Copied to
clipboard

G = Q8⋊Q8  order 64 = 26

1st semidirect product of Q8 and Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: Q8⋊1Q8, C4.12SD16, C42.23C22, C4⋊C8.9C2, C4⋊Q8.5C2, (C4×Q8).6C2, C4.12(C2×Q8), C4.Q8.4C2, (C2×C4).132D4, C4.24(C4○D4), C4⋊C4.12C22, (C2×C8).33C22, Q8⋊C4.4C2, C2.10(C2×SD16), C22.96(C2×D4), (C2×C4).100C23, C2.13(C22⋊Q8), (C2×Q8).51C22, C2.15(C8.C22), SmallGroup(64,156)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — Q8⋊Q8
Chief series
C1 — C2 — C4 — C2×C4 — C2×Q8 — C4×Q8 — Q8⋊Q8
Lower central
C1 — C2 — C2×C4 — Q8⋊Q8
Upper central
C1 — C22 — C42 — Q8⋊Q8
Jennings
C1 — C2 — C2 — C2×C4 — Q8⋊Q8

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 >

C1
C2
C2
C2
C4
C4
C4
C22
C4
2C4
2C4
2C4
4C4
4C4
4C4
Q8
Q8
C2×C4
C2×C4
C2×C4
2C2×C4
2C8
2C2×C4
2C8
2C2×C4
2Q8
2C2×C4
4Q8
4Q8
C4⋊C4
C2×C8
C2×C8
C2×Q8
C42
C4⋊C4
C4⋊C4
2C4⋊C4
2C4⋊C4
2C42
2C2×Q8
C4⋊Q8
C4.Q8
C4.Q8
C4⋊C8
Q8⋊C4
Q8⋊C4
C4×Q8
Q8⋊Q8

Character table of Q8⋊Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 1111222244444884444
ρ11111111111111111111    trivial
ρ211111111-11-1-1-1-1-11111    linear of order 2
ρ31111-111-1-1-1-1111-11-1-11    linear of order 2
ρ41111-111-11-11-1-1-111-1-11    linear of order 2
ρ51111-111-1-1-1-111-11-111-1    linear of order 2
ρ61111-111-11-11-1-11-1-111-1    linear of order 2
ρ71111111111111-1-1-1-1-1-1    linear of order 2
ρ811111111-11-1-1-111-1-1-1-1    linear of order 2
ρ922222-2-220-2000000000    orthogonal lifted from D4
ρ102222-2-2-2-202000000000    orthogonal lifted from D4
ρ112-22-202-200002-2000000    symplectic lifted from Q8, Schur index 2
ρ122-22-202-20000-22000000    symplectic lifted from Q8, Schur index 2
ρ132-22-20-2202i0-2i00000000    complex lifted from C4○D4
ρ142-22-20-220-2i02i00000000    complex lifted from C4○D4
ρ152-2-22-20020000000√-2-√-2√-2-√-2    complex lifted from SD16
ρ162-2-22200-20000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ172-2-22200-20000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ182-2-22-20020000000-√-2√-2-√-2√-2    complex lifted from SD16
ρ1944-4-4000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of Q8⋊Q8
►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 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  Q8×SD16  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 ...
 (Cp×Q8)⋊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)  (C2×C8).170D4  (C2×C4).28D8
 C42.D2p: C42.99D4  C42.122D4  Dic6⋊4Q8  Q8⋊4Dic6  Dic6⋊6Q8  Dic10⋊4Q8  C20.48SD16  Dic10⋊6Q8 ...
 (Cp×Q8)⋊Q8: (C2×Q8)⋊Q8  Q8⋊2Dic6  Q8⋊Dic10  Q8⋊Dic14 ...
 C4⋊C4.D2p: Q8⋊C4⋊C4  C42.30Q8  (C2×C8)⋊Q8  (C2×Q8).8Q8  Dic6⋊Q8  Dic10⋊Q8  Dic14⋊Q8 ...

Matrix representation of Q8⋊Q8 ►in GL4(𝔽17) generated by

16000
01600
0040
00013
,
0100
1000
0001
00160
,
0400
4000
0010
0001
,
101300
4700
0009
0020
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

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