direct product, metabelian, supersoluble, monomial
Aliases: C3×D8⋊S3, C24⋊12D6, C8⋊2(S3×C6), C24⋊4(C2×C6), D4⋊S3⋊2C6, D4⋊2(S3×C6), D8⋊2(C3×S3), (C3×D8)⋊6S3, (S3×D4)⋊2C6, (C3×D8)⋊4C6, C8⋊S3⋊3C6, C24⋊C2⋊3C6, (C3×D4)⋊13D6, D4.S3⋊1C6, D6.6(C3×D4), C6.28(C6×D4), D4⋊2S3⋊1C6, Dic6⋊1(C2×C6), (S3×C6).42D4, D12.1(C2×C6), C6.188(S3×D4), (C32×D8)⋊8C2, (C3×C24)⋊13C22, C12.2(C22×C6), Dic3.8(C3×D4), C32⋊18(C8⋊C22), (C3×C12).73C23, (C3×Dic3).45D4, (D4×C32)⋊6C22, (S3×C12).26C22, C12.153(C22×S3), (C3×Dic6)⋊11C22, (C3×D12).25C22, C3⋊C8⋊1(C2×C6), (C3×S3×D4)⋊5C2, C4.2(S3×C2×C6), C2.16(C3×S3×D4), (C3×D4)⋊2(C2×C6), C3⋊2(C3×C8⋊C22), (C3×C8⋊S3)⋊7C2, (C3×C24⋊C2)⋊7C2, (C3×D4⋊S3)⋊11C2, (C3×C3⋊C8)⋊18C22, (C4×S3).1(C2×C6), (C3×D4.S3)⋊9C2, (C3×D4⋊2S3)⋊4C2, (C3×C6).216(C2×D4), SmallGroup(288,682)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D8⋊S3
G = < a,b,c,d,e | a3=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >
Subgroups: 426 in 147 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, M4(2), D8, D8, SD16, C2×D4, C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C22×C6, C8⋊C22, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×M4(2), C3×D8, C3×D8, C3×SD16, S3×D4, D4⋊2S3, C6×D4, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, D4×C32, S3×C2×C6, D8⋊S3, C3×C8⋊C22, C3×C8⋊S3, C3×C24⋊C2, C3×D4⋊S3, C3×D4.S3, C32×D8, C3×S3×D4, C3×D4⋊2S3, C3×D8⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C8⋊C22, S3×C6, S3×D4, C6×D4, S3×C2×C6, D8⋊S3, C3×C8⋊C22, C3×S3×D4, C3×D8⋊S3
►On 48 points
(1 37 32)(2 38 25)(3 39 26)(4 40 27)(5 33 28)(6 34 29)(7 35 30)(8 36 31)(9 21 41)(10 22 42)(11 23 43)(12 24 44)(13 17 45)(14 18 46)(15 19 47)(16 20 48)
(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)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 19)(20 24)(21 23)(25 31)(26 30)(27 29)(34 40)(35 39)(36 38)(41 43)(44 48)(45 47)
(1 32 37)(2 25 38)(3 26 39)(4 27 40)(5 28 33)(6 29 34)(7 30 35)(8 31 36)(9 21 41)(10 22 42)(11 23 43)(12 24 44)(13 17 45)(14 18 46)(15 19 47)(16 20 48)
(1 18)(2 23)(3 20)(4 17)(5 22)(6 19)(7 24)(8 21)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)(33 42)(34 47)(35 44)(36 41)(37 46)(38 43)(39 48)(40 45)
G:=sub<Sym(48)| (1,37,32)(2,38,25)(3,39,26)(4,40,27)(5,33,28)(6,34,29)(7,35,30)(8,36,31)(9,21,41)(10,22,42)(11,23,43)(12,24,44)(13,17,45)(14,18,46)(15,19,47)(16,20,48), (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,31)(26,30)(27,29)(34,40)(35,39)(36,38)(41,43)(44,48)(45,47), (1,32,37)(2,25,38)(3,26,39)(4,27,40)(5,28,33)(6,29,34)(7,30,35)(8,31,36)(9,21,41)(10,22,42)(11,23,43)(12,24,44)(13,17,45)(14,18,46)(15,19,47)(16,20,48), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(33,42)(34,47)(35,44)(36,41)(37,46)(38,43)(39,48)(40,45)>;
