4  C12.1S4  C22⋊D36  A4×C36
C4×C3.A4 is a maximal quotient of   Q8.C36

48 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F9A···9F12A12B12C12D12E12F12G12H18A···18F36A···36L
order12223344446666669···9121212121212121218···1836···36
size11331111331133334···4111133334···44···4

48 irreducible representations

dim111111111333333
type++++
imageC1C2C3C4C6C9C12C18C36A4C2×A4C3.A4C4×A4C2×C3.A4C4×C3.A4
kernelC4×C3.A4C2×C3.A4C22×C12C3.A4C22×C6C22×C4C2×C6C23C22C12C6C4C3C2C1
# reps1122264612112224

Matrix representation of C4×C3.A4 in GL3(𝔽13) generated by

500
050
005
,
900
090
009
,
100
0120
0012
,
1200
0120
001
,
0100
0010
100
G:=sub<GL(3,GF(13))| [5,0,0,0,5,0,0,0,5],[9,0,0,0,9,0,0,0,9],[1,0,0,0,12,0,0,0,12],[12,0,0,0,12,0,0,0,1],[0,0,1,10,0,0,0,10,0] >;

C4×C3.A4 in GAP, Magma, Sage, TeX

C_4\times C_3.A_4
% in TeX

G:=Group("C4xC3.A4");
// GroupNames label

G:=SmallGroup(144,34);
// by ID

G=gap.SmallGroup(144,34);
# by ID

G:=PCGroup([6,-2,-3,-2,-3,-2,2,36,79,1090,1955]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^3=c^2=d^2=1,e^3=b,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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Subgroup lattice of C4×C3.A4 in TeX

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