metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8⋊3S3, Dic3○D8, C8.8D6, D4.1D6, D6.1D4, Dic12⋊4C2, C24.6C22, C12.3C23, Dic3.12D4, Dic6.1C22, (S3×C8)⋊2C2, (C3×D8)⋊3C2, C3⋊2(C4○D8), D4.S3⋊2C2, C2.17(S3×D4), C6.29(C2×D4), C3⋊C8.5C22, D4⋊2S3⋊2C2, C4.3(C22×S3), (C4×S3).8C22, (C3×D4).1C22, SmallGroup(96,119)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8⋊3S3
G = < a,b,c,d | a8=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >
Subgroups: 146 in 62 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, D4, Q8, Dic3, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3×D4, C4○D8, S3×C8, Dic12, D4.S3, C3×D8, D4⋊2S3, D8⋊3S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C4○D8, S3×D4, D8⋊3S3
Character table of D8⋊3S3
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12 | 24A | 24B | |
| size | 1 | 1 | 4 | 4 | 6 | 2 | 2 | 3 | 3 | 12 | 12 | 2 | 8 | 8 | 2 | 2 | 6 | 6 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | -2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | 2 | -2 | 2 | 0 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | √-2 | -√-2 | 0 | √2 | -√2 | complex lifted from C4○D8 |
| ρ16 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | √-2 | -√-2 | 0 | -√2 | √2 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | -√-2 | √-2 | 0 | -√2 | √2 | complex lifted from C4○D8 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | -√-2 | √-2 | 0 | √2 | -√2 | complex lifted from C4○D8 |
| ρ19 | 4 | 4 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | -√2 | √2 | symplectic faithful, Schur index 2 |
| ρ21 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | √2 | -√2 | symplectic faithful, Schur index 2 |
►On 48 points
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 24)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 40)(16 39)(25 42)(26 41)(27 48)(28 47)(29 46)(30 45)(31 44)(32 43)
(1 35 46)(2 36 47)(3 37 48)(4 38 41)(5 39 42)(6 40 43)(7 33 44)(8 34 45)(9 26 20)(10 27 21)(11 28 22)(12 29 23)(13 30 24)(14 31 17)(15 32 18)(16 25 19)
(9 30)(10 31)(11 32)(12 25)(13 26)(14 27)(15 28)(16 29)(17 21)(18 22)(19 23)(20 24)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
G:=sub<Sym(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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43), (1,35,46)(2,36,47)(3,37,48)(4,38,41)(5,39,42)(6,40,43)(7,33,44)(8,34,45)(9,26,20)(10,27,21)(11,28,22)(12,29,23)(13,30,24)(14,31,17)(15,32,18)(16,25,19), (9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,21)(18,22)(19,23)(20,24)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43)>;
G:=Group( (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,24)(9,38)(10,37)(11,36)(12,35)(13,34)(14,33)(15,40)(16,39)(25,42)(26,41)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43), (1,35,46)(2,36,47)(3,37,48)(4,38,41)(5,39,42)(6,40,43)(7,33,44)(8,34,45)(9,26,20)(10,27,21)(11,28,22)(12,29,23)(13,30,24)(14,31,17)(15,32,18)(16,25,19), (9,30)(10,31)(11,32)(12,25)(13,26)(14,27)(15,28)(16,29)(17,21)(18,22)(19,23)(20,24)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43) );
G=PermutationGroup([[(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,24),(9,38),(10,37),(11,36),(12,35),(13,34),(14,33),(15,40),(16,39),(25,42),(26,41),(27,48),(28,47),(29,46),(30,45),(31,44),(32,43)], [(1,35,46),(2,36,47),(3,37,48),(4,38,41),(5,39,42),(6,40,43),(7,33,44),(8,34,45),(9,26,20),(10,27,21),(11,28,22),(12,29,23),(13,30,24),(14,31,17),(15,32,18),(16,25,19)], [(9,30),(10,31),(11,32),(12,25),(13,26),(14,27),(15,28),(16,29),(17,21),(18,22),(19,23),(20,24),(33,44),(34,45),(35,46),(36,47),(37,48),(38,41),(39,42),(40,43)]])
D8⋊3S3 is a maximal subgroup of
D8⋊D6 D16⋊3S3 SD32⋊S3 D6.2D8 D8⋊13D6 S3×C4○D8 D8.10D6 D8⋊4D6 D8⋊6D6 D8⋊3D9 D24⋊7S3 D24⋊5S3 D12.22D6 D12.8D6 C24.26D6 D40⋊7S3 D40⋊5S3 D20.24D6 D20.10D6 D8⋊3D15
D8⋊3S3 is a maximal quotient of
Dic3⋊6SD16 D4.2Dic6 Dic6.D4 (C2×C8).200D6 D4⋊2S3⋊C4 D6⋊SD16 D4.D12 C24⋊1C4⋊C2 Dic3⋊5Q16 Dic6.2Q8 C8.6Dic6 C8.27(C4×S3) D6⋊2Q16 C2.D8⋊7S3 Dic3×D8 (C6×D8).C2 C24.22D4 D6⋊3D8 Dic6⋊D4 D8⋊3D9 D24⋊7S3 D24⋊5S3 D12.22D6 D12.8D6 C24.26D6 D40⋊7S3 D40⋊5S3 D20.24D6 D20.10D6 D8⋊3D15
Matrix representation of D8⋊3S3 ►in GL4(𝔽7) generated by
| 0 | 2 | 5 | 5 |
| 3 | 5 | 6 | 5 |
| 1 | 1 | 5 | 5 |
| 6 | 1 | 4 | 3 |
| 4 | 4 | 2 | 0 |
| 2 | 0 | 4 | 6 |
| 6 | 6 | 2 | 2 |
| 4 | 3 | 5 | 1 |
| 0 | 0 | 5 | 2 |
| 5 | 3 | 0 | 3 |
| 2 | 5 | 0 | 3 |
| 5 | 5 | 1 | 2 |
| 6 | 0 | 5 | 3 |
| 5 | 5 | 2 | 2 |
| 1 | 4 | 6 | 1 |
| 3 | 5 | 1 | 4 |
G:=sub<GL(4,GF(7))| [0,3,1,6,2,5,1,1,5,6,5,4,5,5,5,3],[4,2,6,4,4,0,6,3,2,4,2,5,0,6,2,1],[0,5,2,5,0,3,5,5,5,0,0,1,2,3,3,2],[6,5,1,3,0,5,4,5,5,2,6,1,3,2,1,4] >;
D8⋊3S3 in GAP, Magma, Sage, TeX
D_8\rtimes_3S_3
% in TeX
G:=Group("D8:3S3"); // GroupNames label
G:=SmallGroup(96,119);
// by ID
G=gap.SmallGroup(96,119);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,362,116,297,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations
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