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
| C22 — C4×C3.A4 |
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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