direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.S3, D4.7D6, C6⋊2SD16, C12.16D4, C12.13C23, Dic6⋊6C22, C3⋊C8⋊8C22, C3⋊3(C2×SD16), (C2×D4).4S3, (C6×D4).3C2, (C2×C6).40D4, (C2×C4).48D6, C6.46(C2×D4), (C2×Dic6)⋊9C2, C4.6(C3⋊D4), C4.13(C22×S3), (C3×D4).7C22, (C2×C12).31C22, C22.22(C3⋊D4), (C2×C3⋊C8)⋊5C2, C2.10(C2×C3⋊D4), SmallGroup(96,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4.S3
G = < a,b,c,d,e | a2=b4=c2=d3=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >
Subgroups: 146 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×D4, C3×D4, C22×C6, C2×SD16, C2×C3⋊C8, D4.S3, C2×Dic6, C6×D4, C2×D4.S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C2×SD16, D4.S3, C2×C3⋊D4, C2×D4.S3
Character table of C2×D4.S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 12 | 12 | 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 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -2 | -2 | -2 | 2 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | -2 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | -2 | -2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 1 | -1 | 1 | -√-3 | √-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -√-3 | √-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | √-3 | -√-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 1 | -1 | 1 | √-3 | -√-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | 1 | -1 | complex lifted from C3⋊D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | complex lifted from SD16 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | 0 | 0 | complex lifted from SD16 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | 0 | 0 | complex lifted from SD16 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | 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 | symplectic lifted from D4.S3, Schur index 2 |
| 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 | symplectic lifted from D4.S3, Schur index 2 |
►On 48 points
(1 22)(2 23)(3 24)(4 21)(5 39)(6 40)(7 37)(8 38)(9 43)(10 44)(11 41)(12 42)(13 27)(14 28)(15 25)(16 26)(17 31)(18 32)(19 29)(20 30)(33 45)(34 46)(35 47)(36 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 4)(2 3)(6 8)(9 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)(25 26)(27 28)(29 32)(30 31)(34 36)(38 40)(41 43)(46 48)
(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 47)(6 11 48)(7 12 45)(8 9 46)(21 27 32)(22 28 29)(23 25 30)(24 26 31)(33 37 42)(34 38 43)(35 39 44)(36 40 41)
(1 48 3 46)(2 47 4 45)(5 18 7 20)(6 17 8 19)(9 14 11 16)(10 13 12 15)(21 33 23 35)(22 36 24 34)(25 44 27 42)(26 43 28 41)(29 40 31 38)(30 39 32 37)
G:=sub<Sym(48)| (1,22)(2,23)(3,24)(4,21)(5,39)(6,40)(7,37)(8,38)(9,43)(10,44)(11,41)(12,42)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,45)(34,46)(35,47)(36,48), (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), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,32)(30,31)(34,36)(38,40)(41,43)(46,48), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,47)(6,11,48)(7,12,45)(8,9,46)(21,27,32)(22,28,29)(23,25,30)(24,26,31)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,48,3,46)(2,47,4,45)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15)(21,33,23,35)(22,36,24,34)(25,44,27,42)(26,43,28,41)(29,40,31,38)(30,39,32,37)>;
G:=Group( (1,22)(2,23)(3,24)(4,21)(5,39)(6,40)(7,37)(8,38)(9,43)(10,44)(11,41)(12,42)(13,27)(14,28)(15,25)(16,26)(17,31)(18,32)(19,29)(20,30)(33,45)(34,46)(35,47)(36,48), (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), (1,4)(2,3)(6,8)(9,11)(13,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,32)(30,31)(34,36)(38,40)(41,43)(46,48), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,47)(6,11,48)(7,12,45)(8,9,46)(21,27,32)(22,28,29)(23,25,30)(24,26,31)(33,37,42)(34,38,43)(35,39,44)(36,40,41), (1,48,3,46)(2,47,4,45)(5,18,7,20)(6,17,8,19)(9,14,11,16)(10,13,12,15)(21,33,23,35)(22,36,24,34)(25,44,27,42)(26,43,28,41)(29,40,31,38)(30,39,32,37) );
G=PermutationGroup([[(1,22),(2,23),(3,24),(4,21),(5,39),(6,40),(7,37),(8,38),(9,43),(10,44),(11,41),(12,42),(13,27),(14,28),(15,25),(16,26),(17,31),(18,32),(19,29),(20,30),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,4),(2,3),(6,8),(9,11),(13,14),(15,16),(17,20),(18,19),(21,22),(23,24),(25,26),(27,28),(29,32),(30,31),(34,36),(38,40),(41,43),(46,48)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,10,47),(6,11,48),(7,12,45),(8,9,46),(21,27,32),(22,28,29),(23,25,30),(24,26,31),(33,37,42),(34,38,43),(35,39,44),(36,40,41)], [(1,48,3,46),(2,47,4,45),(5,18,7,20),(6,17,8,19),(9,14,11,16),(10,13,12,15),(21,33,23,35),(22,36,24,34),(25,44,27,42),(26,43,28,41),(29,40,31,38),(30,39,32,37)]])
C2×D4.S3 is a maximal subgroup of
D12.2D4 D4.S3⋊C4 Dic3⋊6SD16 Dic6⋊2D4 Dic6.D4 D6⋊5SD16 D6⋊SD16 C3⋊C8⋊1D4 D4.D12 D4.1D12 C42.51D6 D4.2D12 D12⋊17D4 Dic6⋊17D4 C3⋊C8⋊23D4 C3⋊C8⋊5D4 C42.61D6 C42.214D6 C42.65D6 C42.74D6 Dic6⋊9D4 C12⋊4SD16 (C6×D8).C2 C24⋊11D4 C24.22D4 Dic6⋊D4 Dic3⋊3SD16 C24.31D4 D6⋊8SD16 C24⋊15D4 M4(2).13D6 (C3×D4).31D4 (C3×D4).32D4 C2×S3×SD16 D8⋊6D6 D12.33C23
C2×D4.S3 is a maximal quotient of
C4⋊C4.231D6 C12.38SD16 D4.2D12 C4⋊D4.S3 Dic6⋊17D4 C3⋊C8⋊23D4 C12.16D8 Dic6⋊9D4 C12⋊4SD16 C12.SD16 C12.Q16 Dic6⋊6Q8 (C3×D4).31D4
Matrix representation of C2×D4.S3 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 67 | 6 |
| 0 | 0 | 6 | 6 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[64,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,67,6,0,0,6,6] >;
C2×D4.S3 in GAP, Magma, Sage, TeX
C_2\times D_4.S_3
% in TeX
G:=Group("C2xD4.S3"); // GroupNames label
G:=SmallGroup(96,140);
// by ID
G=gap.SmallGroup(96,140);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,218,579,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^3=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations
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