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G = Q8⋊Q8order 64 = 26

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

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

Aliases: Q81Q8, 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

C1C2×C4 — Q8⋊Q8
C1C2C4C2×C4C2×Q8C4×Q8 — Q8⋊Q8
C1C2C2×C4 — Q8⋊Q8
C1C22C42 — Q8⋊Q8
C1C2C2C2×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 >

2C4
2C4
2C4
4C4
4C4
4C4
2C2×C4
2C8
2C2×C4
2C8
2C2×C4
2Q8
2C2×C4
4Q8
4Q8
2C4⋊C4
2C4⋊C4
2C42
2C2×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  Q83SD16  D4.5SD16  Q83Q16  Q84Q16  C42.211C23  Q84SD16  Q8.SD16  C814SD16  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  D48SD16  C42.469C23  C42.51C23  C42.56C23  C42.477C23  C42.481C23  D49SD16  C42.491C23  C42.58C23  C42.62C23  C42.494C23  C42.497C23  Q87SD16  C42.506C23  C42.510C23  C42.514C23  C42.515C23  C42.517C23  Q8×SD16  Q166Q8  SD162Q8  SD163Q8  Q8⋊Dic6
 C4p.SD16: Q81Q16  C8.SD16  Dic64Q8  Q84Dic6  Dic66Q8  Dic104Q8  C20.48SD16  Dic106Q8 ...
 (Cp×Q8)⋊Q8: C42.220D4  C42.21C23  Q82Dic6  Q8⋊Dic10  Q8⋊Dic14 ...
 C4p⋊Q8.C2: Q164Q8  Dic6⋊Q8  Dic10⋊Q8  Dic14⋊Q8 ...
Q8⋊Q8 is a maximal quotient of
C4.Q89C4  C4.(C4⋊Q8)  (C2×C8).170D4  (C2×C4).28D8
 C42.D2p: C42.99D4  C42.122D4  Dic64Q8  Q84Dic6  Dic66Q8  Dic104Q8  C20.48SD16  Dic106Q8 ...
 (Cp×Q8)⋊Q8: (C2×Q8)⋊Q8  Q82Dic6  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

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