direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4⋊2S3, D4⋊5D6, C6.6C24, D6.2C23, C23.24D6, C12.20C23, Dic6⋊7C22, Dic3.3C23, (C6×D4)⋊6C2, (C2×D4)⋊8S3, D4○(C2×Dic3), Dic3○(C2×D4), C6⋊2(C4○D4), (C2×C4).60D6, (C4×S3)⋊4C22, (C3×D4)⋊6C22, C3⋊D4⋊2C22, (C2×C6).1C23, C2.7(S3×C23), (C2×Dic6)⋊12C2, C4.20(C22×S3), (C2×C12).45C22, (C2×Dic3)⋊9C22, (C22×Dic3)⋊8C2, C22.1(C22×S3), (C22×C6).23C22, (C22×S3).29C22, (S3×C2×C4)⋊4C2, C3⋊2(C2×C4○D4), (C2×C3⋊D4)⋊10C2, SmallGroup(96,210)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4⋊2S3
G = < a,b,c,d,e | a2=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 306 in 164 conjugacy classes, 89 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C2×C4○D4, C2×Dic6, S3×C2×C4, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, C2×D4⋊2S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, D4⋊2S3, S3×C23, C2×D4⋊2S3
Character table of C2×D4⋊2S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 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 | 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 | 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 | 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 | 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 | -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 | -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 | -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 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | 1 | -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 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | 1 | -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 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | 1 | -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 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | 1 | -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 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | -1 | 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 |
| ρ14 | 1 | 1 | -1 | -1 | -1 | -1 | 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 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | -1 | 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 |
| ρ16 | 1 | 1 | 1 | 1 | 1 | -1 | 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 |
| ρ17 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ20 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | orthogonal lifted from D6 |
| ρ23 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ24 | 2 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ26 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ27 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ30 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
►On 48 points
(1 35)(2 36)(3 33)(4 34)(5 29)(6 30)(7 31)(8 32)(9 38)(10 39)(11 40)(12 37)(13 25)(14 26)(15 27)(16 28)(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 20)(2 19)(3 18)(4 17)(5 26)(6 25)(7 28)(8 27)(9 24)(10 23)(11 22)(12 21)(13 30)(14 29)(15 32)(16 31)(33 42)(34 41)(35 44)(36 43)(37 45)(38 48)(39 47)(40 46)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 47 43)(14 48 44)(15 45 41)(16 46 42)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(29 38 35)(30 39 36)(31 40 33)(32 37 34)
(5 9)(6 10)(7 11)(8 12)(13 45)(14 46)(15 47)(16 48)(17 19)(18 20)(21 25)(22 26)(23 27)(24 28)(29 38)(30 39)(31 40)(32 37)(41 43)(42 44)
