direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C4⋊D4, C12⋊9D4, C4⋊C4⋊2C6, C4⋊2(C3×D4), (C2×C6)⋊4D4, (C2×D4)⋊2C6, C2.5(C6×D4), (C6×D4)⋊11C2, C22⋊C4⋊3C6, (C22×C4)⋊6C6, C6.68(C2×D4), C22⋊2(C3×D4), (C22×C12)⋊11C2, C6.41(C4○D4), (C2×C6).76C23, C23.11(C2×C6), (C2×C12).123C22, C22.11(C22×C6), (C22×C6).27C22, (C3×C4⋊C4)⋊11C2, (C2×C4).3(C2×C6), C2.4(C3×C4○D4), (C3×C22⋊C4)⋊11C2, SmallGroup(96,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4⋊D4
G = < a,b,c,d | a3=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 148 in 94 conjugacy classes, 48 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C4⋊D4, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×D4, C3×C4⋊D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4⋊D4, C6×D4, C3×C4○D4, C3×C4⋊D4
►On 48 points
(1 11 31)(2 12 32)(3 9 29)(4 10 30)(5 13 33)(6 14 34)(7 15 35)(8 16 36)(17 37 41)(18 38 42)(19 39 43)(20 40 44)(21 28 45)(22 25 46)(23 26 47)(24 27 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 35 39 23)(2 34 40 22)(3 33 37 21)(4 36 38 24)(5 41 28 9)(6 44 25 12)(7 43 26 11)(8 42 27 10)(13 17 45 29)(14 20 46 32)(15 19 47 31)(16 18 48 30)
(2 4)(5 28)(6 27)(7 26)(8 25)(10 12)(13 45)(14 48)(15 47)(16 46)(18 20)(21 33)(22 36)(23 35)(24 34)(30 32)(38 40)(42 44)
G:=sub<Sym(48)| (1,11,31)(2,12,32)(3,9,29)(4,10,30)(5,13,33)(6,14,34)(7,15,35)(8,16,36)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,28,45)(22,25,46)(23,26,47)(24,27,48), (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), (1,35,39,23)(2,34,40,22)(3,33,37,21)(4,36,38,24)(5,41,28,9)(6,44,25,12)(7,43,26,11)(8,42,27,10)(13,17,45,29)(14,20,46,32)(15,19,47,31)(16,18,48,30), (2,4)(5,28)(6,27)(7,26)(8,25)(10,12)(13,45)(14,48)(15,47)(16,46)(18,20)(21,33)(22,36)(23,35)(24,34)(30,32)(38,40)(42,44)>;
G:=Group( (1,11,31)(2,12,32)(3,9,29)(4,10,30)(5,13,33)(6,14,34)(7,15,35)(8,16,36)(17,37,41)(18,38,42)(19,39,43)(20,40,44)(21,28,45)(22,25,46)(23,26,47)(24,27,48), (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), (1,35,39,23)(2,34,40,22)(3,33,37,21)(4,36,38,24)(5,41,28,9)(6,44,25,12)(7,43,26,11)(8,42,27,10)(13,17,45,29)(14,20,46,32)(15,19,47,31)(16,18,48,30), (2,4)(5,28)(6,27)(7,26)(8,25)(10,12)(13,45)(14,48)(15,47)(16,46)(18,20)(21,33)(22,36)(23,35)(24,34)(30,32)(38,40)(42,44) );
G=PermutationGroup([[(1,11,31),(2,12,32),(3,9,29),(4,10,30),(5,13,33),(6,14,34),(7,15,35),(8,16,36),(17,37,41),(18,38,42),(19,39,43),(20,40,44),(21,28,45),(22,25,46),(23,26,47),(24,27,48)], [(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)], [(1,35,39,23),(2,34,40,22),(3,33,37,21),(4,36,38,24),(5,41,28,9),(6,44,25,12),(7,43,26,11),(8,42,27,10),(13,17,45,29),(14,20,46,32),(15,19,47,31),(16,18,48,30)], [(2,4),(5,28),(6,27),(7,26),(8,25),(10,12),(13,45),(14,48),(15,47),(16,46),(18,20),(21,33),(22,36),(23,35),(24,34),(30,32),(38,40),(42,44)]])
C3×C4⋊D4 is a maximal subgroup of
(C6×D4)⋊C4 (C2×C6).D8 C4⋊D4.S3 C6.Q16⋊C2 D12⋊16D4 D12⋊17D4 C3⋊C8⋊22D4 C4⋊D4⋊S3 Dic6⋊17D4 C3⋊C8⋊23D4 C3⋊C8⋊5D4 C12⋊(C4○D4) C6.322+ 1+4 Dic6⋊19D4 Dic6⋊20D4 C4⋊C4.178D6 C6.342+ 1+4 C6.702- 1+4 C6.712- 1+4 C6.372+ 1+4 C4⋊C4⋊21D6 C6.382+ 1+4 C6.722- 1+4 D12⋊19D4 C6.402+ 1+4 C6.732- 1+4 D12⋊20D4 C6.422+ 1+4 C6.432+ 1+4 C6.442+ 1+4 C6.452+ 1+4 C6.462+ 1+4 C6.1152+ 1+4 C6.472+ 1+4 C6.482+ 1+4 C6.492+ 1+4 C3×D42
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 12A | ··· | 12H | 12I | 12J | 12K | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | D4 | C4○D4 | C3×D4 | C3×D4 | C3×C4○D4 |
| kernel | C3×C4⋊D4 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 2 | 2 | 6 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C3×C4⋊D4 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 |
| 4 | 8 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 9 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 4 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[5,4,0,0,0,8,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,9,1,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,4,12,0,0,0,0,1,0,0,0,0,12] >;
C3×C4⋊D4 in GAP, Magma, Sage, TeX
C_3\times C_4\rtimes D_4
% in TeX
G:=Group("C3xC4:D4"); // GroupNames label
G:=SmallGroup(96,168);
// by ID
G=gap.SmallGroup(96,168);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,151,938]);
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations