direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8⋊2S3, Q8⋊4D6, C6⋊3SD16, C12.18D4, C12.14C23, D12.9C22, C3⋊C8⋊9C22, (C2×Q8)⋊3S3, (C6×Q8)⋊1C2, C3⋊4(C2×SD16), C6.53(C2×D4), (C2×C6).41D4, (C2×C4).53D6, (C2×D12).8C2, C4.8(C3⋊D4), (C3×Q8)⋊3C22, C4.14(C22×S3), (C2×C12).36C22, C22.23(C3⋊D4), (C2×C3⋊C8)⋊6C2, C2.17(C2×C3⋊D4), SmallGroup(96,148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8⋊2S3
G = < a,b,c,d,e | a2=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >
Subgroups: 178 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C12, C12, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3⋊C8, D12, D12, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C2×SD16, C2×C3⋊C8, Q8⋊2S3, C2×D12, C6×Q8, C2×Q8⋊2S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C2×SD16, Q8⋊2S3, C2×C3⋊D4, C2×Q8⋊2S3
Character table of C2×Q8⋊2S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 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 | 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 | 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 | -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 | -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 | 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 | 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 | -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 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 2 | -2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | -2 | 2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | -√-3 | -√-3 | √-3 | √-3 | -1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | √-3 | √-3 | -√-3 | -√-3 | -1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | √-3 | -√-3 | -√-3 | √-3 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | -√-3 | √-3 | √-3 | -√-3 | 1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | √-2 | -√-2 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -√-2 | √-2 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ23 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊2S3 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊2S3 |
►On 48 points
(1 34)(2 35)(3 36)(4 33)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 48 3 46)(2 47 4 45)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)(21 35 23 33)(22 34 24 36)(25 44 27 42)(26 43 28 41)(29 40 31 38)(30 39 32 37)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 45 10)(6 46 11)(7 47 12)(8 48 9)(21 30 25)(22 31 26)(23 32 27)(24 29 28)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(2 4)(5 9)(6 12)(7 11)(8 10)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 29)(26 32)(27 31)(28 30)(33 35)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)
G:=sub<Sym(48)| (1,34)(2,35)(3,36)(4,33)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,48,3,46)(2,47,4,45)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)(21,35,23,33)(22,34,24,36)(25,44,27,42)(26,43,28,41)(29,40,31,38)(30,39,32,37), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (2,4)(5,9)(6,12)(7,11)(8,10)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,29)(26,32)(27,31)(28,30)(33,35)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)>;
G:=Group( (1,34)(2,35)(3,36)(4,33)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48), (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), (1,48,3,46)(2,47,4,45)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)(21,35,23,33)(22,34,24,36)(25,44,27,42)(26,43,28,41)(29,40,31,38)(30,39,32,37), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,45,10)(6,46,11)(7,47,12)(8,48,9)(21,30,25)(22,31,26)(23,32,27)(24,29,28)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (2,4)(5,9)(6,12)(7,11)(8,10)(13,20)(14,19)(15,18)(16,17)(21,24)(22,23)(25,29)(26,32)(27,31)(28,30)(33,35)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,33),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48)], [(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)], [(1,48,3,46),(2,47,4,45),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13),(21,35,23,33),(22,34,24,36),(25,44,27,42),(26,43,28,41),(29,40,31,38),(30,39,32,37)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,45,10),(6,46,11),(7,47,12),(8,48,9),(21,30,25),(22,31,26),(23,32,27),(24,29,28),(33,37,42),(34,38,43),(35,39,44),(36,40,41)], [(2,4),(5,9),(6,12),(7,11),(8,10),(13,20),(14,19),(15,18),(16,17),(21,24),(22,23),(25,29),(26,32),(27,31),(28,30),(33,35),(37,44),(38,43),(39,42),(40,41),(45,48),(46,47)]])
C2×Q8⋊2S3 is a maximal subgroup of
D12.6D4 Dic3⋊7SD16 Q8⋊3D12 D6⋊2SD16 Q8⋊4D12 C3⋊(C8⋊D4) Q8⋊3(C4×S3) Dic3⋊SD16 D12.12D4 C42.56D6 Q8⋊2D12 Q8.6D12 D12.36D4 D12.37D4 C3⋊C8⋊24D4 C3⋊C8⋊6D4 D12.23D4 C42.64D6 C42.214D6 C12⋊5SD16 C12⋊6SD16 C42.80D6 Dic3⋊5SD16 D6⋊6SD16 C24⋊15D4 C24⋊9D4 (C2×Q16)⋊S3 D12.17D4 C24.37D4 C24.28D4 M4(2).15D6 (C3×Q8)⋊13D4 (C3×D4)⋊14D4 C2×S3×SD16 C24.C23 D12.34C23
C2×Q8⋊2S3 is a maximal quotient of
C4⋊C4.228D6 Q8⋊4Dic6 Q8⋊2D12 (C2×Q8).49D6 D12.36D4 C3⋊C8⋊24D4 C12.9Q16 C12.SD16 C12⋊5SD16 D12⋊5Q8 C12⋊6SD16 C12.D8 (C3×Q8)⋊13D4
Matrix representation of C2×Q8⋊2S3 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 48 |
| 0 | 0 | 3 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 61 | 4 |
| 0 | 0 | 55 | 12 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 3 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,1,3,0,0,48,72],[1,0,0,0,0,1,0,0,0,0,61,55,0,0,4,12],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,1,3,0,0,0,72] >;
C2×Q8⋊2S3 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes_2S_3
% in TeX
G:=Group("C2xQ8:2S3"); // GroupNames label
G:=SmallGroup(96,148);
// by ID
G=gap.SmallGroup(96,148);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,86,579,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^3=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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations
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