metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D8:2D6, C16:2D6, D16:2S3, D6.6D8, C48:4C22, Dic3.8D8, C24.14C23, D24.1C22, Dic12:4C22, C3:C8.2D4, (S3xD8):4C2, C4.2(S3xD4), (C3xD16):4C2, C3:D16:2C2, C3:C16:1C22, D8.S3:1C2, D6.C8:3C2, (C4xS3).7D4, C48:C2:3C2, C2.17(S3xD8), C12.8(C2xD4), C6.33(C2xD8), D8:3S3:3C2, C3:2(C16:C22), (C3xD8):6C22, (S3xC8).3C22, C8.20(C22xS3), SmallGroup(192,470)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8:D6
G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=dad=a-1, cbc-1=a5b, dbd=ab, dcd=c-1 >
Subgroups: 380 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2xC6, C16, C16, C2xC8, D8, D8, SD16, Q16, C2xD4, C4oD4, C3:C8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C3xD4, C22xS3, M5(2), D16, D16, SD32, C2xD8, C4oD8, C3:C16, C48, S3xC8, D24, Dic12, D4:S3, D4.S3, C3xD8, S3xD4, D4:2S3, C16:C22, D6.C8, C48:C2, C3:D16, D8.S3, C3xD16, S3xD8, D8:3S3, D8:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C22xS3, C2xD8, S3xD4, C16:C22, S3xD8, D8:D6
Character table of D8:D6
class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 12 | 16A | 16B | 16C | 16D | 24A | 24B | 48A | 48B | 48C | 48D | |
size | 1 | 1 | 6 | 8 | 8 | 24 | 2 | 2 | 6 | 24 | 2 | 16 | 16 | 2 | 2 | 12 | 4 | 4 | 4 | 12 | 12 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
ρ9 | 2 | 2 | 0 | 2 | 2 | 0 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ10 | 2 | 2 | 0 | -2 | -2 | 0 | -1 | 2 | 0 | 0 | -1 | 1 | 1 | 2 | 2 | 0 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
ρ11 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ12 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ13 | 2 | 2 | 0 | -2 | 2 | 0 | -1 | 2 | 0 | 0 | -1 | 1 | -1 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ14 | 2 | 2 | 0 | 2 | -2 | 0 | -1 | 2 | 0 | 0 | -1 | -1 | 1 | 2 | 2 | 0 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ15 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | √2 | -√2 | -√2 | √2 | 0 | 0 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
ρ16 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | √2 | -√2 | √2 | -√2 | 0 | 0 | √2 | -√2 | √2 | -√2 | orthogonal lifted from D8 |
ρ17 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -√2 | √2 | -√2 | √2 | 0 | 0 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
ρ18 | 2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -√2 | √2 | √2 | -√2 | 0 | 0 | -√2 | √2 | -√2 | √2 | orthogonal lifted from D8 |
ρ19 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | 4 | 0 | 0 | -2 | 0 | 0 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xD4 |
ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | orthogonal lifted from C16:C22 |
ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | orthogonal lifted from C16:C22 |
ρ22 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | orthogonal lifted from S3xD8 |
ρ23 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -4 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | orthogonal lifted from S3xD8 |
ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | 2ζ167ζ3+ζ167-2ζ16ζ3-ζ16 | -2ζ1613ζ32-ζ1613+2ζ1611ζ32+ζ1611 | -2ζ167ζ3-ζ167+2ζ16ζ3+ζ16 | -2ζ165ζ32-ζ165+2ζ163ζ32+ζ163 | complex faithful |
ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -2ζ1613ζ32-ζ1613+2ζ1611ζ32+ζ1611 | -2ζ167ζ3-ζ167+2ζ16ζ3+ζ16 | -2ζ165ζ32-ζ165+2ζ163ζ32+ζ163 | 2ζ167ζ3+ζ167-2ζ16ζ3-ζ16 | complex faithful |
ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -2ζ167ζ3-ζ167+2ζ16ζ3+ζ16 | -2ζ165ζ32-ζ165+2ζ163ζ32+ζ163 | 2ζ167ζ3+ζ167-2ζ16ζ3-ζ16 | -2ζ1613ζ32-ζ1613+2ζ1611ζ32+ζ1611 | complex faithful |
ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | -2ζ165ζ32-ζ165+2ζ163ζ32+ζ163 | 2ζ167ζ3+ζ167-2ζ16ζ3-ζ16 | -2ζ1613ζ32-ζ1613+2ζ1611ζ32+ζ1611 | -2ζ167ζ3-ζ167+2ζ16ζ3+ζ16 | complex faithful |
(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 10)(2 9)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 32)(33 44)(34 43)(35 42)(36 41)(37 48)(38 47)(39 46)(40 45)
(1 36 25)(2 35 26 8 37 32)(3 34 27 7 38 31)(4 33 28 6 39 30)(5 40 29)(9 45 22 14 48 19)(10 44 23 13 41 18)(11 43 24 12 42 17)(15 47 20 16 46 21)
(1 25)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(16 24)(33 39)(34 38)(35 37)(41 48)(42 47)(43 46)(44 45)
G:=sub<Sym(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), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32)(33,44)(34,43)(35,42)(36,41)(37,48)(38,47)(39,46)(40,45), (1,36,25)(2,35,26,8,37,32)(3,34,27,7,38,31)(4,33,28,6,39,30)(5,40,29)(9,45,22,14,48,19)(10,44,23,13,41,18)(11,43,24,12,42,17)(15,47,20,16,46,21), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(16,24)(33,39)(34,38)(35,37)(41,48)(42,47)(43,46)(44,45)>;
G:=Group( (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), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,32)(33,44)(34,43)(35,42)(36,41)(37,48)(38,47)(39,46)(40,45), (1,36,25)(2,35,26,8,37,32)(3,34,27,7,38,31)(4,33,28,6,39,30)(5,40,29)(9,45,22,14,48,19)(10,44,23,13,41,18)(11,43,24,12,42,17)(15,47,20,16,46,21), (1,25)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,23)(10,22)(11,21)(12,20)(13,19)(14,18)(15,17)(16,24)(33,39)(34,38)(35,37)(41,48)(42,47)(43,46)(44,45) );
G=PermutationGroup([[(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)], [(1,10),(2,9),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,32),(33,44),(34,43),(35,42),(36,41),(37,48),(38,47),(39,46),(40,45)], [(1,36,25),(2,35,26,8,37,32),(3,34,27,7,38,31),(4,33,28,6,39,30),(5,40,29),(9,45,22,14,48,19),(10,44,23,13,41,18),(11,43,24,12,42,17),(15,47,20,16,46,21)], [(1,25),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,23),(10,22),(11,21),(12,20),(13,19),(14,18),(15,17),(16,24),(33,39),(34,38),(35,37),(41,48),(42,47),(43,46),(44,45)]])
Matrix representation of D8:D6 ►in GL4(F97) generated by
7 | 0 | 90 | 0 |
0 | 7 | 0 | 90 |
7 | 0 | 7 | 0 |
0 | 7 | 0 | 7 |
96 | 95 | 15 | 30 |
2 | 1 | 67 | 82 |
15 | 30 | 1 | 2 |
67 | 82 | 95 | 96 |
96 | 96 | 0 | 0 |
1 | 0 | 0 | 0 |
0 | 0 | 1 | 1 |
0 | 0 | 96 | 0 |
96 | 96 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 1 | 1 |
0 | 0 | 0 | 96 |
G:=sub<GL(4,GF(97))| [7,0,7,0,0,7,0,7,90,0,7,0,0,90,0,7],[96,2,15,67,95,1,30,82,15,67,1,95,30,82,2,96],[96,1,0,0,96,0,0,0,0,0,1,96,0,0,1,0],[96,0,0,0,96,1,0,0,0,0,1,0,0,0,1,96] >;
D8:D6 in GAP, Magma, Sage, TeX
D_8\rtimes D_6
% in TeX
G:=Group("D8:D6");
// GroupNames label
G:=SmallGroup(192,470);
// by ID
G=gap.SmallGroup(192,470);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,135,346,185,192,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^5*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export