direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C4.4D4, C42⋊8C6, C12.39D4, (C2×Q8)⋊4C6, (C6×Q8)⋊9C2, C2.8(C6×D4), C4.4(C3×D4), (C4×C12)⋊12C2, C22⋊C4⋊5C6, (C2×D4).5C6, C6.71(C2×D4), (C6×D4).12C2, C23.2(C2×C6), C6.44(C4○D4), (C2×C6).79C23, (C2×C12).66C22, (C22×C6).2C22, C22.14(C22×C6), (C2×C4).6(C2×C6), C2.7(C3×C4○D4), (C3×C22⋊C4)⋊13C2, SmallGroup(96,171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.4D4
G = < a,b,c,d | a3=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 116 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4.4D4, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C3×C4.4D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4.4D4, C6×D4, C3×C4○D4, C3×C4.4D4
►On 48 points
(1 39 47)(2 40 48)(3 37 45)(4 38 46)(5 33 41)(6 34 42)(7 35 43)(8 36 44)(9 25 15)(10 26 16)(11 27 13)(12 28 14)(17 21 31)(18 22 32)(19 23 29)(20 24 30)
(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 34 13 17)(2 35 14 18)(3 36 15 19)(4 33 16 20)(5 26 30 46)(6 27 31 47)(7 28 32 48)(8 25 29 45)(9 23 37 44)(10 24 38 41)(11 21 39 42)(12 22 40 43)
(1 18 3 20)(2 17 4 19)(5 27 7 25)(6 26 8 28)(9 41 11 43)(10 44 12 42)(13 35 15 33)(14 34 16 36)(21 38 23 40)(22 37 24 39)(29 48 31 46)(30 47 32 45)
G:=sub<Sym(48)| (1,39,47)(2,40,48)(3,37,45)(4,38,46)(5,33,41)(6,34,42)(7,35,43)(8,36,44)(9,25,15)(10,26,16)(11,27,13)(12,28,14)(17,21,31)(18,22,32)(19,23,29)(20,24,30), (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), (1,34,13,17)(2,35,14,18)(3,36,15,19)(4,33,16,20)(5,26,30,46)(6,27,31,47)(7,28,32,48)(8,25,29,45)(9,23,37,44)(10,24,38,41)(11,21,39,42)(12,22,40,43), (1,18,3,20)(2,17,4,19)(5,27,7,25)(6,26,8,28)(9,41,11,43)(10,44,12,42)(13,35,15,33)(14,34,16,36)(21,38,23,40)(22,37,24,39)(29,48,31,46)(30,47,32,45)>;
G:=Group( (1,39,47)(2,40,48)(3,37,45)(4,38,46)(5,33,41)(6,34,42)(7,35,43)(8,36,44)(9,25,15)(10,26,16)(11,27,13)(12,28,14)(17,21,31)(18,22,32)(19,23,29)(20,24,30), (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), (1,34,13,17)(2,35,14,18)(3,36,15,19)(4,33,16,20)(5,26,30,46)(6,27,31,47)(7,28,32,48)(8,25,29,45)(9,23,37,44)(10,24,38,41)(11,21,39,42)(12,22,40,43), (1,18,3,20)(2,17,4,19)(5,27,7,25)(6,26,8,28)(9,41,11,43)(10,44,12,42)(13,35,15,33)(14,34,16,36)(21,38,23,40)(22,37,24,39)(29,48,31,46)(30,47,32,45) );
G=PermutationGroup([[(1,39,47),(2,40,48),(3,37,45),(4,38,46),(5,33,41),(6,34,42),(7,35,43),(8,36,44),(9,25,15),(10,26,16),(11,27,13),(12,28,14),(17,21,31),(18,22,32),(19,23,29),(20,24,30)], [(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)], [(1,34,13,17),(2,35,14,18),(3,36,15,19),(4,33,16,20),(5,26,30,46),(6,27,31,47),(7,28,32,48),(8,25,29,45),(9,23,37,44),(10,24,38,41),(11,21,39,42),(12,22,40,43)], [(1,18,3,20),(2,17,4,19),(5,27,7,25),(6,26,8,28),(9,41,11,43),(10,44,12,42),(13,35,15,33),(14,34,16,36),(21,38,23,40),(22,37,24,39),(29,48,31,46),(30,47,32,45)]])
C3×C4.4D4 is a maximal subgroup of
C42.7D6 C42⋊4Dic3 C42.Dic3 C42.61D6 C42.62D6 C42.213D6 D12.23D4 C42.64D6 C42.214D6 C42.65D6 C42⋊7D6 D12.14D4 C42.233D6 C42.137D6 C42.138D6 C42.139D6 C42.140D6 C42⋊20D6 C42.141D6 D12⋊10D4 Dic6⋊10D4 C42⋊22D6 C42⋊23D6 C42.234D6 C42.143D6 C42.144D6 C42⋊24D6 C42.145D6
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 12A | ··· | 12L | 12M | 12N | 12O | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | ··· | 1 | 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 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | D4 | C4○D4 | C3×D4 | C3×C4○D4 |
| kernel | C3×C4.4D4 | C4×C12 | C3×C22⋊C4 | C6×D4 | C6×Q8 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of C3×C4.4D4 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 3 | 0 | 0 |
| 8 | 12 | 0 | 0 |
| 0 | 0 | 8 | 10 |
| 0 | 0 | 0 | 5 |
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 11 |
| 0 | 0 | 0 | 12 |
| 8 | 11 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 8 | 5 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[1,8,0,0,3,12,0,0,0,0,8,0,0,0,10,5],[8,0,0,0,0,8,0,0,0,0,1,0,0,0,11,12],[8,0,0,0,11,5,0,0,0,0,8,8,0,0,0,5] >;
C3×C4.4D4 in GAP, Magma, Sage, TeX
C_3\times C_4._4D_4
% in TeX
G:=Group("C3xC4.4D4"); // GroupNames label
G:=SmallGroup(96,171);
// by ID
G=gap.SmallGroup(96,171);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,295,938,122]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations