metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22⋊C4⋊4D6, (C22×C4)⋊5D6, (C2×D4).43D6, (C2×C12).10D4, C6.50C22≀C2, D4⋊6D6.3C2, (C2×Dic3).4D4, (C22×S3).4D4, (C22×C6).21D4, C22.34(S3×D4), (C22×C12)⋊2C22, (C6×D4).59C22, C22.D4⋊1S3, C23.7D6⋊6C2, C23.6D6⋊6C2, C3⋊2(C23.7D4), C23.9(C3⋊D4), C2.18(C23⋊2D6), C23.28D6⋊1C2, C6.D4⋊5C22, C23.85(C22×S3), (C22×C6).114C23, (C2×C6).31(C2×D4), (C2×C4).9(C3⋊D4), (C2×C3⋊D4).6C22, C22.30(C2×C3⋊D4), (C3×C22⋊C4)⋊35C22, (C3×C22.D4)⋊1C2, SmallGroup(192,612)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4⋊D6
G = < a,b,c,d,e | a2=b2=c4=d6=e2=1, cac-1=dad-1=ab=ba, ae=ea, bc=cb, bd=db, be=eb, dcd-1=bc-1, ece=abc, ede=d-1 >
Subgroups: 528 in 160 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C23⋊C4, C22.D4, C22.D4, 2+ 1+4, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C23.7D4, C23.6D6, C23.7D6, C23.28D6, C3×C22.D4, D4⋊6D6, C22⋊C4⋊D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, C23.7D4, C23⋊2D6, C22⋊C4⋊D6
Character table of C22⋊C4⋊D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 12 | 12 | 2 | 4 | 4 | 4 | 8 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | |
| 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 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ16 | 2 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | -√-3 | √-3 | √-3 | -√-3 | -1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | √-3 | -√-3 | -√-3 | √-3 | 1 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | √-3 | -√-3 | -√-3 | √-3 | -1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -√-3 | √-3 | √-3 | -√-3 | 1 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ23 | 4 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | complex lifted from C23.7D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | complex lifted from C23.7D4 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 2ζ4ζ3 | 2ζ43ζ32 | 2ζ4ζ32 | 2ζ43ζ3 | 0 | 0 | 0 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 2ζ43ζ32 | 2ζ4ζ3 | 2ζ43ζ3 | 2ζ4ζ32 | 0 | 0 | 0 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 2ζ43ζ3 | 2ζ4ζ32 | 2ζ43ζ32 | 2ζ4ζ3 | 0 | 0 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 2ζ4ζ32 | 2ζ43ζ3 | 2ζ4ζ3 | 2ζ43ζ32 | 0 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(1 22)(2 20)(3 24)(4 19)(5 23)(6 21)(7 13)(8 17)(9 15)(10 16)(11 14)(12 18)(25 37)(26 46)(27 39)(28 48)(29 41)(30 44)(31 38)(32 47)(33 40)(34 43)(35 42)(36 45)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)
(1 31 10 29)(2 35 11 27)(3 33 12 25)(4 26 7 34)(5 30 8 32)(6 28 9 36)(13 41 19 38)(14 47 20 44)(15 37 21 40)(16 43 22 46)(17 39 23 42)(18 45 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 3)(4 6)(7 12)(8 11)(9 10)(13 18)(14 17)(15 16)(19 21)(22 24)(25 43)(26 48)(27 47)(28 46)(29 45)(30 44)(31 40)(32 39)(33 38)(34 37)(35 42)(36 41)
G:=sub<Sym(48)| (1,22)(2,20)(3,24)(4,19)(5,23)(6,21)(7,13)(8,17)(9,15)(10,16)(11,14)(12,18)(25,37)(26,46)(27,39)(28,48)(29,41)(30,44)(31,38)(32,47)(33,40)(34,43)(35,42)(36,45), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44), (1,31,10,29)(2,35,11,27)(3,33,12,25)(4,26,7,34)(5,30,8,32)(6,28,9,36)(13,41,19,38)(14,47,20,44)(15,37,21,40)(16,43,22,46)(17,39,23,42)(18,45,24,48), (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), (1,3)(4,6)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,21)(22,24)(25,43)(26,48)(27,47)(28,46)(29,45)(30,44)(31,40)(32,39)(33,38)(34,37)(35,42)(36,41)>;
G:=Group( (1,22)(2,20)(3,24)(4,19)(5,23)(6,21)(7,13)(8,17)(9,15)(10,16)(11,14)(12,18)(25,37)(26,46)(27,39)(28,48)(29,41)(30,44)(31,38)(32,47)(33,40)(34,43)(35,42)(36,45), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,36)(26,31)(27,32)(28,33)(29,34)(30,35)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44), (1,31,10,29)(2,35,11,27)(3,33,12,25)(4,26,7,34)(5,30,8,32)(6,28,9,36)(13,41,19,38)(14,47,20,44)(15,37,21,40)(16,43,22,46)(17,39,23,42)(18,45,24,48), (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), (1,3)(4,6)(7,12)(8,11)(9,10)(13,18)(14,17)(15,16)(19,21)(22,24)(25,43)(26,48)(27,47)(28,46)(29,45)(30,44)(31,40)(32,39)(33,38)(34,37)(35,42)(36,41) );
G=PermutationGroup([[(1,22),(2,20),(3,24),(4,19),(5,23),(6,21),(7,13),(8,17),(9,15),(10,16),(11,14),(12,18),(25,37),(26,46),(27,39),(28,48),(29,41),(30,44),(31,38),(32,47),(33,40),(34,43),(35,42),(36,45)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,36),(26,31),(27,32),(28,33),(29,34),(30,35),(37,45),(38,46),(39,47),(40,48),(41,43),(42,44)], [(1,31,10,29),(2,35,11,27),(3,33,12,25),(4,26,7,34),(5,30,8,32),(6,28,9,36),(13,41,19,38),(14,47,20,44),(15,37,21,40),(16,43,22,46),(17,39,23,42),(18,45,24,48)], [(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)], [(1,3),(4,6),(7,12),(8,11),(9,10),(13,18),(14,17),(15,16),(19,21),(22,24),(25,43),(26,48),(27,47),(28,46),(29,45),(30,44),(31,40),(32,39),(33,38),(34,37),(35,42),(36,41)]])
Matrix representation of C22⋊C4⋊D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 9 | 4 | 9 |
| 0 | 0 | 9 | 9 | 9 | 4 |
| 0 | 0 | 9 | 4 | 4 | 4 |
| 0 | 0 | 4 | 9 | 4 | 4 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,9,9,4,0,0,9,9,4,9,0,0,4,9,4,4,0,0,9,4,4,4],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1] >;
C22⋊C4⋊D6 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\rtimes D_6
% in TeX
G:=Group("C2^2:C4:D6"); // GroupNames label
G:=SmallGroup(192,612);
// by ID
G=gap.SmallGroup(192,612);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,570,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^6=e^2=1,c*a*c^-1=d*a*d^-1=a*b=b*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b*c^-1,e*c*e=a*b*c,e*d*e=d^-1>;
// generators/relations
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