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## G = C4×C3.A4order 144 = 24·32

### Direct product of C4 and C3.A4

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

 Derived series C1 — C22 — C4×C3.A4
 Chief series C1 — C22 — C2×C6 — C22×C6 — C2×C3.A4 — C4×C3.A4
 Lower central C22 — C4×C3.A4
 Upper central C1 — C12

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 >

Smallest permutation representation of C4×C3.A4
On 36 points
Generators in S36
(1 16 19 31)(2 17 20 32)(3 18 21 33)(4 10 22 34)(5 11 23 35)(6 12 24 36)(7 13 25 28)(8 14 26 29)(9 15 27 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 20)(3 21)(5 23)(6 24)(8 26)(9 27)(11 35)(12 36)(14 29)(15 30)(17 32)(18 33)
(1 19)(3 21)(4 22)(6 24)(7 25)(9 27)(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)| (1,16,19,31)(2,17,20,32)(3,18,21,33)(4,10,22,34)(5,11,23,35)(6,12,24,36)(7,13,25,28)(8,14,26,29)(9,15,27,30), (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), (2,20)(3,21)(5,23)(6,24)(8,26)(9,27)(11,35)(12,36)(14,29)(15,30)(17,32)(18,33), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(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( (1,16,19,31)(2,17,20,32)(3,18,21,33)(4,10,22,34)(5,11,23,35)(6,12,24,36)(7,13,25,28)(8,14,26,29)(9,15,27,30), (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), (2,20)(3,21)(5,23)(6,24)(8,26)(9,27)(11,35)(12,36)(14,29)(15,30)(17,32)(18,33), (1,19)(3,21)(4,22)(6,24)(7,25)(9,27)(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([[(1,16,19,31),(2,17,20,32),(3,18,21,33),(4,10,22,34),(5,11,23,35),(6,12,24,36),(7,13,25,28),(8,14,26,29),(9,15,27,30)], [(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)], [(2,20),(3,21),(5,23),(6,24),(8,26),(9,27),(11,35),(12,36),(14,29),(15,30),(17,32),(18,33)], [(1,19),(3,21),(4,22),(6,24),(7,25),(9,27),(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 2A 2B 2C 3A 3B 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 9A ··· 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18F 36A ··· 36L order 1 2 2 2 3 3 4 4 4 4 6 6 6 6 6 6 9 ··· 9 12 12 12 12 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 3 3 1 1 1 1 3 3 1 1 3 3 3 3 4 ··· 4 1 1 1 1 3 3 3 3 4 ··· 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + + + image C1 C2 C3 C4 C6 C9 C12 C18 C36 A4 C2×A4 C3.A4 C4×A4 C2×C3.A4 C4×C3.A4 kernel C4×C3.A4 C2×C3.A4 C22×C12 C3.A4 C22×C6 C22×C4 C2×C6 C23 C22 C12 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 1 1 2 2 2 4

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

 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 10 0 0 0 10 1 0 0
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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