p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8.Q8, C42.27C22, C4⋊C8.8C2, (C2×C4).36D4, (C4×Q8).8C2, C4.16(C2×Q8), C2.D8.5C2, C4.Q8.5C2, C2.14(C4○D8), C4.28(C4○D4), C4⋊C4.15C22, (C2×C8).36C22, Q8⋊C4.5C2, C42.C2.2C2, (C2×C4).104C23, C22.100(C2×D4), C2.17(C22⋊Q8), (C2×Q8).53C22, C2.16(C8.C22), SmallGroup(64,160)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8.Q8
G = < a,b,c,d | a4=c4=1, b2=a2, d2=a2c2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >
Character table of Q8.Q8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | 2i | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ15 | 2 | -2 | -2 | 2 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √2 | -√2 | √-2 | complex lifted from C4○D8 |
| ρ16 | 2 | -2 | -2 | 2 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √2 | -√2 | -√-2 | complex lifted from C4○D8 |
| ρ17 | 2 | -2 | -2 | 2 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√2 | √2 | -√-2 | complex lifted from C4○D8 |
| ρ18 | 2 | -2 | -2 | 2 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√2 | √2 | √-2 | complex lifted from C4○D8 |
| ρ19 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►Regular action on 64 points
(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 11 3 9)(2 10 4 12)(5 13 7 15)(6 16 8 14)(17 25 19 27)(18 28 20 26)(21 29 23 31)(22 32 24 30)(33 44 35 42)(34 43 36 41)(37 48 39 46)(38 47 40 45)(49 60 51 58)(50 59 52 57)(53 64 55 62)(54 63 56 61)
(1 21 5 19)(2 22 6 20)(3 23 7 17)(4 24 8 18)(9 31 15 25)(10 32 16 26)(11 29 13 27)(12 30 14 28)(33 51 39 53)(34 52 40 54)(35 49 37 55)(36 50 38 56)(41 59 47 61)(42 60 48 62)(43 57 45 63)(44 58 46 64)
(1 37 7 33)(2 40 8 36)(3 39 5 35)(4 38 6 34)(9 47 13 43)(10 46 14 42)(11 45 15 41)(12 48 16 44)(17 55 21 51)(18 54 22 50)(19 53 23 49)(20 56 24 52)(25 63 29 59)(26 62 30 58)(27 61 31 57)(28 64 32 60)
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), (1,11,3,9)(2,10,4,12)(5,13,7,15)(6,16,8,14)(17,25,19,27)(18,28,20,26)(21,29,23,31)(22,32,24,30)(33,44,35,42)(34,43,36,41)(37,48,39,46)(38,47,40,45)(49,60,51,58)(50,59,52,57)(53,64,55,62)(54,63,56,61), (1,21,5,19)(2,22,6,20)(3,23,7,17)(4,24,8,18)(9,31,15,25)(10,32,16,26)(11,29,13,27)(12,30,14,28)(33,51,39,53)(34,52,40,54)(35,49,37,55)(36,50,38,56)(41,59,47,61)(42,60,48,62)(43,57,45,63)(44,58,46,64), (1,37,7,33)(2,40,8,36)(3,39,5,35)(4,38,6,34)(9,47,13,43)(10,46,14,42)(11,45,15,41)(12,48,16,44)(17,55,21,51)(18,54,22,50)(19,53,23,49)(20,56,24,52)(25,63,29,59)(26,62,30,58)(27,61,31,57)(28,64,32,60)>;
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), (1,11,3,9)(2,10,4,12)(5,13,7,15)(6,16,8,14)(17,25,19,27)(18,28,20,26)(21,29,23,31)(22,32,24,30)(33,44,35,42)(34,43,36,41)(37,48,39,46)(38,47,40,45)(49,60,51,58)(50,59,52,57)(53,64,55,62)(54,63,56,61), (1,21,5,19)(2,22,6,20)(3,23,7,17)(4,24,8,18)(9,31,15,25)(10,32,16,26)(11,29,13,27)(12,30,14,28)(33,51,39,53)(34,52,40,54)(35,49,37,55)(36,50,38,56)(41,59,47,61)(42,60,48,62)(43,57,45,63)(44,58,46,64), (1,37,7,33)(2,40,8,36)(3,39,5,35)(4,38,6,34)(9,47,13,43)(10,46,14,42)(11,45,15,41)(12,48,16,44)(17,55,21,51)(18,54,22,50)(19,53,23,49)(20,56,24,52)(25,63,29,59)(26,62,30,58)(27,61,31,57)(28,64,32,60) );
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)], [(1,11,3,9),(2,10,4,12),(5,13,7,15),(6,16,8,14),(17,25,19,27),(18,28,20,26),(21,29,23,31),(22,32,24,30),(33,44,35,42),(34,43,36,41),(37,48,39,46),(38,47,40,45),(49,60,51,58),(50,59,52,57),(53,64,55,62),(54,63,56,61)], [(1,21,5,19),(2,22,6,20),(3,23,7,17),(4,24,8,18),(9,31,15,25),(10,32,16,26),(11,29,13,27),(12,30,14,28),(33,51,39,53),(34,52,40,54),(35,49,37,55),(36,50,38,56),(41,59,47,61),(42,60,48,62),(43,57,45,63),(44,58,46,64)], [(1,37,7,33),(2,40,8,36),(3,39,5,35),(4,38,6,34),(9,47,13,43),(10,46,14,42),(11,45,15,41),(12,48,16,44),(17,55,21,51),(18,54,22,50),(19,53,23,49),(20,56,24,52),(25,63,29,59),(26,62,30,58),(27,61,31,57),(28,64,32,60)]])
Q8.Q8 is a maximal subgroup of
Q8.Dic6
C42.D2p: C42.447D4 C42.220D4 C42.449D4 C42.384D4 C42.226D4 C42.229D4 C42.235D4 C42.283D4 ...
C4⋊C4.D2p: C42.21C23 C42.22C23 C42.23C23 C42.353C23 C42.354C23 C42.360C23 C42.361C23 C42.424C23 ...
Q8.Q8 is a maximal quotient of
C2.D8⋊4C4 C4.Q8⋊10C4 (C2×C4).26D8 C4.(C4⋊Q8) (C2×C8).171D4 C4⋊C4.Q8
C42.D2p: C42.101D4 C42.123D4 Dic6.3Q8 Q8.5Dic6 Dic6.4Q8 Dic10.3Q8 Q8.3Dic10 Dic10.4Q8 ...
C4⋊C4.D2p: Q8⋊(C4⋊C4) Q8⋊C4⋊C4 C42.31Q8 C4⋊C4.85D4 C2.(C8⋊Q8) (C2×Q8).8Q8 (C2×C8).52D4 Q8.3Dic6 ...
Matrix representation of Q8.Q8 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 1 |
| 0 | 0 | 1 | 7 |
| 0 | 13 | 0 | 0 |
| 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 0 | 13 | 11 |
| 0 | 0 | 11 | 4 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,10,1,0,0,1,7],[0,13,0,0,13,0,0,0,0,0,13,0,0,0,0,13],[0,16,0,0,1,0,0,0,0,0,13,11,0,0,11,4] >;
Q8.Q8 in GAP, Magma, Sage, TeX
Q_8.Q_8
% in TeX
G:=Group("Q8.Q8"); // GroupNames label
G:=SmallGroup(64,160);
// by ID
G=gap.SmallGroup(64,160);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,247,362,158,1444,376,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=1,b^2=a^2,d^2=a^2*c^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;
// generators/relations
Export
Subgroup lattice of Q8.Q8 in TeX
Character table of Q8.Q8 in TeX