direct product, non-abelian, soluble
Aliases: C3×C4.A4, C12.3A4, SL2(𝔽3)⋊2C6, C4.(C3×A4), C4○D4⋊C32, Q8.(C3×C6), C2.3(C6×A4), C6.10(C2×A4), (C3×Q8).4C6, (C3×SL2(𝔽3))⋊5C2, (C3×C4○D4)⋊C3, SmallGroup(144,157)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — C3×C4.A4 |
| Q8 — C3×C4.A4 |
Generators and relations for C3×C4.A4
G = < a,b,c,d,e | a3=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >
Smallest permutation representation of C3×C4.A4
►On 48 points
Generators in S48
(1 31 35)(2 32 36)(3 29 33)(4 30 34)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(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)(37 38 39 40)(41 42 43 44)(45 46 47 48)
(1 17 3 19)(2 18 4 20)(5 48 7 46)(6 45 8 47)(9 39 11 37)(10 40 12 38)(13 43 15 41)(14 44 16 42)(21 33 23 35)(22 34 24 36)(25 31 27 29)(26 32 28 30)
(1 37 3 39)(2 38 4 40)(5 28 7 26)(6 25 8 27)(9 17 11 19)(10 18 12 20)(13 21 15 23)(14 22 16 24)(29 47 31 45)(30 48 32 46)(33 43 35 41)(34 44 36 42)
(5 28 46)(6 25 47)(7 26 48)(8 27 45)(9 17 37)(10 18 38)(11 19 39)(12 20 40)(13 21 41)(14 22 42)(15 23 43)(16 24 44)
G:=sub<Sym(48)| (1,31,35)(2,32,36)(3,29,33)(4,30,34)(5,14,10)(6,15,11)(7,16,12)(8,13,9)(17,27,21)(18,28,22)(19,25,23)(20,26,24)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,17,3,19)(2,18,4,20)(5,48,7,46)(6,45,8,47)(9,39,11,37)(10,40,12,38)(13,43,15,41)(14,44,16,42)(21,33,23,35)(22,34,24,36)(25,31,27,29)(26,32,28,30), (1,37,3,39)(2,38,4,40)(5,28,7,26)(6,25,8,27)(9,17,11,19)(10,18,12,20)(13,21,15,23)(14,22,16,24)(29,47,31,45)(30,48,32,46)(33,43,35,41)(34,44,36,42), (5,28,46)(6,25,47)(7,26,48)(8,27,45)(9,17,37)(10,18,38)(11,19,39)(12,20,40)(13,21,41)(14,22,42)(15,23,43)(16,24,44)>;
G:=Group( (1,31,35)(2,32,36)(3,29,33)(4,30,34)(5,14,10)(6,15,11)(7,16,12)(8,13,9)(17,27,21)(18,28,22)(19,25,23)(20,26,24)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (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)(37,38,39,40)(41,42,43,44)(45,46,47,48), (1,17,3,19)(2,18,4,20)(5,48,7,46)(6,45,8,47)(9,39,11,37)(10,40,12,38)(13,43,15,41)(14,44,16,42)(21,33,23,35)(22,34,24,36)(25,31,27,29)(26,32,28,30), (1,37,3,39)(2,38,4,40)(5,28,7,26)(6,25,8,27)(9,17,11,19)(10,18,12,20)(13,21,15,23)(14,22,16,24)(29,47,31,45)(30,48,32,46)(33,43,35,41)(34,44,36,42), (5,28,46)(6,25,47)(7,26,48)(8,27,45)(9,17,37)(10,18,38)(11,19,39)(12,20,40)(13,21,41)(14,22,42)(15,23,43)(16,24,44) );
G=PermutationGroup([[(1,31,35),(2,32,36),(3,29,33),(4,30,34),(5,14,10),(6,15,11),(7,16,12),(8,13,9),(17,27,21),(18,28,22),(19,25,23),(20,26,24),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(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),(37,38,39,40),(41,42,43,44),(45,46,47,48)], [(1,17,3,19),(2,18,4,20),(5,48,7,46),(6,45,8,47),(9,39,11,37),(10,40,12,38),(13,43,15,41),(14,44,16,42),(21,33,23,35),(22,34,24,36),(25,31,27,29),(26,32,28,30)], [(1,37,3,39),(2,38,4,40),(5,28,7,26),(6,25,8,27),(9,17,11,19),(10,18,12,20),(13,21,15,23),(14,22,16,24),(29,47,31,45),(30,48,32,46),(33,43,35,41),(34,44,36,42)], [(5,28,46),(6,25,47),(7,26,48),(8,27,45),(9,17,37),(10,18,38),(11,19,39),(12,20,40),(13,21,41),(14,22,42),(15,23,43),(16,24,44)]])
C3×C4.A4 is a maximal subgroup of
C3⋊U2(𝔽3) SL2(𝔽3).Dic3 C12.6S4 C12.14S4 C12.7S4 Dic6.A4 D12.A4 C36.A4 C4○D4⋊He3
C3×C4.A4 is a maximal quotient of
C12×SL2(𝔽3) C36.A4 Q8⋊C9⋊4C6 C4○D4⋊He3
42 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 12A | 12B | 12C | 12D | 12E | ··· | 12P | 12Q | 12R |
| order | 1 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 6 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 6 | 1 | 1 | 4 | ··· | 4 | 6 | 6 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 6 | 6 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C4.A4 | C3×C4.A4 | A4 | C2×A4 | C3×A4 | C6×A4 |
| kernel | C3×C4.A4 | C3×SL2(𝔽3) | C4.A4 | C3×C4○D4 | SL2(𝔽3) | C3×Q8 | C3 | C1 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 6 | 12 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C4.A4 ►in GL2(𝔽13) generated by
| 9 | 0 |
| 0 | 9 |
| 8 | 0 |
| 0 | 8 |
| 0 | 4 |
| 3 | 0 |
| 5 | 0 |
| 0 | 8 |
| 12 | 7 |
| 10 | 5 |
G:=sub<GL(2,GF(13))| [9,0,0,9],[8,0,0,8],[0,3,4,0],[5,0,0,8],[12,10,7,5] >;
C3×C4.A4 in GAP, Magma, Sage, TeX
C_3\times C_4.A_4
% in TeX
G:=Group("C3xC4.A4"); // GroupNames label
G:=SmallGroup(144,157);
// by ID
G=gap.SmallGroup(144,157);
# by ID
G:=PCGroup([6,-2,-3,-3,-2,2,-2,432,441,117,820,202,88]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=e^3=1,c^2=d^2=b^2,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,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,e*d*e^-1=c>;
// generators/relations
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