G:=Group( (1,37,32)(2,38,25)(3,39,26)(4,40,27)(5,33,28)(6,34,29)(7,35,30)(8,36,31)(9,21,41)(10,22,42)(11,23,43)(12,24,44)(13,17,45)(14,18,46)(15,19,47)(16,20,48), (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), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,19)(20,24)(21,23)(25,31)(26,30)(27,29)(34,40)(35,39)(36,38)(41,43)(44,48)(45,47), (1,32,37)(2,25,38)(3,26,39)(4,27,40)(5,28,33)(6,29,34)(7,30,35)(8,31,36)(9,21,41)(10,22,42)(11,23,43)(12,24,44)(13,17,45)(14,18,46)(15,19,47)(16,20,48), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,31)(10,28)(11,25)(12,30)(13,27)(14,32)(15,29)(16,26)(33,42)(34,47)(35,44)(36,41)(37,46)(38,43)(39,48)(40,45) );
G=PermutationGroup([[(1,37,32),(2,38,25),(3,39,26),(4,40,27),(5,33,28),(6,34,29),(7,35,30),(8,36,31),(9,21,41),(10,22,42),(11,23,43),(12,24,44),(13,17,45),(14,18,46),(15,19,47),(16,20,48)], [(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)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,19),(20,24),(21,23),(25,31),(26,30),(27,29),(34,40),(35,39),(36,38),(41,43),(44,48),(45,47)], [(1,32,37),(2,25,38),(3,26,39),(4,27,40),(5,28,33),(6,29,34),(7,30,35),(8,31,36),(9,21,41),(10,22,42),(11,23,43),(12,24,44),(13,17,45),(14,18,46),(15,19,47),(16,20,48)], [(1,18),(2,23),(3,20),(4,17),(5,22),(6,19),(7,24),(8,21),(9,31),(10,28),(11,25),(12,30),(13,27),(14,32),(15,29),(16,26),(33,42),(34,47),(35,44),(36,41),(37,46),(38,43),(39,48),(40,45)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | ··· | 6Q | 6R | 6S | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 24A | ··· | 24H | 24I | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | 24 |
| size | 1 | 1 | 4 | 4 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 4 | 12 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 4 | ··· | 4 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | C3×S3 | C3×D4 | C3×D4 | S3×C6 | S3×C6 | C8⋊C22 | S3×D4 | D8⋊S3 | C3×C8⋊C22 | C3×S3×D4 | C3×D8⋊S3 |
| kernel | C3×D8⋊S3 | C3×C8⋊S3 | C3×C24⋊C2 | C3×D4⋊S3 | C3×D4.S3 | C32×D8 | C3×S3×D4 | C3×D4⋊2S3 | D8⋊S3 | C8⋊S3 | C24⋊C2 | D4⋊S3 | D4.S3 | C3×D8 | S3×D4 | D4⋊2S3 | C3×D8 | C3×Dic3 | S3×C6 | C24 | C3×D4 | D8 | Dic3 | D6 | C8 | D4 | C32 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×D8⋊S3 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 6 | 2 | 3 |
| 5 | 5 | 3 | 5 |
| 1 | 3 | 4 | 0 |
| 6 | 4 | 0 | 5 |
| 0 | 5 | 2 | 6 |
| 2 | 0 | 2 | 1 |
| 3 | 3 | 0 | 1 |
| 1 | 6 | 3 | 0 |
| 2 | 5 | 1 | 1 |
| 3 | 3 | 3 | 6 |
| 4 | 3 | 4 | 6 |
| 2 | 2 | 6 | 3 |
| 5 | 1 | 2 | 2 |
| 0 | 0 | 6 | 5 |
| 4 | 3 | 5 | 6 |
| 5 | 5 | 1 | 4 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,5,1,6,6,5,3,4,2,3,4,0,3,5,0,5],[0,2,3,1,5,0,3,6,2,2,0,3,6,1,1,0],[2,3,4,2,5,3,3,2,1,3,4,6,1,6,6,3],[5,0,4,5,1,0,3,5,2,6,5,1,2,5,6,4] >;
C3×D8⋊S3 in GAP, Magma, Sage, TeX
C_3\times D_8\rtimes S_3
% in TeX
G:=Group("C3xD8:S3"); // GroupNames label
G:=SmallGroup(288,682);
// by ID
G=gap.SmallGroup(288,682);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,1094,303,1271,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations