direct product, metabelian, supersoluble, monomial
Aliases: S3×C24⋊C2, C24⋊17D6, Dic6⋊7D6, D12.16D6, D6.10D12, Dic3.1D12, C8⋊5S32, C3⋊C8⋊21D6, (S3×C8)⋊4S3, C6.1(S3×D4), (S3×C24)⋊9C2, C3⋊1(S3×SD16), C2.6(S3×D12), C6.1(C2×D12), C24⋊2S3⋊7C2, (S3×Dic6)⋊1C2, (C3×S3)⋊1SD16, (S3×C6).17D4, (S3×D12).1C2, (C4×S3).33D6, C32⋊4(C2×SD16), (C3×C24)⋊14C22, C32⋊5SD16⋊5C2, D12.S3⋊5C2, C12.66(C22×S3), (C3×C12).40C23, (C3×Dic3).20D4, (C3×Dic6)⋊1C22, (C3×D12).1C22, C32⋊4Q8⋊1C22, (S3×C12).41C22, C12⋊S3.1C22, C4.40(C2×S32), C3⋊1(C2×C24⋊C2), (C3×C24⋊C2)⋊9C2, (C3×C3⋊C8)⋊28C22, (C3×C6).24(C2×D4), SmallGroup(288,440)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C24⋊C2
G = < a,b,c,d | a3=b2=c24=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c11 >
Subgroups: 746 in 146 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3×S3, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C2×SD16, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, C24⋊C2, D4.S3, Q8⋊2S3, C2×C24, C3×SD16, C2×Dic6, C2×D12, S3×D4, S3×Q8, C3×C3⋊C8, C3×C24, S3×Dic3, C3⋊D12, C32⋊2Q8, C3×Dic6, S3×C12, C3×D12, C32⋊4Q8, C12⋊S3, C2×S32, C2×C24⋊C2, S3×SD16, D12.S3, C32⋊5SD16, S3×C24, C3×C24⋊C2, C24⋊2S3, S3×Dic6, S3×D12, S3×C24⋊C2
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, D12, C22×S3, C2×SD16, S32, C24⋊C2, C2×D12, S3×D4, C2×S32, C2×C24⋊C2, S3×SD16, S3×D12, S3×C24⋊C2
►On 48 points
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 25)(21 26)(22 27)(23 28)(24 29)
(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 30)(2 41)(3 28)(4 39)(5 26)(6 37)(7 48)(8 35)(9 46)(10 33)(11 44)(12 31)(13 42)(14 29)(15 40)(16 27)(17 38)(18 25)(19 36)(20 47)(21 34)(22 45)(23 32)(24 43)
G:=sub<Sym(48)| (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,25)(21,26)(22,27)(23,28)(24,29), (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), (1,30)(2,41)(3,28)(4,39)(5,26)(6,37)(7,48)(8,35)(9,46)(10,33)(11,44)(12,31)(13,42)(14,29)(15,40)(16,27)(17,38)(18,25)(19,36)(20,47)(21,34)(22,45)(23,32)(24,43)>;
G:=Group( (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,25)(21,26)(22,27)(23,28)(24,29), (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), (1,30)(2,41)(3,28)(4,39)(5,26)(6,37)(7,48)(8,35)(9,46)(10,33)(11,44)(12,31)(13,42)(14,29)(15,40)(16,27)(17,38)(18,25)(19,36)(20,47)(21,34)(22,45)(23,32)(24,43) );
G=PermutationGroup([[(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,25),(21,26),(22,27),(23,28),(24,29)], [(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)], [(1,30),(2,41),(3,28),(4,39),(5,26),(6,37),(7,48),(8,35),(9,46),(10,33),(11,44),(12,31),(13,42),(14,29),(15,40),(16,27),(17,38),(18,25),(19,36),(20,47),(21,34),(22,45),(23,32),(24,43)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 3 | 3 | 12 | 36 | 2 | 2 | 4 | 2 | 6 | 12 | 36 | 2 | 2 | 4 | 6 | 6 | 24 | 2 | 2 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 24 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | D6 | SD16 | D12 | D12 | C24⋊C2 | S32 | S3×D4 | C2×S32 | S3×SD16 | S3×D12 | S3×C24⋊C2 |
| kernel | S3×C24⋊C2 | D12.S3 | C32⋊5SD16 | S3×C24 | C3×C24⋊C2 | C24⋊2S3 | S3×Dic6 | S3×D12 | S3×C8 | C24⋊C2 | C3×Dic3 | S3×C6 | C3⋊C8 | C24 | Dic6 | C4×S3 | D12 | C3×S3 | Dic3 | D6 | S3 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of S3×C24⋊C2 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 67 | 6 | 0 | 0 | 0 | 0 |
| 67 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
S3×C24⋊C2 in GAP, Magma, Sage, TeX
S_3\times C_{24}\rtimes C_2 % in TeX
G:=Group("S3xC24:C2"); // GroupNames label
G:=SmallGroup(288,440);
// by ID
G=gap.SmallGroup(288,440);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,58,346,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^24=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^11>;
// generators/relations