direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2xD4:S3, C6:2D8, D4:3D6, C12.14D4, D12:5C22, C12.11C23, C3:3(C2xD8), (C6xD4):1C2, (C2xD4):1S3, C3:C8:7C22, (C2xD12):8C2, C6.44(C2xD4), (C2xC6).38D4, (C2xC4).47D6, (C3xD4):3C22, C4.5(C3:D4), C4.11(C22xS3), (C2xC12).29C22, C22.21(C3:D4), (C2xC3:C8):4C2, C2.8(C2xC3:D4), SmallGroup(96,138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2xD4:S3
G = < a,b,c,d,e | a2=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >
Subgroups: 210 in 76 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C8, C2xC4, D4, D4, C23, C12, D6, C2xC6, C2xC6, C2xC8, D8, C2xD4, C2xD4, C3:C8, D12, D12, C2xC12, C3xD4, C3xD4, C22xS3, C22xC6, C2xD8, C2xC3:C8, D4:S3, C2xD12, C6xD4, C2xD4:S3
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C3:D4, C22xS3, C2xD8, D4:S3, C2xC3:D4, C2xD4:S3
Character table of C2xD4:S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 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 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | 0 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | √2 | √2 | -√2 | -√2 | 0 | 0 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -√2 | -√2 | √2 | √2 | 0 | 0 | orthogonal lifted from D8 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | 0 | orthogonal lifted from D8 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | -1 | -√-3 | √-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3:D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -1 | -1 | -1 | √-3 | -√-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3:D4 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 1 | -1 | 1 | -√-3 | √-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | -1 | complex lifted from C3:D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | 1 | -1 | 1 | √-3 | -√-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | -1 | complex lifted from C3:D4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4:S3, Schur index 2 |
| ρ24 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4:S3, Schur index 2 |
►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 45)(2 48)(3 47)(4 46)(5 19)(6 18)(7 17)(8 20)(9 15)(10 14)(11 13)(12 16)(21 34)(22 33)(23 36)(24 35)(25 43)(26 42)(27 41)(28 44)(29 39)(30 38)(31 37)(32 40)
(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)
(1 34)(2 33)(3 36)(4 35)(5 29)(6 32)(7 31)(8 30)(9 25)(10 28)(11 27)(12 26)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)
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), (1,45)(2,48)(3,47)(4,46)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16)(21,34)(22,33)(23,36)(24,35)(25,43)(26,42)(27,41)(28,44)(29,39)(30,38)(31,37)(32,40), (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), (1,34)(2,33)(3,36)(4,35)(5,29)(6,32)(7,31)(8,30)(9,25)(10,28)(11,27)(12,26)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45)>;
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), (1,45)(2,48)(3,47)(4,46)(5,19)(6,18)(7,17)(8,20)(9,15)(10,14)(11,13)(12,16)(21,34)(22,33)(23,36)(24,35)(25,43)(26,42)(27,41)(28,44)(29,39)(30,38)(31,37)(32,40), (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), (1,34)(2,33)(3,36)(4,35)(5,29)(6,32)(7,31)(8,30)(9,25)(10,28)(11,27)(12,26)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45) );
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)], [(1,45),(2,48),(3,47),(4,46),(5,19),(6,18),(7,17),(8,20),(9,15),(10,14),(11,13),(12,16),(21,34),(22,33),(23,36),(24,35),(25,43),(26,42),(27,41),(28,44),(29,39),(30,38),(31,37),(32,40)], [(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)], [(1,34),(2,33),(3,36),(4,35),(5,29),(6,32),(7,31),(8,30),(9,25),(10,28),(11,27),(12,26),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,48),(22,47),(23,46),(24,45)]])
C2xD4:S3 is a maximal subgroup of
D12.3D4 Dic3:4D8 D4:D12 D6:D8 D4:3D12 C3:C8:D4 D4:S3:C4 D12:3D4 D12.D4 C42.48D6 C12:7D8 D4.1D12 D12:16D4 D12:17D4 C3:C8:22D4 C4:D4:S3 D12.23D4 C42.64D6 C42.214D6 C12:2D8 C12:D8 C42.74D6 Dic3:D8 C24:5D4 C24:11D4 D12:D4 (C3xD4).D4 C24.43D4 D12:7D4 C24:9D4 M4(2).D6 (C2xC6):8D8 (C3xD4):14D4 C2xS3xD8 D8:5D6 D12.32C23
C2xD4:S3 is a maximal quotient of
(C2xC6).40D8 C12.50D8 C12:7D8 (C2xC6).D8 D12:16D4 C3:C8:22D4 C12.16D8 C12:2D8 C12:D8 C12.17D8 D12:6Q8 C12.D8 D8.D6 C24.27C23 Q16:D6 Q16.D6 D8.9D6 (C2xC6):8D8
Matrix representation of C2xD4:S3 ►in GL4(F73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 57 | 16 |
| 0 | 0 | 16 | 16 |
| 72 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,57,16,0,0,16,16],[72,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,1,0,0,0,0,72] >;
C2xD4:S3 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes S_3
% in TeX
G:=Group("C2xD4:S3"); // GroupNames label
G:=SmallGroup(96,138);
// by ID
G=gap.SmallGroup(96,138);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,218,579,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations
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