metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8.6D6, C24.34D4, C12.21D8, D24⋊14C22, C24.24C23, Dic12⋊11C22, (C2×D8)⋊7S3, (C6×D8)⋊1C2, C4○D24⋊2C2, C3⋊D16⋊5C2, C3⋊C16⋊3C22, D8.S3⋊5C2, (C2×C6).42D8, (C2×C8).83D6, C6.63(C2×D8), C3⋊4(C16⋊C22), C8.2(C3⋊D4), C12.C8⋊2C2, C4.17(D4⋊S3), (C2×C12).180D4, C12.160(C2×D4), C8.30(C22×S3), (C3×D8).6C22, (C2×C24).31C22, C22.10(D4⋊S3), C4.2(C2×C3⋊D4), C2.18(C2×D4⋊S3), (C2×C4).79(C3⋊D4), SmallGroup(192,706)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8.D6
G = < a,b,c,d | a8=b2=1, c6=d2=a4, bab=dad-1=a-1, ac=ca, cbc-1=a4b, dbd-1=ab, dcd-1=c5 >
Subgroups: 312 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C2×C6, C16, C2×C8, D8, D8, SD16, Q16, C2×D4, C4○D4, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C22×C6, M5(2), D16, SD32, C2×D8, C4○D8, C3⋊C16, C24⋊C2, D24, Dic12, C2×C24, C3×D8, C3×D8, C4○D12, C6×D4, C16⋊C22, C12.C8, C3⋊D16, D8.S3, C4○D24, C6×D8, D8.D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C3⋊D4, C22×S3, C2×D8, D4⋊S3, C2×C3⋊D4, C16⋊C22, C2×D4⋊S3, D8.D6
Character table of D8.D6
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 12A | 12B | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 2 | 8 | 8 | 24 | 2 | 2 | 2 | 24 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 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 | 2 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 2 | 2 | -2 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | -2 | 2 | -2 | 0 | -1 | -2 | 2 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 2 | 2 | -2 | -1 | 1 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | -2 | -2 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -√2 | √2 | -√2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ16 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | √2 | -√2 | √2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ17 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | √2 | √2 | -√2 | -√2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ18 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -√2 | -√2 | √2 | √2 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ19 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | -1 | 1 | 1 | -√-3 | -√-3 | √-3 | √-3 | -2 | -2 | 2 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | -√-3 | √-3 | -√-3 | √-3 | -2 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | 0 | -1 | -1 | -1 | √-3 | -√-3 | √-3 | -√-3 | -2 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -2 | 2 | 0 | -1 | 1 | 1 | √-3 | √-3 | -√-3 | -√-3 | -2 | -2 | 2 | -1 | 1 | 0 | 0 | 0 | 0 | -1 | 1 | -1 | 1 | complex lifted from C3⋊D4 |
| ρ23 | 4 | 4 | -4 | 0 | 0 | 0 | -2 | 4 | -4 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ24 | 4 | 4 | 4 | 0 | 0 | 0 | -2 | -4 | -4 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4⋊S3, Schur index 2 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | orthogonal lifted from C16⋊C22 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | orthogonal lifted from C16⋊C22 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | -√2 | -√-6 | √2 | complex faithful |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-6 | √2 | -√-6 | -√2 | complex faithful |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | -2√-3 | 2√-3 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | -√2 | √-6 | √2 | complex faithful |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 2√-3 | -2√-3 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-6 | √2 | √-6 | -√2 | complex faithful |
