direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xQ8:2S3, Q8:4D6, C6:3SD16, C12.18D4, C12.14C23, D12.9C22, C3:C8:9C22, (C2xQ8):3S3, (C6xQ8):1C2, C3:4(C2xSD16), C6.53(C2xD4), (C2xC6).41D4, (C2xC4).53D6, (C2xD12).8C2, C4.8(C3:D4), (C3xQ8):3C22, C4.14(C22xS3), (C2xC12).36C22, C22.23(C3:D4), (C2xC3:C8):6C2, C2.17(C2xC3:D4), SmallGroup(96,148)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xQ8:2S3
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, C2xC4, C2xC4, D4, Q8, Q8, C23, C12, C12, D6, C2xC6, C2xC8, SD16, C2xD4, C2xQ8, C3:C8, D12, D12, C2xC12, C2xC12, C3xQ8, C3xQ8, C22xS3, C2xSD16, C2xC3:C8, Q8:2S3, C2xD12, C6xQ8, C2xQ8:2S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C3:D4, C22xS3, C2xSD16, Q8:2S3, C2xC3:D4, C2xQ8:2S3
Character table of C2xQ8:2S3
| 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)]])
C2xQ8:2S3 is a maximal subgroup of
D12.6D4 Dic3:7SD16 Q8:3D12 D6:2SD16 Q8:4D12 C3:(C8:D4) Q8:3(C4xS3) 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 (C2xQ16):S3 D12.17D4 C24.37D4 C24.28D4 M4(2).15D6 (C3xQ8):13D4 (C3xD4):14D4 C2xS3xSD16 C24.C23 D12.34C23
C2xQ8:2S3 is a maximal quotient of
C4:C4.228D6 Q8:4Dic6 Q8:2D12 (C2xQ8).49D6 D12.36D4 C3:C8:24D4 C12.9Q16 C12.SD16 C12:5SD16 D12:5Q8 C12:6SD16 C12.D8 (C3xQ8):13D4
Matrix representation of C2xQ8:2S3 ►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] >;
C2xQ8:2S3 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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