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## G = Q8.Q8order 64 = 26

### The non-split extension by Q8 of Q8 acting via Q8/C4=C2

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

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=a2c2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >

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 C4○D4 ρ14 2 -2 2 -2 0 -2 2 0 -2i 0 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ15 2 -2 -2 2 2i 0 0 -2i 0 0 0 0 0 0 0 -√-2 √2 -√2 √-2 complex lifted from C4○D8 ρ16 2 -2 -2 2 -2i 0 0 2i 0 0 0 0 0 0 0 √-2 √2 -√2 -√-2 complex lifted from C4○D8 ρ17 2 -2 -2 2 2i 0 0 -2i 0 0 0 0 0 0 0 √-2 -√2 √2 -√-2 complex lifted from C4○D8 ρ18 2 -2 -2 2 -2i 0 0 2i 0 0 0 0 0 0 0 -√-2 -√2 √2 √-2 complex lifted from C4○D8 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

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 21 5 19)(2 22 6 20)(3 23 7 17)(4 24 8 18)(9 31 15 25)(10 32 16 26)(11 29 13 27)(12 30 14 28)(33 51 39 53)(34 52 40 54)(35 49 37 55)(36 50 38 56)(41 59 47 61)(42 60 48 62)(43 57 45 63)(44 58 46 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), (1,21,5,19)(2,22,6,20)(3,23,7,17)(4,24,8,18)(9,31,15,25)(10,32,16,26)(11,29,13,27)(12,30,14,28)(33,51,39,53)(34,52,40,54)(35,49,37,55)(36,50,38,56)(41,59,47,61)(42,60,48,62)(43,57,45,63)(44,58,46,64), (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), (1,21,5,19)(2,22,6,20)(3,23,7,17)(4,24,8,18)(9,31,15,25)(10,32,16,26)(11,29,13,27)(12,30,14,28)(33,51,39,53)(34,52,40,54)(35,49,37,55)(36,50,38,56)(41,59,47,61)(42,60,48,62)(43,57,45,63)(44,58,46,64), (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)], [(1,21,5,19),(2,22,6,20),(3,23,7,17),(4,24,8,18),(9,31,15,25),(10,32,16,26),(11,29,13,27),(12,30,14,28),(33,51,39,53),(34,52,40,54),(35,49,37,55),(36,50,38,56),(41,59,47,61),(42,60,48,62),(43,57,45,63),(44,58,46,64)], [(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
Q8.Dic6
C42.D2p: C42.447D4  C42.220D4  C42.449D4  C42.384D4  C42.226D4  C42.229D4  C42.235D4  C42.283D4 ...
C4⋊C4.D2p: C42.21C23  C42.22C23  C42.23C23  C42.353C23  C42.354C23  C42.360C23  C42.361C23  C42.424C23 ...
Q8.Q8 is a maximal quotient of
C2.D84C4  C4.Q810C4  (C2×C4).26D8  C4.(C4⋊Q8)  (C2×C8).171D4  C4⋊C4.Q8
C42.D2p: C42.101D4  C42.123D4  Dic6.3Q8  Q8.5Dic6  Dic6.4Q8  Dic10.3Q8  Q8.3Dic10  Dic10.4Q8 ...
C4⋊C4.D2p: Q8⋊(C4⋊C4)  Q8⋊C4⋊C4  C42.31Q8  C4⋊C4.85D4  C2.(C8⋊Q8)  (C2×Q8).8Q8  (C2×C8).52D4  Q8.3Dic6 ...

Matrix representation of Q8.Q8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 0 1 0 0 16 0
,
 0 1 0 0 1 0 0 0 0 0 10 1 0 0 1 7
,
 0 13 0 0 13 0 0 0 0 0 13 0 0 0 0 13
,
 0 1 0 0 16 0 0 0 0 0 13 11 0 0 11 4
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,16,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,10,1,0,0,1,7],[0,13,0,0,13,0,0,0,0,0,13,0,0,0,0,13],[0,16,0,0,1,0,0,0,0,0,13,11,0,0,11,4] >;

Q8.Q8 in GAP, Magma, Sage, TeX

Q_8.Q_8
% in TeX

G:=Group("Q8.Q8");
// GroupNames label

G:=SmallGroup(64,160);
// by ID

G=gap.SmallGroup(64,160);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,247,362,158,1444,376,88]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^4=1,b^2=a^2,d^2=a^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=a^2*c^-1>;
// generators/relations

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