G:=sub<Sym(48)| (1,35)(2,36)(3,33)(4,34)(5,29)(6,30)(7,31)(8,32)(9,38)(10,39)(11,40)(12,37)(13,25)(14,26)(15,27)(16,28)(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), (1,20)(2,19)(3,18)(4,17)(5,26)(6,25)(7,28)(8,27)(9,24)(10,23)(11,22)(12,21)(13,30)(14,29)(15,32)(16,31)(33,42)(34,41)(35,44)(36,43)(37,45)(38,48)(39,47)(40,46), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(17,27,21)(18,28,22)(19,25,23)(20,26,24)(29,38,35)(30,39,36)(31,40,33)(32,37,34), (5,9)(6,10)(7,11)(8,12)(13,45)(14,46)(15,47)(16,48)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,38)(30,39)(31,40)(32,37)(41,43)(42,44)>;
G:=Group( (1,35)(2,36)(3,33)(4,34)(5,29)(6,30)(7,31)(8,32)(9,38)(10,39)(11,40)(12,37)(13,25)(14,26)(15,27)(16,28)(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), (1,20)(2,19)(3,18)(4,17)(5,26)(6,25)(7,28)(8,27)(9,24)(10,23)(11,22)(12,21)(13,30)(14,29)(15,32)(16,31)(33,42)(34,41)(35,44)(36,43)(37,45)(38,48)(39,47)(40,46), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,47,43)(14,48,44)(15,45,41)(16,46,42)(17,27,21)(18,28,22)(19,25,23)(20,26,24)(29,38,35)(30,39,36)(31,40,33)(32,37,34), (5,9)(6,10)(7,11)(8,12)(13,45)(14,46)(15,47)(16,48)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,38)(30,39)(31,40)(32,37)(41,43)(42,44) );
G=PermutationGroup([[(1,35),(2,36),(3,33),(4,34),(5,29),(6,30),(7,31),(8,32),(9,38),(10,39),(11,40),(12,37),(13,25),(14,26),(15,27),(16,28),(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)], [(1,20),(2,19),(3,18),(4,17),(5,26),(6,25),(7,28),(8,27),(9,24),(10,23),(11,22),(12,21),(13,30),(14,29),(15,32),(16,31),(33,42),(34,41),(35,44),(36,43),(37,45),(38,48),(39,47),(40,46)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,47,43),(14,48,44),(15,45,41),(16,46,42),(17,27,21),(18,28,22),(19,25,23),(20,26,24),(29,38,35),(30,39,36),(31,40,33),(32,37,34)], [(5,9),(6,10),(7,11),(8,12),(13,45),(14,46),(15,47),(16,48),(17,19),(18,20),(21,25),(22,26),(23,27),(24,28),(29,38),(30,39),(31,40),(32,37),(41,43),(42,44)]])
C2×D4⋊2S3 is a maximal subgroup of
C23⋊C4⋊5S3 M4(2).19D6 D4⋊(C4×S3) D4⋊2S3⋊C4 D4⋊3D12 D4.D12 Dic6⋊D4 Dic6.16D4 C42.108D6 D4⋊5D12 D4⋊6D12 C24.67D6 C24.44D6 C24.45D6 C12⋊(C4○D4) C6.322+ 1+4 Dic6⋊19D4 Dic6⋊20D4 C4⋊C4⋊21D6 C6.722- 1+4 C6.402+ 1+4 C6.732- 1+4 C6.792- 1+4 C6.822- 1+4 C4⋊C4⋊28D6 C42.233D6 C42.141D6 Dic6⋊10D4 C42⋊28D6 C42.238D6 Dic6⋊11D4 D8⋊4D6 C24.53D6 C6.1042- 1+4 C2×S3×C4○D4 D6.C24
C2×D4⋊2S3 is a maximal quotient of
C24.42D6 C6.52- 1+4 C42.102D6 C42.105D6 C42.106D6 D4⋊6Dic6 D4⋊6D12 C42.229D6 C42.117D6 C42.119D6 C24.67D6 C24.43D6 C24.44D6 C24.46D6 C12⋊(C4○D4) Dic6⋊19D4 C4⋊C4.178D6 C6.342+ 1+4 C6.702- 1+4 C6.712- 1+4 C4⋊C4⋊21D6 C6.732- 1+4 C6.432+ 1+4 C6.452+ 1+4 C6.462+ 1+4 C6.1152+ 1+4 C6.472+ 1+4 (Q8×Dic3)⋊C2 C4⋊C4.187D6 C6.152- 1+4 C6.1182+ 1+4 C6.212- 1+4 C6.232- 1+4 C6.772- 1+4 C6.242- 1+4 C4⋊C4.197D6 C6.802- 1+4 C6.1222+ 1+4 C6.852- 1+4 C42.139D6 C42.234D6 C42.143D6 C42.144D6 C42.166D6 C42.238D6 Dic6⋊11D4 C42.168D6 Dic6⋊8Q8 C42.241D6 C42.176D6 C42.177D6 C2×D4×Dic3 C24.53D6
Matrix representation of C2×D4⋊2S3 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 5 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,5,0,0,0,0,0,8],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,8,0],[1,0,0,0,0,0,12,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12] >;
C2×D4⋊2S3 in GAP, Magma, Sage, TeX
C_2\times D_4\rtimes_2S_3
% in TeX
G:=Group("C2xD4:2S3"); // GroupNames label
G:=SmallGroup(96,210);
// by ID
G=gap.SmallGroup(96,210);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,86,579,159,2309]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations
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