p-group, metabelian, nilpotent (class 4), monomial
Aliases: D8.1Q8, C42.149D4, C4⋊C16⋊9C2, C16⋊3C4⋊5C2, C16⋊4C4⋊8C2, (C2×C4).39D8, C8.33(C2×Q8), (C4×D8).13C2, (C2×C8).176D4, C8.5Q8⋊1C2, C2.D16.4C2, C8.67(C4○D4), C2.13(C4○D16), (C2×C8).530C23, (C4×C8).105C22, (C2×C16).44C22, C22.116(C2×D8), C4.49(C22⋊Q8), C2.16(C16⋊C22), C2.D8.15C22, C2.16(D4⋊Q8), (C2×D8).114C22, C4.12(C8.C22), (C2×C4).798(C2×D4), SmallGroup(128,960)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for D8.Q8
G = < a,b,c,d | a8=b2=c4=1, d2=a4c2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a5b, dcd-1=a4c-1 >
Subgroups: 180 in 68 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, C23, C16, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, C22×C4, C2×D4, C4×C8, D4⋊C4, C4.Q8, C2.D8, C2×C16, C4×D4, C42.C2, C2×D8, C2.D16, C4⋊C16, C16⋊3C4, C16⋊4C4, C4×D8, C8.5Q8, D8.Q8
Quotients: C1, C2, C22, D4, Q8, C23, D8, C2×D4, C2×Q8, C4○D4, C22⋊Q8, C2×D8, C8.C22, D4⋊Q8, C4○D16, C16⋊C22, D8.Q8
Character table of D8.Q8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 8A | 8B | 8C | 8D | 8E | 8F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 16 | 16 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | -√2 | √2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | √2 | -√2 | -√2 | -√2 | √2 | orthogonal lifted from D8 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√2 | √2 | √2 | -√2 | orthogonal lifted from D8 |
| ρ15 | 2 | -2 | -2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ16 | 2 | -2 | -2 | 2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | -2i | 2i | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 2i | -2i | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √-2 | -√-2 | ζ165+ζ163 | ζ167+ζ16 | ζ1613+ζ1611 | -ζ165+ζ163 | ζ1615+ζ169 | ζ1615-ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | complex lifted from C4○D16 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√-2 | √-2 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ165+ζ163 | -ζ165+ζ163 | ζ167+ζ16 | ζ1615-ζ169 | -ζ1615+ζ169 | ζ165-ζ163 | complex lifted from C4○D16 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√-2 | √-2 | ζ167+ζ16 | ζ1613+ζ1611 | ζ1615+ζ169 | -ζ1615+ζ169 | ζ165+ζ163 | -ζ165+ζ163 | ζ165-ζ163 | ζ1615-ζ169 | complex lifted from C4○D16 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √-2 | -√-2 | ζ1615+ζ169 | ζ165+ζ163 | ζ167+ζ16 | -ζ1615+ζ169 | ζ1613+ζ1611 | -ζ165+ζ163 | ζ165-ζ163 | ζ1615-ζ169 | complex lifted from C4○D16 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | -√-2 | √-2 | ζ1615+ζ169 | ζ165+ζ163 | ζ167+ζ16 | ζ1615-ζ169 | ζ1613+ζ1611 | ζ165-ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | complex lifted from C4○D16 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | √2 | -√2 | √2 | -√2 | √-2 | -√-2 | ζ167+ζ16 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ1615-ζ169 | ζ165+ζ163 | ζ165-ζ163 | -ζ165+ζ163 | -ζ1615+ζ169 | complex lifted from C4○D16 |
| ρ25 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | √-2 | -√-2 | ζ1613+ζ1611 | ζ1615+ζ169 | ζ165+ζ163 | ζ165-ζ163 | ζ167+ζ16 | -ζ1615+ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | complex lifted from C4○D16 |
| ρ26 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -√2 | √2 | -√2 | √2 | -√-2 | √-2 | ζ165+ζ163 | ζ167+ζ16 | ζ1613+ζ1611 | ζ165-ζ163 | ζ1615+ζ169 | -ζ1615+ζ169 | ζ1615-ζ169 | -ζ165+ζ163 | complex lifted from C4○D16 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | -2√2 | 2√2 | 2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C16⋊C22 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 2√2 | -2√2 | -2√2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C16⋊C22 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►On 64 points
