direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8⋊3D6, C24⋊4C23, SD16⋊8D6, D12⋊2C23, C12.6C24, D24⋊20C22, (C2×C8)⋊10D6, C3⋊C8⋊2C23, C8⋊4(C22×S3), (C2×Q8)⋊24D6, C4.43(S3×D4), (C2×D24)⋊26C2, C6⋊3(C8⋊C22), (C2×SD16)⋊4S3, (C6×SD16)⋊5C2, D6.50(C2×D4), (C4×S3).15D4, C12.81(C2×D4), (S3×D4)⋊6C22, C4.6(S3×C23), Q8⋊3(C22×S3), (C3×Q8)⋊2C23, C8⋊S3⋊8C22, (C2×C24)⋊13C22, D4⋊S3⋊10C22, (C2×D4).182D6, (C4×S3).3C23, (C6×Q8)⋊18C22, D4.4(C22×S3), (C3×D4).4C23, (C2×D12)⋊33C22, Dic3.55(C2×D4), Q8⋊2S3⋊8C22, Q8⋊3S3⋊5C22, (C3×SD16)⋊8C22, (C22×S3).98D4, C6.107(C22×D4), C22.139(S3×D4), (C2×C12).523C23, (C2×Dic3).192D4, (C6×D4).164C22, (C2×S3×D4)⋊23C2, C3⋊3(C2×C8⋊C22), C2.80(C2×S3×D4), (C2×C8⋊S3)⋊4C2, (C2×D4⋊S3)⋊27C2, (C2×C3⋊C8)⋊15C22, (C2×C6).396(C2×D4), (C2×Q8⋊2S3)⋊26C2, (C2×Q8⋊3S3)⋊14C2, (S3×C2×C4).156C22, (C2×C4).612(C22×S3), SmallGroup(192,1318)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8⋊3D6
G = < a,b,c,d,e | a2=b4=d6=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >
Subgroups: 952 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), D8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×S3, C22×C6, C2×M4(2), C2×D8, C2×SD16, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C8⋊S3, D24, C2×C3⋊C8, D4⋊S3, Q8⋊2S3, C2×C24, C3×SD16, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, S3×D4, S3×D4, Q8⋊3S3, Q8⋊3S3, C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C2×C8⋊C22, C2×C8⋊S3, C2×D24, Q8⋊3D6, C2×D4⋊S3, C2×Q8⋊2S3, C6×SD16, C2×S3×D4, C2×Q8⋊3S3, C2×Q8⋊3D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C8⋊C22, C22×D4, S3×D4, S3×C23, C2×C8⋊C22, Q8⋊3D6, C2×S3×D4, C2×Q8⋊3D6
►On 48 points
(1 11)(2 12)(3 10)(4 9)(5 7)(6 8)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)
(1 23 4 20)(2 21 5 24)(3 19 6 22)(7 15 12 18)(8 13 10 16)(9 17 11 14)(25 45 48 28)(26 29 43 46)(27 47 44 30)(31 37 40 34)(32 35 41 38)(33 39 42 36)
(1 30 4 47)(2 28 5 45)(3 26 6 43)(7 36 12 39)(8 34 10 37)(9 32 11 41)(13 40 16 31)(14 35 17 38)(15 42 18 33)(19 46 22 29)(20 27 23 44)(21 48 24 25)
(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 3)(4 6)(8 9)(10 11)(13 14)(15 18)(16 17)(19 20)(21 24)(22 23)(25 45)(26 44)(27 43)(28 48)(29 47)(30 46)(31 41)(32 40)(33 39)(34 38)(35 37)(36 42)
G:=sub<Sym(48)| (1,11)(2,12)(3,10)(4,9)(5,7)(6,8)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,23,4,20)(2,21,5,24)(3,19,6,22)(7,15,12,18)(8,13,10,16)(9,17,11,14)(25,45,48,28)(26,29,43,46)(27,47,44,30)(31,37,40,34)(32,35,41,38)(33,39,42,36), (1,30,4,47)(2,28,5,45)(3,26,6,43)(7,36,12,39)(8,34,10,37)(9,32,11,41)(13,40,16,31)(14,35,17,38)(15,42,18,33)(19,46,22,29)(20,27,23,44)(21,48,24,25), (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), (1,3)(4,6)(8,9)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42)>;
