direct product, metacyclic, supersoluble, monomial
Aliases: C3×C24⋊C2, C24⋊2C6, C24⋊6S3, Dic6⋊1C6, D12.1C6, C6.19D12, C12.60D6, C32⋊6SD16, C8⋊2(C3×S3), (C3×C24)⋊4C2, C4.8(S3×C6), C6.1(C3×D4), C12.8(C2×C6), C3⋊1(C3×SD16), (C3×C6).17D4, C2.3(C3×D12), (C3×D12).4C2, (C3×Dic6)⋊10C2, (C3×C12).37C22, SmallGroup(144,71)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24⋊C2
G = < a,b,c | a3=b24=c2=1, ab=ba, ac=ca, cbc=b11 >
Smallest permutation representation of C3×C24⋊C2
►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 37)(2 48)(3 35)(4 46)(5 33)(6 44)(7 31)(8 42)(9 29)(10 40)(11 27)(12 38)(13 25)(14 36)(15 47)(16 34)(17 45)(18 32)(19 43)(20 30)(21 41)(22 28)(23 39)(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), (1,37)(2,48)(3,35)(4,46)(5,33)(6,44)(7,31)(8,42)(9,29)(10,40)(11,27)(12,38)(13,25)(14,36)(15,47)(16,34)(17,45)(18,32)(19,43)(20,30)(21,41)(22,28)(23,39)(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), (1,37)(2,48)(3,35)(4,46)(5,33)(6,44)(7,31)(8,42)(9,29)(10,40)(11,27)(12,38)(13,25)(14,36)(15,47)(16,34)(17,45)(18,32)(19,43)(20,30)(21,41)(22,28)(23,39)(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)], [(1,37),(2,48),(3,35),(4,46),(5,33),(6,44),(7,31),(8,42),(9,29),(10,40),(11,27),(12,38),(13,25),(14,36),(15,47),(16,34),(17,45),(18,32),(19,43),(20,30),(21,41),(22,28),(23,39),(24,26)]])
C3×C24⋊C2 is a maximal subgroup of
C24⋊1D6 C24⋊9D6 C24⋊6D6 C24.3D6 D6.1D12 D12.2D6 D12.4D6 C3×S3×SD16 He3⋊6SD16 C72⋊2C6 He3⋊7SD16
C3×C24⋊C2 is a maximal quotient of
He3⋊6SD16 C72⋊2C6
45 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 12A | ··· | 12H | 12I | 12J | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 1 | 1 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 2 | ··· | 2 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | SD16 | C3×S3 | D12 | C3×D4 | S3×C6 | C24⋊C2 | C3×SD16 | C3×D12 | C3×C24⋊C2 |
| kernel | C3×C24⋊C2 | C3×C24 | C3×Dic6 | C3×D12 | C24⋊C2 | C24 | Dic6 | D12 | C24 | C3×C6 | C12 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C3×C24⋊C2 ►in GL2(𝔽73) generated by
| 64 | 0 |
| 0 | 64 |
| 7 | 0 |
| 0 | 52 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(73))| [64,0,0,64],[7,0,0,52],[0,1,1,0] >;
C3×C24⋊C2 in GAP, Magma, Sage, TeX
C_3\times C_{24}\rtimes C_2 % in TeX
G:=Group("C3xC24:C2"); // GroupNames label
G:=SmallGroup(144,71);
// by ID
G=gap.SmallGroup(144,71);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,169,79,867,69,3461]);
// 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^11>;
// generators/relations
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