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G = Q83Q8order 64 = 26

The semidirect product of Q8 and Q8 acting through Inn(Q8)

p-group, metabelian, nilpotent (class 2), monomial

Aliases: Q83Q8, C22.51C24, C42.52C22, C2.142- 1+4, Q83(C4⋊C4), C4⋊Q8.13C2, (C4×Q8).9C2, C4.18(C2×Q8), C4.46(C4○D4), C4⋊C4.38C22, (C2×C4).58C23, C42.C2.6C2, C2.11(C22×Q8), (C2×Q8).66C22, C2.30(C2×C4○D4), SmallGroup(64,238)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — Q83Q8
C1C2C22C2×C4C42C4×Q8 — Q83Q8
C1C22 — Q83Q8
C1C22 — Q83Q8
C1C22 — Q83Q8

Generators and relations for Q83Q8
 G = < a,b,c,d | a4=c4=1, b2=a2, d2=c2, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 117 in 100 conjugacy classes, 83 normal (10 characteristic)
C1, C2 [×3], C4 [×8], C4 [×11], C22, C2×C4, C2×C4 [×14], Q8 [×4], Q8 [×6], C42 [×9], C4⋊C4, C4⋊C4 [×21], C2×Q8, C2×Q8 [×3], C4×Q8, C4×Q8 [×5], C42.C2 [×6], C4⋊Q8 [×3], Q83Q8
Quotients: C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], C2×Q8 [×6], C4○D4 [×2], C24, C22×Q8, C2×C4○D4, 2- 1+4, Q83Q8

Character table of Q83Q8

 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
ρ172-22-2000-20220-220-2000000000    symplectic lifted from Q8, Schur index 2
ρ182-22-200020-2-20220-2000000000    symplectic lifted from Q8, Schur index 2
ρ192-22-2000202-20-2-202000000000    symplectic lifted from Q8, Schur index 2
ρ202-22-2000-20-2202-202000000000    symplectic lifted from Q8, Schur index 2
ρ2122-2-2-2i-2i2i0-2002002i0000000000    complex lifted from C4○D4
ρ2222-2-22i-2i-2i0200-2002i0000000000    complex lifted from C4○D4
ρ2322-2-2-2i2i2i0200-200-2i0000000000    complex lifted from C4○D4
ρ2422-2-22i2i-2i0-200200-2i0000000000    complex lifted from C4○D4
ρ254-4-44000000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of Q83Q8
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 24 3 22)(2 23 4 21)(5 48 7 46)(6 47 8 45)(9 27 11 25)(10 26 12 28)(13 35 15 33)(14 34 16 36)(17 31 19 29)(18 30 20 32)(37 59 39 57)(38 58 40 60)(41 55 43 53)(42 54 44 56)(49 61 51 63)(50 64 52 62)
(1 14 12 20)(2 15 9 17)(3 16 10 18)(4 13 11 19)(5 54 62 58)(6 55 63 59)(7 56 64 60)(8 53 61 57)(21 35 25 29)(22 36 26 30)(23 33 27 31)(24 34 28 32)(37 45 41 51)(38 46 42 52)(39 47 43 49)(40 48 44 50)
(1 38 12 42)(2 39 9 43)(3 40 10 44)(4 37 11 41)(5 36 62 30)(6 33 63 31)(7 34 64 32)(8 35 61 29)(13 51 19 45)(14 52 20 46)(15 49 17 47)(16 50 18 48)(21 57 25 53)(22 58 26 54)(23 59 27 55)(24 60 28 56)

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), (1,24,3,22)(2,23,4,21)(5,48,7,46)(6,47,8,45)(9,27,11,25)(10,26,12,28)(13,35,15,33)(14,34,16,36)(17,31,19,29)(18,30,20,32)(37,59,39,57)(38,58,40,60)(41,55,43,53)(42,54,44,56)(49,61,51,63)(50,64,52,62), (1,14,12,20)(2,15,9,17)(3,16,10,18)(4,13,11,19)(5,54,62,58)(6,55,63,59)(7,56,64,60)(8,53,61,57)(21,35,25,29)(22,36,26,30)(23,33,27,31)(24,34,28,32)(37,45,41,51)(38,46,42,52)(39,47,43,49)(40,48,44,50), (1,38,12,42)(2,39,9,43)(3,40,10,44)(4,37,11,41)(5,36,62,30)(6,33,63,31)(7,34,64,32)(8,35,61,29)(13,51,19,45)(14,52,20,46)(15,49,17,47)(16,50,18,48)(21,57,25,53)(22,58,26,54)(23,59,27,55)(24,60,28,56)>;

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), (1,24,3,22)(2,23,4,21)(5,48,7,46)(6,47,8,45)(9,27,11,25)(10,26,12,28)(13,35,15,33)(14,34,16,36)(17,31,19,29)(18,30,20,32)(37,59,39,57)(38,58,40,60)(41,55,43,53)(42,54,44,56)(49,61,51,63)(50,64,52,62), (1,14,12,20)(2,15,9,17)(3,16,10,18)(4,13,11,19)(5,54,62,58)(6,55,63,59)(7,56,64,60)(8,53,61,57)(21,35,25,29)(22,36,26,30)(23,33,27,31)(24,34,28,32)(37,45,41,51)(38,46,42,52)(39,47,43,49)(40,48,44,50), (1,38,12,42)(2,39,9,43)(3,40,10,44)(4,37,11,41)(5,36,62,30)(6,33,63,31)(7,34,64,32)(8,35,61,29)(13,51,19,45)(14,52,20,46)(15,49,17,47)(16,50,18,48)(21,57,25,53)(22,58,26,54)(23,59,27,55)(24,60,28,56) );

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)], [(1,24,3,22),(2,23,4,21),(5,48,7,46),(6,47,8,45),(9,27,11,25),(10,26,12,28),(13,35,15,33),(14,34,16,36),(17,31,19,29),(18,30,20,32),(37,59,39,57),(38,58,40,60),(41,55,43,53),(42,54,44,56),(49,61,51,63),(50,64,52,62)], [(1,14,12,20),(2,15,9,17),(3,16,10,18),(4,13,11,19),(5,54,62,58),(6,55,63,59),(7,56,64,60),(8,53,61,57),(21,35,25,29),(22,36,26,30),(23,33,27,31),(24,34,28,32),(37,45,41,51),(38,46,42,52),(39,47,43,49),(40,48,44,50)], [(1,38,12,42),(2,39,9,43),(3,40,10,44),(4,37,11,41),(5,36,62,30),(6,33,63,31),(7,34,64,32),(8,35,61,29),(13,51,19,45),(14,52,20,46),(15,49,17,47),(16,50,18,48),(21,57,25,53),(22,58,26,54),(23,59,27,55),(24,60,28,56)])

Q83Q8 is a maximal subgroup of
Q86SD16  Q8.Q16  Q84Q16  Q8.SD16  Q89SD16  Q86Q16  C42.73C23  C42.75C23  Q8×C4○D4  C22.69C25  C22.93C25  C22.105C25  C22.106C25  C22.113C25  SL2(𝔽3)⋊3Q8
 C2p.2- 1+4: Q87SD16  Q85Q16  C42.505C23  C42.506C23  C42.509C23  C42.512C23  C42.513C23  C42.515C23 ...
Q83Q8 is a maximal quotient of
Q8×C4⋊C4  C23.233C24  C23.237C24  C23.247C24  C23.252C24  C23.346C24  C23.351C24  C23.353C24  C23.362C24  C23.406C24  C23.408C24  C23.409C24  C23.411C24  C23.420C24  C426Q8  C42.35Q8  C23.485C24  C23.488C24  C23.490C24  C23.613C24  C23.626C24  C23.655C24  C23.674C24  C23.691C24  C23.699C24  C23.702C24  C23.705C24  C23.706C24  C23.709C24
 C42.D2p: C42.169D4  C42.174D4  C42.177D4  C42.179D4  C42.180D4  C42.181D4  Dic610Q8  Q86Dic6 ...

Matrix representation of Q83Q8 in GL4(𝔽5) generated by

1000
0100
0020
0023
,
4000
0400
0013
0014
,
2000
3300
0040
0004
,
4300
1100
0010
0014
G:=sub<GL(4,GF(5))| [1,0,0,0,0,1,0,0,0,0,2,2,0,0,0,3],[4,0,0,0,0,4,0,0,0,0,1,1,0,0,3,4],[2,3,0,0,0,3,0,0,0,0,4,0,0,0,0,4],[4,1,0,0,3,1,0,0,0,0,1,1,0,0,0,4] >;

Q83Q8 in GAP, Magma, Sage, TeX

Q_8\rtimes_3Q_8
% in TeX

G:=Group("Q8:3Q8");
// GroupNames label

G:=SmallGroup(64,238);
// by ID

G=gap.SmallGroup(64,238);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103,650,158,297,69]);
// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of Q83Q8 in TeX

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