G:=Group( (1,11)(2,12)(3,10)(4,9)(5,7)(6,8)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45), (1,23,4,20)(2,21,5,24)(3,19,6,22)(7,15,12,18)(8,13,10,16)(9,17,11,14)(25,45,48,28)(26,29,43,46)(27,47,44,30)(31,37,40,34)(32,35,41,38)(33,39,42,36), (1,30,4,47)(2,28,5,45)(3,26,6,43)(7,36,12,39)(8,34,10,37)(9,32,11,41)(13,40,16,31)(14,35,17,38)(15,42,18,33)(19,46,22,29)(20,27,23,44)(21,48,24,25), (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), (1,3)(4,6)(8,9)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,23)(25,45)(26,44)(27,43)(28,48)(29,47)(30,46)(31,41)(32,40)(33,39)(34,38)(35,37)(36,42) );
G=PermutationGroup([[(1,11),(2,12),(3,10),(4,9),(5,7),(6,8),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21),(25,42),(26,37),(27,38),(28,39),(29,40),(30,41),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45)], [(1,23,4,20),(2,21,5,24),(3,19,6,22),(7,15,12,18),(8,13,10,16),(9,17,11,14),(25,45,48,28),(26,29,43,46),(27,47,44,30),(31,37,40,34),(32,35,41,38),(33,39,42,36)], [(1,30,4,47),(2,28,5,45),(3,26,6,43),(7,36,12,39),(8,34,10,37),(9,32,11,41),(13,40,16,31),(14,35,17,38),(15,42,18,33),(19,46,22,29),(20,27,23,44),(21,48,24,25)], [(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)], [(1,3),(4,6),(8,9),(10,11),(13,14),(15,18),(16,17),(19,20),(21,24),(22,23),(25,45),(26,44),(27,43),(28,48),(29,47),(30,46),(31,41),(32,40),(33,39),(34,38),(35,37),(36,42)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | C8⋊C22 | S3×D4 | S3×D4 | Q8⋊3D6 |
| kernel | C2×Q8⋊3D6 | C2×C8⋊S3 | C2×D24 | Q8⋊3D6 | C2×D4⋊S3 | C2×Q8⋊2S3 | C6×SD16 | C2×S3×D4 | C2×Q8⋊3S3 | C2×SD16 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 1 | 1 | 4 |
Matrix representation of C2×Q8⋊3D6 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 72 | 71 | 72 |
| 0 | 0 | 25 | 25 | 1 | 0 |
| 0 | 0 | 24 | 25 | 1 | 0 |
| 72 | 70 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 10 | 44 | 44 |
| 0 | 0 | 34 | 5 | 0 | 44 |
| 0 | 0 | 68 | 68 | 68 | 63 |
| 0 | 0 | 39 | 68 | 39 | 34 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 25 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 2 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 | 0 | 72 |
| 0 | 0 | 0 | 72 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 48 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 0 | 0 | 72 | 72 | 72 | 71 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,25,24,0,0,0,72,25,25,0,0,72,71,1,1,0,0,1,72,0,0],[72,0,0,0,0,0,70,1,0,0,0,0,0,0,39,34,68,39,0,0,10,5,68,68,0,0,44,0,68,39,0,0,44,44,63,34],[72,25,0,0,0,0,0,1,0,0,0,0,0,0,1,0,72,0,0,0,1,0,0,72,0,0,1,72,0,0,0,0,2,1,72,72],[1,48,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,72,71,1,1] >;
C2×Q8⋊3D6 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes_3D_6
% in TeX
G:=Group("C2xQ8:3D6"); // GroupNames label
G:=SmallGroup(192,1318);
// by ID
G=gap.SmallGroup(192,1318);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1123,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^6=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations