direct product, metacyclic, supersoluble, monomial
Aliases: C3×Dic12, C24.1C6, C24.5S3, C32⋊4Q16, C12.62D6, C6.21D12, Dic6.1C6, C8.(C3×S3), C3⋊1(C3×Q16), C6.3(C3×D4), C4.10(S3×C6), (C3×C24).2C2, (C3×C6).19D4, C2.5(C3×D12), C12.10(C2×C6), (C3×Dic6).4C2, (C3×C12).39C22, SmallGroup(144,73)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic12
G = < a,b,c | a3=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic12
►On 48 points
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(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 25 13 37)(2 48 14 36)(3 47 15 35)(4 46 16 34)(5 45 17 33)(6 44 18 32)(7 43 19 31)(8 42 20 30)(9 41 21 29)(10 40 22 28)(11 39 23 27)(12 38 24 26)
G:=sub<Sym(48)| (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(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), (1,25,13,37)(2,48,14,36)(3,47,15,35)(4,46,16,34)(5,45,17,33)(6,44,18,32)(7,43,19,31)(8,42,20,30)(9,41,21,29)(10,40,22,28)(11,39,23,27)(12,38,24,26)>;
G:=Group( (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(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), (1,25,13,37)(2,48,14,36)(3,47,15,35)(4,46,16,34)(5,45,17,33)(6,44,18,32)(7,43,19,31)(8,42,20,30)(9,41,21,29)(10,40,22,28)(11,39,23,27)(12,38,24,26) );
G=PermutationGroup([(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(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)], [(1,25,13,37),(2,48,14,36),(3,47,15,35),(4,46,16,34),(5,45,17,33),(6,44,18,32),(7,43,19,31),(8,42,20,30),(9,41,21,29),(10,40,22,28),(11,39,23,27),(12,38,24,26)])
C3×Dic12 is a maximal subgroup of
D24.S3 C24.49D6 C32⋊2Q32 C32⋊3Q32 Dic12⋊S3 C24.23D6 D6.3D12 D24⋊5S3 D12.4D6 C3×S3×Q16 He3⋊4Q16 C72.C6 He3⋊5Q16
C3×Dic12 is a maximal quotient of
He3⋊4Q16 C72.C6
45 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | Q16 | C3×S3 | D12 | C3×D4 | S3×C6 | Dic12 | C3×Q16 | C3×D12 | C3×Dic12 |
| kernel | C3×Dic12 | C3×C24 | C3×Dic6 | Dic12 | C24 | Dic6 | C24 | C3×C6 | C12 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C3×Dic12 ►in GL2(𝔽73) generated by
| 64 | 0 |
| 0 | 64 |
| 43 | 0 |
| 0 | 17 |
| 0 | 1 |
| 72 | 0 |
G:=sub<GL(2,GF(73))| [64,0,0,64],[43,0,0,17],[0,72,1,0] >;
C3×Dic12 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{12} % in TeX
G:=Group("C3xDic12"); // GroupNames label
G:=SmallGroup(144,73);
// by ID
G=gap.SmallGroup(144,73);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,223,867,69,3461]);
// Polycyclic
G:=Group<a,b,c|a^3=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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