direct product, p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: Q82, C42.53C22, C22.52C24, C2.192+ 1+4, C4⋊Q8.14C2, C4.19(C2×Q8), (C4×Q8).10C2, C4⋊C4.76C22, (C2×C4).34C23, C2.12(C22×Q8), (C2×Q8).34C22, 2-Sylow(Spin+(4,3)), SmallGroup(64,239)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q82
G = < a,b,c,d | a4=c4=1, b2=a2, d2=c2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 133 in 106 conjugacy classes, 91 normal (4 characteristic)
C1, C2, C2, C4, C4, C22, C2×C4, Q8, Q8, C42, C4⋊C4, C2×Q8, C4×Q8, C4⋊Q8, Q82
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, 2+ 1+4, Q82
Character table of Q82
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ9 | 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 |
| ρ10 | 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 |
| ρ11 | 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 |
| ρ12 | 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 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 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 |
| ρ16 | 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 |
| ρ17 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 2 | 2 | 0 | -2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ20 | 2 | 2 | -2 | -2 | -2 | 2 | 2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | 2 | -2 | 0 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | -2 | 2 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
►Regular action 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 37 3 39)(2 40 4 38)(5 50 7 52)(6 49 8 51)(9 31 11 29)(10 30 12 32)(13 44 15 42)(14 43 16 41)(17 47 19 45)(18 46 20 48)(21 62 23 64)(22 61 24 63)(25 54 27 56)(26 53 28 55)(33 59 35 57)(34 58 36 60)
(1 13 7 17)(2 14 8 18)(3 15 5 19)(4 16 6 20)(9 58 55 61)(10 59 56 62)(11 60 53 63)(12 57 54 64)(21 32 33 27)(22 29 34 28)(23 30 35 25)(24 31 36 26)(37 44 52 47)(38 41 49 48)(39 42 50 45)(40 43 51 46)
(1 26 7 31)(2 27 8 32)(3 28 5 29)(4 25 6 30)(9 39 55 50)(10 40 56 51)(11 37 53 52)(12 38 54 49)(13 36 17 24)(14 33 18 21)(15 34 19 22)(16 35 20 23)(41 57 48 64)(42 58 45 61)(43 59 46 62)(44 60 47 63)
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), (1,37,3,39)(2,40,4,38)(5,50,7,52)(6,49,8,51)(9,31,11,29)(10,30,12,32)(13,44,15,42)(14,43,16,41)(17,47,19,45)(18,46,20,48)(21,62,23,64)(22,61,24,63)(25,54,27,56)(26,53,28,55)(33,59,35,57)(34,58,36,60), (1,13,7,17)(2,14,8,18)(3,15,5,19)(4,16,6,20)(9,58,55,61)(10,59,56,62)(11,60,53,63)(12,57,54,64)(21,32,33,27)(22,29,34,28)(23,30,35,25)(24,31,36,26)(37,44,52,47)(38,41,49,48)(39,42,50,45)(40,43,51,46), (1,26,7,31)(2,27,8,32)(3,28,5,29)(4,25,6,30)(9,39,55,50)(10,40,56,51)(11,37,53,52)(12,38,54,49)(13,36,17,24)(14,33,18,21)(15,34,19,22)(16,35,20,23)(41,57,48,64)(42,58,45,61)(43,59,46,62)(44,60,47,63)>;
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), (1,37,3,39)(2,40,4,38)(5,50,7,52)(6,49,8,51)(9,31,11,29)(10,30,12,32)(13,44,15,42)(14,43,16,41)(17,47,19,45)(18,46,20,48)(21,62,23,64)(22,61,24,63)(25,54,27,56)(26,53,28,55)(33,59,35,57)(34,58,36,60), (1,13,7,17)(2,14,8,18)(3,15,5,19)(4,16,6,20)(9,58,55,61)(10,59,56,62)(11,60,53,63)(12,57,54,64)(21,32,33,27)(22,29,34,28)(23,30,35,25)(24,31,36,26)(37,44,52,47)(38,41,49,48)(39,42,50,45)(40,43,51,46), (1,26,7,31)(2,27,8,32)(3,28,5,29)(4,25,6,30)(9,39,55,50)(10,40,56,51)(11,37,53,52)(12,38,54,49)(13,36,17,24)(14,33,18,21)(15,34,19,22)(16,35,20,23)(41,57,48,64)(42,58,45,61)(43,59,46,62)(44,60,47,63) );
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)], [(1,37,3,39),(2,40,4,38),(5,50,7,52),(6,49,8,51),(9,31,11,29),(10,30,12,32),(13,44,15,42),(14,43,16,41),(17,47,19,45),(18,46,20,48),(21,62,23,64),(22,61,24,63),(25,54,27,56),(26,53,28,55),(33,59,35,57),(34,58,36,60)], [(1,13,7,17),(2,14,8,18),(3,15,5,19),(4,16,6,20),(9,58,55,61),(10,59,56,62),(11,60,53,63),(12,57,54,64),(21,32,33,27),(22,29,34,28),(23,30,35,25),(24,31,36,26),(37,44,52,47),(38,41,49,48),(39,42,50,45),(40,43,51,46)], [(1,26,7,31),(2,27,8,32),(3,28,5,29),(4,25,6,30),(9,39,55,50),(10,40,56,51),(11,37,53,52),(12,38,54,49),(13,36,17,24),(14,33,18,21),(15,34,19,22),(16,35,20,23),(41,57,48,64),(42,58,45,61),(43,59,46,62),(44,60,47,63)]])
Q82 is a maximal subgroup of
Q8⋊SD16 Q8⋊Q16 Q8⋊3Q16 Q8⋊8SD16 Q8⋊5Q16 C42.510C23 C42.516C23 SD16⋊Q8 Q16⋊4Q8 C22.72C25 C22.91C25 C22.92C25 C22.107C25 C22.111C25 C22.133C25 C22.139C25 C22.144C25 Q8⋊SL2(𝔽3)
C42.D2p: Q8≀C2 Dic6⋊9Q8 Dic10⋊9Q8 Dic14⋊9Q8 ...
Q82 is a maximal quotient of
C23.251C24 C23.351C24 C23.407C24 C42⋊7Q8 C23.486C24 C23.634C24 C23.692C24 C23.706C24 C23.711C24
C42.D2p: C42.176D4 C42.180D4 Dic6⋊9Q8 Dic10⋊9Q8 Dic14⋊9Q8 ...
Matrix representation of Q82 ►in GL4(𝔽5) generated by
| 0 | 1 | 0 | 0 |
| 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 0 | 2 | 0 | 0 |
| 2 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 4 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 3 |
| 0 | 0 | 3 | 0 |
G:=sub<GL(4,GF(5))| [0,4,0,0,1,0,0,0,0,0,4,0,0,0,0,4],[0,2,0,0,2,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,4,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,3,0,0,3,0] >;
Q82 in GAP, Magma, Sage, TeX
Q_8^2
% in TeX
G:=Group("Q8^2"); // GroupNames label
G:=SmallGroup(64,239);
// by ID
G=gap.SmallGroup(64,239);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,288,217,103,650,158,297,69]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^4=1,b^2=a^2,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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