direct product, metabelian, supersoluble, monomial
Aliases: C2×C32⋊2Q8, C6⋊1Dic6, Dic3.7D6, C62.12C22, (C3×C6)⋊2Q8, C32⋊4(C2×Q8), C22.12S32, C3⋊2(C2×Dic6), (C2×C6).17D6, (C3×C6).16C23, C6.16(C22×S3), (C2×Dic3).3S3, (C6×Dic3).4C2, C3⋊Dic3.15C22, (C3×Dic3).7C22, C2.16(C2×S32), (C2×C3⋊Dic3).8C2, SmallGroup(144,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C32⋊2Q8
G = < a,b,c,d,e | a2=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 224 in 84 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, Q8, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×Dic3, C3⋊Dic3, C62, C2×Dic6, C32⋊2Q8, C6×Dic3, C2×C3⋊Dic3, C2×C32⋊2Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, C32⋊2Q8, C2×S32, C2×C32⋊2Q8
Character table of C2×C32⋊2Q8
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | -1 | 2 | -1 | 0 | 2 | 0 | -2 | 0 | 0 | 1 | -2 | -1 | 1 | -2 | 2 | 1 | -1 | 1 | 1 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | 2 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | -2 | 0 | 2 | 0 | 0 | 0 | -2 | 1 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | 0 | -1 | -1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 1 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | 0 | 1 | 1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -2 | 0 | -2 | 0 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 0 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 0 | 2 | 0 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 0 | -1 | -1 | 0 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ15 | 2 | -2 | 2 | -2 | -1 | 2 | -1 | 0 | -2 | 0 | 2 | 0 | 0 | 1 | -2 | -1 | 1 | -2 | 2 | 1 | -1 | 1 | -1 | 0 | 0 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | -2 | 0 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 2 | -1 | 1 | 1 | 1 | -1 | 0 | √3 | -√3 | 0 | √3 | -√3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -2 | -2 | 1 | 1 | -1 | 1 | 1 | 0 | -√3 | √3 | 0 | √3 | -√3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ21 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 1 | 1 | -2 | -2 | -1 | 1 | 1 | -√3 | 0 | 0 | √3 | 0 | 0 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ22 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 2 | -2 | 1 | 1 | -1 | √3 | 0 | 0 | -√3 | 0 | 0 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ23 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | -2 | -2 | 1 | 1 | -1 | 1 | 1 | 0 | √3 | -√3 | 0 | -√3 | √3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ24 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 2 | -2 | 1 | 1 | -1 | -√3 | 0 | 0 | √3 | 0 | 0 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ25 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2 | 1 | 1 | -2 | -2 | -1 | 1 | 1 | √3 | 0 | 0 | -√3 | 0 | 0 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ26 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 2 | -1 | 1 | 1 | 1 | -1 | 0 | -√3 | √3 | 0 | -√3 | √3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ27 | 4 | 4 | 4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ28 | 4 | -4 | 4 | -4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | 2 | -2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S32 |
| ρ29 | 4 | 4 | -4 | -4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2Q8, Schur index 2 |
| ρ30 | 4 | -4 | -4 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C32⋊2Q8, Schur index 2 |
►On 48 points
Generators in S48
(1 5)(2 6)(3 7)(4 8)(9 34)(10 35)(11 36)(12 33)(13 38)(14 39)(15 40)(16 37)(17 29)(18 30)(19 31)(20 32)(21 27)(22 28)(23 25)(24 26)(41 48)(42 45)(43 46)(44 47)
(1 39 35)(2 36 40)(3 37 33)(4 34 38)(5 14 10)(6 11 15)(7 16 12)(8 9 13)(17 22 41)(18 42 23)(19 24 43)(20 44 21)(25 30 45)(26 46 31)(27 32 47)(28 48 29)
(1 35 39)(2 36 40)(3 33 37)(4 34 38)(5 10 14)(6 11 15)(7 12 16)(8 9 13)(17 41 22)(18 42 23)(19 43 24)(20 44 21)(25 30 45)(26 31 46)(27 32 47)(28 29 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 47 3 45)(2 46 4 48)(5 44 7 42)(6 43 8 41)(9 17 11 19)(10 20 12 18)(13 22 15 24)(14 21 16 23)(25 39 27 37)(26 38 28 40)(29 36 31 34)(30 35 32 33)
G:=sub<Sym(48)| (1,5)(2,6)(3,7)(4,8)(9,34)(10,35)(11,36)(12,33)(13,38)(14,39)(15,40)(16,37)(17,29)(18,30)(19,31)(20,32)(21,27)(22,28)(23,25)(24,26)(41,48)(42,45)(43,46)(44,47), (1,39,35)(2,36,40)(3,37,33)(4,34,38)(5,14,10)(6,11,15)(7,16,12)(8,9,13)(17,22,41)(18,42,23)(19,24,43)(20,44,21)(25,30,45)(26,46,31)(27,32,47)(28,48,29), (1,35,39)(2,36,40)(3,33,37)(4,34,38)(5,10,14)(6,11,15)(7,12,16)(8,9,13)(17,41,22)(18,42,23)(19,43,24)(20,44,21)(25,30,45)(26,31,46)(27,32,47)(28,29,48), (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), (1,47,3,45)(2,46,4,48)(5,44,7,42)(6,43,8,41)(9,17,11,19)(10,20,12,18)(13,22,15,24)(14,21,16,23)(25,39,27,37)(26,38,28,40)(29,36,31,34)(30,35,32,33)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,34)(10,35)(11,36)(12,33)(13,38)(14,39)(15,40)(16,37)(17,29)(18,30)(19,31)(20,32)(21,27)(22,28)(23,25)(24,26)(41,48)(42,45)(43,46)(44,47), (1,39,35)(2,36,40)(3,37,33)(4,34,38)(5,14,10)(6,11,15)(7,16,12)(8,9,13)(17,22,41)(18,42,23)(19,24,43)(20,44,21)(25,30,45)(26,46,31)(27,32,47)(28,48,29), (1,35,39)(2,36,40)(3,33,37)(4,34,38)(5,10,14)(6,11,15)(7,12,16)(8,9,13)(17,41,22)(18,42,23)(19,43,24)(20,44,21)(25,30,45)(26,31,46)(27,32,47)(28,29,48), (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), (1,47,3,45)(2,46,4,48)(5,44,7,42)(6,43,8,41)(9,17,11,19)(10,20,12,18)(13,22,15,24)(14,21,16,23)(25,39,27,37)(26,38,28,40)(29,36,31,34)(30,35,32,33) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,34),(10,35),(11,36),(12,33),(13,38),(14,39),(15,40),(16,37),(17,29),(18,30),(19,31),(20,32),(21,27),(22,28),(23,25),(24,26),(41,48),(42,45),(43,46),(44,47)], [(1,39,35),(2,36,40),(3,37,33),(4,34,38),(5,14,10),(6,11,15),(7,16,12),(8,9,13),(17,22,41),(18,42,23),(19,24,43),(20,44,21),(25,30,45),(26,46,31),(27,32,47),(28,48,29)], [(1,35,39),(2,36,40),(3,33,37),(4,34,38),(5,10,14),(6,11,15),(7,12,16),(8,9,13),(17,41,22),(18,42,23),(19,43,24),(20,44,21),(25,30,45),(26,31,46),(27,32,47),(28,29,48)], [(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)], [(1,47,3,45),(2,46,4,48),(5,44,7,42),(6,43,8,41),(9,17,11,19),(10,20,12,18),(13,22,15,24),(14,21,16,23),(25,39,27,37),(26,38,28,40),(29,36,31,34),(30,35,32,33)]])
C2×C32⋊2Q8 is a maximal subgroup of
C62.4D4 Dic3⋊5Dic6 C62.8C23 C62.9C23 C62.10C23 D6⋊Dic6 Dic3.D12 C62.35C23 D6⋊2Dic6 C62.65C23 D6⋊4Dic6 C62.85C23 C12⋊3Dic6 C12⋊Dic6 C62.95C23 C62.101C23 C62⋊3Q8 C62⋊4Q8 C62.15D4 C2×S3×Dic6 Dic6.24D6
C2×C32⋊2Q8 is a maximal quotient of
C62.39C23 C62.42C23 C12⋊3Dic6 C12⋊Dic6 C62⋊3Q8 C62⋊4Q8
Matrix representation of C2×C32⋊2Q8 ►in GL6(𝔽13)
| 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 4 | 10 | 0 | 0 | 0 | 0 |
| 10 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[4,10,0,0,0,0,10,9,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C2×C32⋊2Q8 in GAP, Magma, Sage, TeX
C_2\times C_3^2\rtimes_2Q_8
% in TeX
G:=Group("C2xC3^2:2Q8"); // GroupNames label
G:=SmallGroup(144,152);
// by ID
G=gap.SmallGroup(144,152);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,121,55,490,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations
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