metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊7D6, D12.13D4, Dic6.13D4, (C2×Q8)⋊5D6, C4.51(S3×D4), (C2×D4).51D6, C4.4D4⋊4S3, C12.28(C2×D4), (C6×Q8)⋊2C22, (C4×C12)⋊13C22, C6.51C22≀C2, D4⋊6D6.4C2, C12.D4⋊5C2, C3⋊3(D4.9D4), C42⋊4S3⋊11C2, (C22×C6).22D4, Q8.11D6⋊2C2, (C6×D4).67C22, C4.Dic3⋊6C22, C2.19(C23⋊2D6), (C2×C12).379C23, C4○D12.19C22, C23.10(C3⋊D4), (C2×C6).510(C2×D4), (C3×C4.4D4)⋊4C2, C22.31(C2×C3⋊D4), (C2×C4).116(C22×S3), SmallGroup(192,620)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊7D6
G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=a-1b2, dad=a-1b, cbc-1=b-1, bd=db, dcd=c-1 >
Subgroups: 496 in 152 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, M4(2), SD16, Q16, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4.D4, C4≀C2, C4.4D4, C8.C22, 2+ 1+4, C4.Dic3, Q8⋊2S3, C3⋊Q16, C4×C12, C3×C22⋊C4, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, C6×Q8, D4.9D4, C42⋊4S3, C12.D4, Q8.11D6, C3×C4.4D4, D4⋊6D6, C42⋊7D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S3×D4, C2×C3⋊D4, D4.9D4, C23⋊2D6, C42⋊7D6
Character table of C42⋊7D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 2 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 0 | 0 | √-3 | √-3 | 1 | 1 | -√-3 | -√-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 0 | 0 | -√-3 | -√-3 | 1 | 1 | √-3 | √-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | -1 | 0 | 0 | √-3 | √-3 | 1 | 1 | -√-3 | -√-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 0 | 0 | -√-3 | -√-3 | 1 | 1 | √-3 | √-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 2i | -2i | 0 | 0 | complex lifted from D4.9D4 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -2i | 2i | 0 | 0 | complex lifted from D4.9D4 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 2ζ4ζ32 | 2ζ43ζ32 | 0 | 0 | 2ζ4ζ3 | 2ζ43ζ3 | 0 | 0 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 2ζ43ζ3 | 2ζ4ζ3 | 0 | 0 | 2ζ43ζ32 | 2ζ4ζ32 | 0 | 0 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 2ζ43ζ32 | 2ζ4ζ32 | 0 | 0 | 2ζ43ζ3 | 2ζ4ζ3 | 0 | 0 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 2ζ4ζ3 | 2ζ43ζ3 | 0 | 0 | 2ζ4ζ32 | 2ζ43ζ32 | 0 | 0 | complex faithful |
►On 48 points
(1 24 13 32)(2 47 14 38)(3 20 15 34)(4 43 16 40)(5 22 17 36)(6 45 18 42)(7 23 28 31)(8 46 29 37)(9 19 30 33)(10 48 25 39)(11 21 26 35)(12 44 27 41)
(1 16 29 11)(2 12 30 17)(3 18 25 7)(4 8 26 13)(5 14 27 9)(6 10 28 15)(19 22 38 41)(20 42 39 23)(21 24 40 37)(31 34 45 48)(32 43 46 35)(33 36 47 44)
(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 6)(2 5)(3 4)(7 13)(8 18)(9 17)(10 16)(11 15)(12 14)(19 38)(20 37)(21 42)(22 41)(23 40)(24 39)(25 26)(27 30)(28 29)(31 35)(32 34)(43 45)(46 48)
G:=sub<Sym(48)| (1,24,13,32)(2,47,14,38)(3,20,15,34)(4,43,16,40)(5,22,17,36)(6,45,18,42)(7,23,28,31)(8,46,29,37)(9,19,30,33)(10,48,25,39)(11,21,26,35)(12,44,27,41), (1,16,29,11)(2,12,30,17)(3,18,25,7)(4,8,26,13)(5,14,27,9)(6,10,28,15)(19,22,38,41)(20,42,39,23)(21,24,40,37)(31,34,45,48)(32,43,46,35)(33,36,47,44), (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), (1,6)(2,5)(3,4)(7,13)(8,18)(9,17)(10,16)(11,15)(12,14)(19,38)(20,37)(21,42)(22,41)(23,40)(24,39)(25,26)(27,30)(28,29)(31,35)(32,34)(43,45)(46,48)>;
G:=Group( (1,24,13,32)(2,47,14,38)(3,20,15,34)(4,43,16,40)(5,22,17,36)(6,45,18,42)(7,23,28,31)(8,46,29,37)(9,19,30,33)(10,48,25,39)(11,21,26,35)(12,44,27,41), (1,16,29,11)(2,12,30,17)(3,18,25,7)(4,8,26,13)(5,14,27,9)(6,10,28,15)(19,22,38,41)(20,42,39,23)(21,24,40,37)(31,34,45,48)(32,43,46,35)(33,36,47,44), (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), (1,6)(2,5)(3,4)(7,13)(8,18)(9,17)(10,16)(11,15)(12,14)(19,38)(20,37)(21,42)(22,41)(23,40)(24,39)(25,26)(27,30)(28,29)(31,35)(32,34)(43,45)(46,48) );
G=PermutationGroup([[(1,24,13,32),(2,47,14,38),(3,20,15,34),(4,43,16,40),(5,22,17,36),(6,45,18,42),(7,23,28,31),(8,46,29,37),(9,19,30,33),(10,48,25,39),(11,21,26,35),(12,44,27,41)], [(1,16,29,11),(2,12,30,17),(3,18,25,7),(4,8,26,13),(5,14,27,9),(6,10,28,15),(19,22,38,41),(20,42,39,23),(21,24,40,37),(31,34,45,48),(32,43,46,35),(33,36,47,44)], [(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)], [(1,6),(2,5),(3,4),(7,13),(8,18),(9,17),(10,16),(11,15),(12,14),(19,38),(20,37),(21,42),(22,41),(23,40),(24,39),(25,26),(27,30),(28,29),(31,35),(32,34),(43,45),(46,48)]])
Matrix representation of C42⋊7D6 ►in GL4(𝔽73) generated by
| 17 | 66 | 63 | 66 |
| 7 | 10 | 7 | 56 |
| 10 | 7 | 17 | 66 |
| 66 | 17 | 7 | 10 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 13 | 30 |
| 0 | 0 | 43 | 43 |
| 13 | 30 | 0 | 0 |
| 43 | 43 | 0 | 0 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 30 | 30 |
| 30 | 13 | 0 | 0 |
| 43 | 43 | 0 | 0 |
G:=sub<GL(4,GF(73))| [17,7,10,66,66,10,7,17,63,7,17,7,66,56,66,10],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[0,0,13,43,0,0,30,43,13,43,0,0,30,43,0,0],[0,0,30,43,0,0,13,43,43,30,0,0,60,30,0,0] >;
C42⋊7D6 in GAP, Magma, Sage, TeX
C_4^2\rtimes_7D_6
% in TeX
G:=Group("C4^2:7D6"); // GroupNames label
G:=SmallGroup(192,620);
// by ID
G=gap.SmallGroup(192,620);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,219,184,1123,570,297,136,1684,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1*b,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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