direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C4⋊1D4, C12⋊6D4, C42⋊9C6, C4⋊1(C3×D4), (C2×D4)⋊3C6, C2.9(C6×D4), (C4×C12)⋊13C2, (C6×D4)⋊12C2, C6.72(C2×D4), C23.4(C2×C6), (C2×C6).82C23, (C22×C6).4C22, (C2×C12).125C22, C22.17(C22×C6), (C2×C4).23(C2×C6), SmallGroup(96,174)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊1D4
G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 180 in 108 conjugacy classes, 52 normal (8 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, D4, C23, C12, C2×C6, C2×C6, C42, C2×D4, C2×C12, C3×D4, C22×C6, C4⋊1D4, C4×C12, C6×D4, C3×C4⋊1D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C4⋊1D4, C6×D4, C3×C4⋊1D4
►On 48 points
(1 26 14)(2 27 15)(3 28 16)(4 25 13)(5 34 42)(6 35 43)(7 36 44)(8 33 41)(9 37 46)(10 38 47)(11 39 48)(12 40 45)(17 22 30)(18 23 31)(19 24 32)(20 21 29)
(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 31 10 35)(2 32 11 36)(3 29 12 33)(4 30 9 34)(5 13 22 46)(6 14 23 47)(7 15 24 48)(8 16 21 45)(17 37 42 25)(18 38 43 26)(19 39 44 27)(20 40 41 28)
(1 30)(2 29)(3 32)(4 31)(5 47)(6 46)(7 45)(8 48)(9 35)(10 34)(11 33)(12 36)(13 23)(14 22)(15 21)(16 24)(17 26)(18 25)(19 28)(20 27)(37 43)(38 42)(39 41)(40 44)
G:=sub<Sym(48)| (1,26,14)(2,27,15)(3,28,16)(4,25,13)(5,34,42)(6,35,43)(7,36,44)(8,33,41)(9,37,46)(10,38,47)(11,39,48)(12,40,45)(17,22,30)(18,23,31)(19,24,32)(20,21,29), (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), (1,31,10,35)(2,32,11,36)(3,29,12,33)(4,30,9,34)(5,13,22,46)(6,14,23,47)(7,15,24,48)(8,16,21,45)(17,37,42,25)(18,38,43,26)(19,39,44,27)(20,40,41,28), (1,30)(2,29)(3,32)(4,31)(5,47)(6,46)(7,45)(8,48)(9,35)(10,34)(11,33)(12,36)(13,23)(14,22)(15,21)(16,24)(17,26)(18,25)(19,28)(20,27)(37,43)(38,42)(39,41)(40,44)>;
G:=Group( (1,26,14)(2,27,15)(3,28,16)(4,25,13)(5,34,42)(6,35,43)(7,36,44)(8,33,41)(9,37,46)(10,38,47)(11,39,48)(12,40,45)(17,22,30)(18,23,31)(19,24,32)(20,21,29), (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), (1,31,10,35)(2,32,11,36)(3,29,12,33)(4,30,9,34)(5,13,22,46)(6,14,23,47)(7,15,24,48)(8,16,21,45)(17,37,42,25)(18,38,43,26)(19,39,44,27)(20,40,41,28), (1,30)(2,29)(3,32)(4,31)(5,47)(6,46)(7,45)(8,48)(9,35)(10,34)(11,33)(12,36)(13,23)(14,22)(15,21)(16,24)(17,26)(18,25)(19,28)(20,27)(37,43)(38,42)(39,41)(40,44) );
G=PermutationGroup([[(1,26,14),(2,27,15),(3,28,16),(4,25,13),(5,34,42),(6,35,43),(7,36,44),(8,33,41),(9,37,46),(10,38,47),(11,39,48),(12,40,45),(17,22,30),(18,23,31),(19,24,32),(20,21,29)], [(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)], [(1,31,10,35),(2,32,11,36),(3,29,12,33),(4,30,9,34),(5,13,22,46),(6,14,23,47),(7,15,24,48),(8,16,21,45),(17,37,42,25),(18,38,43,26),(19,39,44,27),(20,40,41,28)], [(1,30),(2,29),(3,32),(4,31),(5,47),(6,46),(7,45),(8,48),(9,35),(10,34),(11,33),(12,36),(13,23),(14,22),(15,21),(16,24),(17,26),(18,25),(19,28),(20,27),(37,43),(38,42),(39,41),(40,44)]])
C3×C4⋊1D4 is a maximal subgroup of
C12.9D8 C42⋊5Dic3 C12.16D8 C42.72D6 C12⋊2D8 C12⋊D8 C42.74D6 Dic6⋊9D4 C12⋊4SD16 C42⋊8D6 C42.166D6 C42⋊28D6 C42.238D6 D12⋊11D4 Dic6⋊11D4 C42.168D6 C42⋊30D6 C3×D42 C42⋊2C18
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4F | 6A | ··· | 6F | 6G | ··· | 6N | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D4 | C3×D4 |
| kernel | C3×C4⋊1D4 | C4×C12 | C6×D4 | C4⋊1D4 | C42 | C2×D4 | C12 | C4 |
| # reps | 1 | 1 | 6 | 2 | 2 | 12 | 6 | 12 |
Matrix representation of C3×C4⋊1D4 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 12 | 0 | 0 |
| 2 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 2 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[1,2,0,0,12,12,0,0,0,0,12,0,0,0,0,12],[1,2,0,0,0,12,0,0,0,0,0,1,0,0,1,0] >;
C3×C4⋊1D4 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes_1D_4
% in TeX
G:=Group("C3xC4:1D4"); // GroupNames label
G:=SmallGroup(96,174);
// by ID
G=gap.SmallGroup(96,174);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,151,938,230]);
// 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^-1,d*c*d=c^-1>;
// generators/relations