Generators in S64
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 10)(2 9)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(17 28)(18 27)(19 26)(20 25)(21 32)(22 31)(23 30)(24 29)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 48)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(56 64)
(1 31 15 19)(2 32 16 20)(3 25 9 21)(4 26 10 22)(5 27 11 23)(6 28 12 24)(7 29 13 17)(8 30 14 18)(33 53 45 57)(34 54 46 58)(35 55 47 59)(36 56 48 60)(37 49 41 61)(38 50 42 62)(39 51 43 63)(40 52 44 64)
(1 45 11 37)(2 44 12 36)(3 43 13 35)(4 42 14 34)(5 41 15 33)(6 48 16 40)(7 47 9 39)(8 46 10 38)(17 63 25 55)(18 62 26 54)(19 61 27 53)(20 60 28 52)(21 59 29 51)(22 58 30 50)(23 57 31 49)(24 64 32 56)
G:=sub<Sym(64)| (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,48)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,64), (1,31,15,19)(2,32,16,20)(3,25,9,21)(4,26,10,22)(5,27,11,23)(6,28,12,24)(7,29,13,17)(8,30,14,18)(33,53,45,57)(34,54,46,58)(35,55,47,59)(36,56,48,60)(37,49,41,61)(38,50,42,62)(39,51,43,63)(40,52,44,64), (1,45,11,37)(2,44,12,36)(3,43,13,35)(4,42,14,34)(5,41,15,33)(6,48,16,40)(7,47,9,39)(8,46,10,38)(17,63,25,55)(18,62,26,54)(19,61,27,53)(20,60,28,52)(21,59,29,51)(22,58,30,50)(23,57,31,49)(24,64,32,56)>;
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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,10)(2,9)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)(33,47)(34,46)(35,45)(36,44)(37,43)(38,42)(39,41)(40,48)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(56,64), (1,31,15,19)(2,32,16,20)(3,25,9,21)(4,26,10,22)(5,27,11,23)(6,28,12,24)(7,29,13,17)(8,30,14,18)(33,53,45,57)(34,54,46,58)(35,55,47,59)(36,56,48,60)(37,49,41,61)(38,50,42,62)(39,51,43,63)(40,52,44,64), (1,45,11,37)(2,44,12,36)(3,43,13,35)(4,42,14,34)(5,41,15,33)(6,48,16,40)(7,47,9,39)(8,46,10,38)(17,63,25,55)(18,62,26,54)(19,61,27,53)(20,60,28,52)(21,59,29,51)(22,58,30,50)(23,57,31,49)(24,64,32,56) );
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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,10),(2,9),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(17,28),(18,27),(19,26),(20,25),(21,32),(22,31),(23,30),(24,29),(33,47),(34,46),(35,45),(36,44),(37,43),(38,42),(39,41),(40,48),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(56,64)], [(1,31,15,19),(2,32,16,20),(3,25,9,21),(4,26,10,22),(5,27,11,23),(6,28,12,24),(7,29,13,17),(8,30,14,18),(33,53,45,57),(34,54,46,58),(35,55,47,59),(36,56,48,60),(37,49,41,61),(38,50,42,62),(39,51,43,63),(40,52,44,64)], [(1,45,11,37),(2,44,12,36),(3,43,13,35),(4,42,14,34),(5,41,15,33),(6,48,16,40),(7,47,9,39),(8,46,10,38),(17,63,25,55),(18,62,26,54),(19,61,27,53),(20,60,28,52),(21,59,29,51),(22,58,30,50),(23,57,31,49),(24,64,32,56)]])
Matrix representation of D8.Q8 ►in GL4(𝔽17) generated by
| 11 | 11 | 0 | 0 |
| 3 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 11 | 11 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 15 | 13 |
| 0 | 0 | 14 | 2 |
| 15 | 5 | 0 | 0 |
| 13 | 2 | 0 | 0 |
| 0 | 0 | 4 | 6 |
| 0 | 0 | 0 | 13 |
G:=sub<GL(4,GF(17))| [11,3,0,0,11,0,0,0,0,0,1,0,0,0,0,1],[11,3,0,0,11,6,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,15,14,0,0,13,2],[15,13,0,0,5,2,0,0,0,0,4,0,0,0,6,13] >;
D8.Q8 in GAP, Magma, Sage, TeX
D_8.Q_8
% in TeX
G:=Group("D8.Q8"); // GroupNames label
G:=SmallGroup(128,960);
// by ID
G=gap.SmallGroup(128,960);
# by ID
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,56,141,512,422,1684,438,242,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^4*c^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=a^4*c^-1>;
// generators/relations
Export