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G = Q82order 64 = 26

Direct product of Q8 and Q8

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

C1C22 — Q82
C1C2C22C2×C4C42C4×Q8 — Q82
C1C22 — Q82
C1C22 — Q82
C1C22 — Q82

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 [×2], C4 [×12], C4 [×9], C22, C2×C4 [×15], Q8 [×8], Q8 [×6], C42 [×9], C4⋊C4 [×18], C2×Q8 [×8], C4×Q8 [×6], C4⋊Q8 [×9], Q82
Quotients: C1, C2 [×15], C22 [×35], Q8 [×8], C23 [×15], C2×Q8 [×12], C24, C22×Q8 [×2], 2+ 1+4, Q82

Character table of Q82

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S4T4U
 size 1111222222222222444444444
ρ11111111111111111111111111    trivial
ρ21111-1-1-11111111-111-11-11-1-1-1-1    linear of order 2
ρ311111-11-1-11-1-11-1-1-1111-1-11-1-11    linear of order 2
ρ41111-11-1-1-11-1-11-11-11-111-1-111-1    linear of order 2
ρ51111-1-1-1-111-111-1-1-1-11-1111-11-1    linear of order 2
ρ61111111-111-111-11-1-1-1-1-11-11-11    linear of order 2
ρ71111-11-11-111-11111-11-1-1-111-1-1    linear of order 2
ρ811111-111-111-111-11-1-1-11-1-1-111    linear of order 2
ρ91111-11-11-1-11-1-1-11-111-1-11-1-111    linear of order 2
ρ1011111-111-1-11-1-1-1-1-11-1-11111-1-1    linear of order 2
ρ111111-1-1-1-11-1-11-11-1111-11-1-11-11    linear of order 2
ρ121111111-11-1-11-11111-1-1-1-11-11-1    linear of order 2
ρ1311111-11-1-1-1-1-1-11-11-111-11-111-1    linear of order 2
ρ141111-11-1-1-1-1-1-1-1111-1-11111-1-11    linear of order 2
ρ15111111111-111-1-11-1-1111-1-1-1-1-1    linear of order 2
ρ161111-1-1-111-111-1-1-1-1-1-11-1-11111    linear of order 2
ρ1722-2-222-20-200200-20000000000    symplectic lifted from Q8, Schur index 2
ρ1822-2-2-2-220-20020020000000000    symplectic lifted from Q8, Schur index 2
ρ192-22-2000-20220-220-2000000000    symplectic lifted from Q8, Schur index 2
ρ2022-2-2-2220200-200-20000000000    symplectic lifted from Q8, Schur index 2
ρ212-22-200020-2-20220-2000000000    symplectic lifted from Q8, Schur index 2
ρ2222-2-22-2-20200-20020000000000    symplectic lifted from Q8, Schur index 2
ρ232-22-2000202-20-2-202000000000    symplectic lifted from Q8, Schur index 2
ρ242-22-2000-20-2202-202000000000    symplectic lifted from Q8, Schur index 2
ρ254-4-44000000000000000000000    orthogonal lifted from 2+ 1+4

Smallest permutation representation of Q82
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  Q83Q16  Q88SD16  Q85Q16  C42.510C23  C42.516C23  SD16⋊Q8  Q164Q8  C22.72C25  C22.91C25  C22.92C25  C22.107C25  C22.111C25  C22.133C25  C22.139C25  C22.144C25  Q8⋊SL2(𝔽3)
 C42.D2p: Q8≀C2  Dic69Q8  Dic109Q8  Dic149Q8 ...
Q82 is a maximal quotient of
C23.251C24  C23.351C24  C23.407C24  C427Q8  C23.486C24  C23.634C24  C23.692C24  C23.706C24  C23.711C24
 C42.D2p: C42.176D4  C42.180D4  Dic69Q8  Dic109Q8  Dic149Q8 ...

Matrix representation of Q82 in GL4(𝔽5) generated by

0100
4000
0040
0004
,
0200
2000
0010
0001
,
1000
0100
0001
0040
,
1000
0100
0003
0030
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

Export

Character table of Q82 in TeX

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