►On 48 points
Generators in S48
(1 22 10 19 7 16 4 13)(2 23 11 20 8 17 5 14)(3 24 12 21 9 18 6 15)(25 37 28 40 31 43 34 46)(26 38 29 41 32 44 35 47)(27 39 30 42 33 45 36 48)
(1 13)(2 20)(3 15)(4 22)(5 17)(6 24)(7 19)(8 14)(9 21)(10 16)(11 23)(12 18)(25 34)(26 29)(27 36)(28 31)(30 33)(32 35)(37 43)(39 45)(41 47)
(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 34 7 28)(2 27 8 33)(3 32 9 26)(4 25 10 31)(5 30 11 36)(6 35 12 29)(13 46 19 40)(14 39 20 45)(15 44 21 38)(16 37 22 43)(17 42 23 48)(18 47 24 41)
G:=sub<Sym(48)| (1,22,10,19,7,16,4,13)(2,23,11,20,8,17,5,14)(3,24,12,21,9,18,6,15)(25,37,28,40,31,43,34,46)(26,38,29,41,32,44,35,47)(27,39,30,42,33,45,36,48), (1,13)(2,20)(3,15)(4,22)(5,17)(6,24)(7,19)(8,14)(9,21)(10,16)(11,23)(12,18)(25,34)(26,29)(27,36)(28,31)(30,33)(32,35)(37,43)(39,45)(41,47), (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), (1,34,7,28)(2,27,8,33)(3,32,9,26)(4,25,10,31)(5,30,11,36)(6,35,12,29)(13,46,19,40)(14,39,20,45)(15,44,21,38)(16,37,22,43)(17,42,23,48)(18,47,24,41)>;
G:=Group( (1,22,10,19,7,16,4,13)(2,23,11,20,8,17,5,14)(3,24,12,21,9,18,6,15)(25,37,28,40,31,43,34,46)(26,38,29,41,32,44,35,47)(27,39,30,42,33,45,36,48), (1,13)(2,20)(3,15)(4,22)(5,17)(6,24)(7,19)(8,14)(9,21)(10,16)(11,23)(12,18)(25,34)(26,29)(27,36)(28,31)(30,33)(32,35)(37,43)(39,45)(41,47), (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), (1,34,7,28)(2,27,8,33)(3,32,9,26)(4,25,10,31)(5,30,11,36)(6,35,12,29)(13,46,19,40)(14,39,20,45)(15,44,21,38)(16,37,22,43)(17,42,23,48)(18,47,24,41) );
G=PermutationGroup([[(1,22,10,19,7,16,4,13),(2,23,11,20,8,17,5,14),(3,24,12,21,9,18,6,15),(25,37,28,40,31,43,34,46),(26,38,29,41,32,44,35,47),(27,39,30,42,33,45,36,48)], [(1,13),(2,20),(3,15),(4,22),(5,17),(6,24),(7,19),(8,14),(9,21),(10,16),(11,23),(12,18),(25,34),(26,29),(27,36),(28,31),(30,33),(32,35),(37,43),(39,45),(41,47)], [(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)], [(1,34,7,28),(2,27,8,33),(3,32,9,26),(4,25,10,31),(5,30,11,36),(6,35,12,29),(13,46,19,40),(14,39,20,45),(15,44,21,38),(16,37,22,43),(17,42,23,48),(18,47,24,41)]])
Matrix representation of D8.D6 ►in GL4(𝔽7) generated by
| 0 | 2 | 5 | 5 |
| 3 | 5 | 6 | 5 |
| 1 | 1 | 5 | 5 |
| 6 | 1 | 4 | 3 |
| 4 | 5 | 1 | 4 |
| 1 | 0 | 3 | 2 |
| 1 | 1 | 2 | 5 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 1 | 6 |
| 2 | 5 | 1 | 4 |
| 1 | 4 | 6 | 1 |
| 2 | 2 | 2 | 3 |
| 5 | 3 | 0 | 4 |
| 2 | 5 | 1 | 4 |
| 5 | 5 | 5 | 4 |
| 6 | 3 | 1 | 6 |
G:=sub<GL(4,GF(7))| [0,3,1,6,2,5,1,1,5,6,5,4,5,5,5,3],[4,1,1,0,5,0,1,0,1,3,2,0,4,2,5,1],[0,2,1,2,1,5,4,2,1,1,6,2,6,4,1,3],[5,2,5,6,3,5,5,3,0,1,5,1,4,4,4,6] >;
D8.D6 in GAP, Magma, Sage, TeX
D_8.D_6
% in TeX
G:=Group("D8.D6"); // GroupNames label
G:=SmallGroup(192,706);
// by ID
G=gap.SmallGroup(192,706);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,387,675,185,192,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=1,c^6=d^2=a^4,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^4*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;
// generators/relations
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