p-group, metabelian, nilpotent (class 3), monomial
Aliases: Q8:1Q8, C4.12SD16, C42.23C22, C4:C8.9C2, C4:Q8.5C2, (C4xQ8).6C2, C4.12(C2xQ8), C4.Q8.4C2, (C2xC4).132D4, C4.24(C4oD4), C4:C4.12C22, (C2xC8).33C22, Q8:C4.4C2, C2.10(C2xSD16), C22.96(C2xD4), (C2xC4).100C23, C2.13(C22:Q8), (C2xQ8).51C22, C2.15(C8.C22), SmallGroup(64,156)
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=c2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=c-1 >
Quotients: C1, C2, C22, D4, Q8, C23, SD16, C2xD4, C2xQ8, C4oD4, C22:Q8, C2xSD16, C8.C22, Q8:Q8
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 C4oD4 |
| ρ14 | 2 | -2 | 2 | -2 | 0 | -2 | 2 | 0 | -2i | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4oD4 |
| ρ15 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | -√-2 | √-2 | -√-2 | complex lifted from SD16 |
| ρ16 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | -√-2 | √-2 | √-2 | complex lifted from SD16 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √-2 | √-2 | -√-2 | -√-2 | complex lifted from SD16 |
| ρ18 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√-2 | √-2 | -√-2 | √-2 | complex lifted from SD16 |
| 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 23 7 19)(2 24 8 20)(3 21 5 17)(4 22 6 18)(9 29 13 25)(10 30 14 26)(11 31 15 27)(12 32 16 28)(33 49 37 53)(34 50 38 54)(35 51 39 55)(36 52 40 56)(41 57 45 61)(42 58 46 62)(43 59 47 63)(44 60 48 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), (1,23,7,19)(2,24,8,20)(3,21,5,17)(4,22,6,18)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28)(33,49,37,53)(34,50,38,54)(35,51,39,55)(36,52,40,56)(41,57,45,61)(42,58,46,62)(43,59,47,63)(44,60,48,64), (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), (1,23,7,19)(2,24,8,20)(3,21,5,17)(4,22,6,18)(9,29,13,25)(10,30,14,26)(11,31,15,27)(12,32,16,28)(33,49,37,53)(34,50,38,54)(35,51,39,55)(36,52,40,56)(41,57,45,61)(42,58,46,62)(43,59,47,63)(44,60,48,64), (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)], [(1,23,7,19),(2,24,8,20),(3,21,5,17),(4,22,6,18),(9,29,13,25),(10,30,14,26),(11,31,15,27),(12,32,16,28),(33,49,37,53),(34,50,38,54),(35,51,39,55),(36,52,40,56),(41,57,45,61),(42,58,46,62),(43,59,47,63),(44,60,48,64)], [(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
C42.201C23 Q8:3SD16 D4.5SD16 Q8:3Q16 Q8:4Q16 C42.211C23 Q8:4SD16 Q8.SD16 C8:14SD16 Q8.3SD16 Q8.2Q16 C8:SD16 C42.251C23 C42.255C23 C42.447D4 C42.448D4 C42.23C23 C42.223D4 C42.451D4 C42.231D4 C42.233D4 C42.355C23 C42.359C23 C42.281D4 C42.285D4 C42.289D4 C42.291D4 C42.424C23 C42.426C23 C42.294D4 C42.296D4 C42.299D4 C42.303D4 C42.25C23 C42.30C23 D4:8SD16 C42.469C23 C42.51C23 C42.56C23 C42.477C23 C42.481C23 D4:9SD16 C42.491C23 C42.58C23 C42.62C23 C42.494C23 C42.497C23 Q8:7SD16 C42.506C23 C42.510C23 C42.514C23 C42.515C23 C42.517C23 Q8xSD16 Q16:6Q8 SD16:2Q8 SD16:3Q8 Q8:Dic6
C4p.SD16: Q8:1Q16 C8.SD16 Dic6:4Q8 Q8:4Dic6 Dic6:6Q8 Dic10:4Q8 C20.48SD16 Dic10:6Q8 ...
(CpxQ8):Q8: C42.220D4 C42.21C23 Q8:2Dic6 Q8:Dic10 Q8:Dic14 ...
C4p:Q8.C2: Q16:4Q8 Dic6:Q8 Dic10:Q8 Dic14:Q8 ...
Q8:Q8 is a maximal quotient of
C4.Q8:9C4 C4.(C4:Q8) (C2xC8).170D4 (C2xC4).28D8
C42.D2p: C42.99D4 C42.122D4 Dic6:4Q8 Q8:4Dic6 Dic6:6Q8 Dic10:4Q8 C20.48SD16 Dic10:6Q8 ...
(CpxQ8):Q8: (C2xQ8):Q8 Q8:2Dic6 Q8:Dic10 Q8:Dic14 ...
C4:C4.D2p: Q8:C4:C4 C42.30Q8 (C2xC8):Q8 (C2xQ8).8Q8 Dic6:Q8 Dic10:Q8 Dic14:Q8 ...
Matrix representation of Q8:Q8 ►in GL4(F17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 13 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 |
| 0 | 4 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 10 | 13 | 0 | 0 |
| 4 | 7 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 2 | 0 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[0,1,0,0,1,0,0,0,0,0,0,16,0,0,1,0],[0,4,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[10,4,0,0,13,7,0,0,0,0,0,2,0,0,9,0] >;
Q8:Q8 in GAP, Magma, Sage, TeX
Q_8\rtimes Q_8
% in TeX
G:=Group("Q8:Q8"); // GroupNames label
G:=SmallGroup(64,156);
// by ID
G=gap.SmallGroup(64,156);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,55,362,158,1444,376,88]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=1,b^2=a^2,d^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=c^-1>;
// generators/relations
Export
Subgroup lattice of Q8:Q8 in TeX
Character table of Q8:Q8 in TeX