direct product, metacyclic, supersoluble, monomial
Aliases: C3×D24, C24⋊1C6, C24⋊3S3, C32⋊4D8, D12⋊1C6, C12.61D6, C6.20D12, C8⋊1(C3×S3), C3⋊1(C3×D8), (C3×C24)⋊2C2, C4.9(S3×C6), C6.2(C3×D4), C12.9(C2×C6), C2.4(C3×D12), (C3×C6).18D4, (C3×D12)⋊10C2, (C3×C12).38C22, SmallGroup(144,72)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D24
G = < a,b,c | a3=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D24
►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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)>;
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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25) );
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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,40),(10,39),(11,38),(12,37),(13,36),(14,35),(15,34),(16,33),(17,32),(18,31),(19,30),(20,29),(21,28),(22,27),(23,26),(24,25)]])
C3×D24 is a maximal subgroup of
C32⋊2D16 C3⋊D48 D24.S3 C32⋊3SD32 D24⋊S3 C24⋊4D6 C24⋊6D6 D24⋊7S3 D24⋊5S3 C3×S3×D8 He3⋊4D8 D72⋊C3 He3⋊5D8
C3×D24 is a maximal quotient of
He3⋊4D8 D72⋊C3
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | ··· | 2 | 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 | D8 | C3×S3 | D12 | C3×D4 | S3×C6 | D24 | C3×D8 | C3×D12 | C3×D24 |
| kernel | C3×D24 | C3×C24 | C3×D12 | D24 | C24 | D12 | 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×D24 ►in GL2(𝔽73) generated by
| 64 | 0 |
| 0 | 64 |
| 30 | 12 |
| 0 | 56 |
| 30 | 12 |
| 65 | 43 |
G:=sub<GL(2,GF(73))| [64,0,0,64],[30,0,12,56],[30,65,12,43] >;
C3×D24 in GAP, Magma, Sage, TeX
C_3\times D_{24} % in TeX
G:=Group("C3xD24"); // GroupNames label
G:=SmallGroup(144,72);
// by ID
G=gap.SmallGroup(144,72);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,169,223,867,69,3461]);
// Polycyclic
G:=Group<a,b,c|a^3=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export