direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3xC42:C2, C42:4C6, C4:C4:6C6, C12o(C4:C4), (C2xC4):4C12, (C4xC12):2C2, (C2xC12):9C4, C4.9(C2xC12), C12o(C22:C4), C12.46(C2xC4), C22:C4.3C6, (C22xC4).6C6, C23.9(C2xC6), C6.38(C4oD4), C22.5(C2xC12), (C2xC6).72C23, C2.3(C22xC12), C6.31(C22xC4), (C22xC12).14C2, (C2xC12).79C22, C22.6(C22xC6), (C22xC6).25C22, C4o(C3xC4:C4), C12o(C3xC4:C4), C4o(C3xC22:C4), (C3xC4:C4):15C2, C12o(C3xC22:C4), C2.1(C3xC4oD4), (C2xC6).22(C2xC4), (C2xC4).14(C2xC6), (C3xC22:C4).6C2, SmallGroup(96,164)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xC42:C2
G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 92 in 76 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2xC4, C2xC4, C23, C12, C12, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C2xC12, C2xC12, C22xC6, C42:C2, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, C3xC42:C2
Quotients: C1, C2, C3, C4, C22, C6, C2xC4, C23, C12, C2xC6, C22xC4, C4oD4, C2xC12, C22xC6, C42:C2, C22xC12, C3xC4oD4, C3xC42:C2
►On 48 points
(1 35 11)(2 36 12)(3 33 9)(4 34 10)(5 30 14)(6 31 15)(7 32 16)(8 29 13)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 27 45)(22 28 46)(23 25 47)(24 26 48)
(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 23 39 15)(2 24 40 16)(3 21 37 13)(4 22 38 14)(5 34 28 42)(6 35 25 43)(7 36 26 44)(8 33 27 41)(9 45 17 29)(10 46 18 30)(11 47 19 31)(12 48 20 32)
(1 3)(2 38)(4 40)(5 26)(6 8)(7 28)(9 11)(10 20)(12 18)(13 15)(14 24)(16 22)(17 19)(21 23)(25 27)(29 31)(30 48)(32 46)(33 35)(34 44)(36 42)(37 39)(41 43)(45 47)
G:=sub<Sym(48)| (1,35,11)(2,36,12)(3,33,9)(4,34,10)(5,30,14)(6,31,15)(7,32,16)(8,29,13)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,27,45)(22,28,46)(23,25,47)(24,26,48), (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), (1,23,39,15)(2,24,40,16)(3,21,37,13)(4,22,38,14)(5,34,28,42)(6,35,25,43)(7,36,26,44)(8,33,27,41)(9,45,17,29)(10,46,18,30)(11,47,19,31)(12,48,20,32), (1,3)(2,38)(4,40)(5,26)(6,8)(7,28)(9,11)(10,20)(12,18)(13,15)(14,24)(16,22)(17,19)(21,23)(25,27)(29,31)(30,48)(32,46)(33,35)(34,44)(36,42)(37,39)(41,43)(45,47)>;
G:=Group( (1,35,11)(2,36,12)(3,33,9)(4,34,10)(5,30,14)(6,31,15)(7,32,16)(8,29,13)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,27,45)(22,28,46)(23,25,47)(24,26,48), (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), (1,23,39,15)(2,24,40,16)(3,21,37,13)(4,22,38,14)(5,34,28,42)(6,35,25,43)(7,36,26,44)(8,33,27,41)(9,45,17,29)(10,46,18,30)(11,47,19,31)(12,48,20,32), (1,3)(2,38)(4,40)(5,26)(6,8)(7,28)(9,11)(10,20)(12,18)(13,15)(14,24)(16,22)(17,19)(21,23)(25,27)(29,31)(30,48)(32,46)(33,35)(34,44)(36,42)(37,39)(41,43)(45,47) );
G=PermutationGroup([[(1,35,11),(2,36,12),(3,33,9),(4,34,10),(5,30,14),(6,31,15),(7,32,16),(8,29,13),(17,37,41),(18,38,42),(19,39,43),(20,40,44),(21,27,45),(22,28,46),(23,25,47),(24,26,48)], [(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)], [(1,23,39,15),(2,24,40,16),(3,21,37,13),(4,22,38,14),(5,34,28,42),(6,35,25,43),(7,36,26,44),(8,33,27,41),(9,45,17,29),(10,46,18,30),(11,47,19,31),(12,48,20,32)], [(1,3),(2,38),(4,40),(5,26),(6,8),(7,28),(9,11),(10,20),(12,18),(13,15),(14,24),(16,22),(17,19),(21,23),(25,27),(29,31),(30,48),(32,46),(33,35),(34,44),(36,42),(37,39),(41,43),(45,47)]])
C3xC42:C2 is a maximal subgroup of
C42:3Dic3 C12.2C42 (C2xC12).Q8 C4:C4.232D6 C4:C4.233D6 C12.5C42 C4:C4.234D6 C42.43D6 C42.187D6 C4:C4:36D6 C4.(C2xD12) C4:C4.236D6 C4:C4.237D6 C42:6D6 (C2xD12):13C4 C42.87D6 C42.88D6 C42.89D6 C42.90D6 C42:9D6 C42.188D6 C42.91D6 C42:10D6 C42:11D6 C42.92D6 C42:12D6 C42.93D6 C42.94D6 C42.95D6 C42.96D6 C42.97D6 C42.98D6 C42.99D6 C42.100D6 C12xC4oD4
C3xC42:C2 is a maximal quotient of
C12xC22:C4 C12xC4:C4
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 12A | ··· | 12H | 12I | ··· | 12AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | C4oD4 | C3xC4oD4 |
| kernel | C3xC42:C2 | C4xC12 | C3xC22:C4 | C3xC4:C4 | C22xC12 | C42:C2 | C2xC12 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xC4 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 4 | 8 |
Matrix representation of C3xC42:C2 ►in GL3(F13) generated by
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 5 | 0 | 0 |
| 0 | 8 | 11 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 1 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 5 | 1 |
G:=sub<GL(3,GF(13))| [1,0,0,0,3,0,0,0,3],[5,0,0,0,8,0,0,11,5],[1,0,0,0,5,0,0,0,5],[1,0,0,0,12,5,0,0,1] >;
C3xC42:C2 in GAP, Magma, Sage, TeX
C_3\times C_4^2\rtimes C_2
% in TeX
G:=Group("C3xC4^2:C2"); // GroupNames label
G:=SmallGroup(96,164);
// by ID
G=gap.SmallGroup(96,164);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,122]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations