direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×D4⋊C4, D4⋊1C12, C6.13D8, C12.60D4, C6.9SD16, C4⋊C4⋊1C6, (C2×C8)⋊2C6, (C2×C24)⋊4C2, (C3×D4)⋊4C4, C2.1(C3×D8), C4.1(C2×C12), (C2×D4).3C6, (C6×D4).9C2, C4.11(C3×D4), (C2×C6).46D4, C12.28(C2×C4), C2.1(C3×SD16), C22.8(C3×D4), C6.24(C22⋊C4), (C2×C12).114C22, (C3×C4⋊C4)⋊10C2, (C2×C4).17(C2×C6), C2.6(C3×C22⋊C4), SmallGroup(96,52)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊C4
G = < a,b,c,d | a3=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >
Smallest permutation representation of C3×D4⋊C4
►On 48 points
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 42 34)(6 43 35)(7 44 36)(8 41 33)(9 28 17)(10 25 18)(11 26 19)(12 27 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(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 10)(2 9)(3 12)(4 11)(5 48)(6 47)(7 46)(8 45)(13 17)(14 20)(15 19)(16 18)(21 28)(22 27)(23 26)(24 25)(29 33)(30 36)(31 35)(32 34)(37 41)(38 44)(39 43)(40 42)
(1 36 11 32)(2 35 12 31)(3 34 9 30)(4 33 10 29)(5 28 46 22)(6 27 47 21)(7 26 48 24)(8 25 45 23)(13 43 20 39)(14 42 17 38)(15 41 18 37)(16 44 19 40)
G:=sub<Sym(48)| (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,42,34)(6,43,35)(7,44,36)(8,41,33)(9,28,17)(10,25,18)(11,26,19)(12,27,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (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), (1,10)(2,9)(3,12)(4,11)(5,48)(6,47)(7,46)(8,45)(13,17)(14,20)(15,19)(16,18)(21,28)(22,27)(23,26)(24,25)(29,33)(30,36)(31,35)(32,34)(37,41)(38,44)(39,43)(40,42), (1,36,11,32)(2,35,12,31)(3,34,9,30)(4,33,10,29)(5,28,46,22)(6,27,47,21)(7,26,48,24)(8,25,45,23)(13,43,20,39)(14,42,17,38)(15,41,18,37)(16,44,19,40)>;
G:=Group( (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,42,34)(6,43,35)(7,44,36)(8,41,33)(9,28,17)(10,25,18)(11,26,19)(12,27,20)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (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), (1,10)(2,9)(3,12)(4,11)(5,48)(6,47)(7,46)(8,45)(13,17)(14,20)(15,19)(16,18)(21,28)(22,27)(23,26)(24,25)(29,33)(30,36)(31,35)(32,34)(37,41)(38,44)(39,43)(40,42), (1,36,11,32)(2,35,12,31)(3,34,9,30)(4,33,10,29)(5,28,46,22)(6,27,47,21)(7,26,48,24)(8,25,45,23)(13,43,20,39)(14,42,17,38)(15,41,18,37)(16,44,19,40) );
G=PermutationGroup([[(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,42,34),(6,43,35),(7,44,36),(8,41,33),(9,28,17),(10,25,18),(11,26,19),(12,27,20),(29,45,37),(30,46,38),(31,47,39),(32,48,40)], [(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)], [(1,10),(2,9),(3,12),(4,11),(5,48),(6,47),(7,46),(8,45),(13,17),(14,20),(15,19),(16,18),(21,28),(22,27),(23,26),(24,25),(29,33),(30,36),(31,35),(32,34),(37,41),(38,44),(39,43),(40,42)], [(1,36,11,32),(2,35,12,31),(3,34,9,30),(4,33,10,29),(5,28,46,22),(6,27,47,21),(7,26,48,24),(8,25,45,23),(13,43,20,39),(14,42,17,38),(15,41,18,37),(16,44,19,40)]])
C3×D4⋊C4 is a maximal subgroup of
Dic3⋊4D8 D4.S3⋊C4 Dic3⋊6SD16 Dic3.D8 Dic3.SD16 D4⋊Dic6 Dic6⋊2D4 D4.Dic6 C4⋊C4.D6 C12⋊Q8⋊C2 D4.2Dic6 Dic6.D4 (C2×C8).200D6 C4⋊C4⋊19D6 D4⋊(C4×S3) D4⋊2S3⋊C4 D4⋊D12 D6.D8 D6⋊D8 D6⋊5SD16 D6.SD16 D6⋊SD16 D6⋊C8⋊11C2 C3⋊C8⋊1D4 D4⋊3D12 C3⋊C8⋊D4 D4.D12 C24⋊1C4⋊C2 D4⋊S3⋊C4 D12⋊3D4 D12.D4 C12×D8 C12×SD16
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | D8 | SD16 | C3×D4 | C3×D4 | C3×D8 | C3×SD16 |
| kernel | C3×D4⋊C4 | C3×C4⋊C4 | C2×C24 | C6×D4 | D4⋊C4 | C3×D4 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C12 | C2×C6 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C3×D4⋊C4 ►in GL3(𝔽73) generated by
| 1 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 72 | 72 |
| 0 | 2 | 1 |
| 1 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 2 | 1 |
| 27 | 0 | 0 |
| 0 | 0 | 67 |
| 0 | 61 | 0 |
G:=sub<GL(3,GF(73))| [1,0,0,0,64,0,0,0,64],[1,0,0,0,72,2,0,72,1],[1,0,0,0,72,2,0,0,1],[27,0,0,0,0,61,0,67,0] >;
C3×D4⋊C4 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes C_4
% in TeX
G:=Group("C3xD4:C4"); // GroupNames label
G:=SmallGroup(96,52);
// by ID
G=gap.SmallGroup(96,52);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,1443,729,117]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations
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