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G = C82Q8order 64 = 26

1st semidirect product of C8 and Q8 acting via Q8/C4=C2

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

Aliases: C82Q8, C4.5D8, C4.4Q16, C42.85C22, (C4×C8).8C2, C4.7(C2×Q8), C2.11(C2×D8), (C2×C4).80D4, C4⋊Q8.11C2, C2.8(C4⋊Q8), C2.D8.7C2, C2.11(C2×Q16), C4⋊C4.24C22, (C2×C8).81C22, (C2×C4).125C23, C22.121(C2×D4), SmallGroup(64,181)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C82Q8
C1C2C4C2×C4C2×C8C4×C8 — C82Q8
C1C2C2×C4 — C82Q8
C1C22C42 — C82Q8
C1C2C2C2×C4 — C82Q8

Generators and relations for C82Q8
 G = < a,b,c | a8=b4=1, c2=b2, ab=ba, cac-1=a-1, cbc-1=b-1 >

4C4
4C4
4C4
4C4
2C2×C4
2C2×C4
2C2×C4
2C2×C4
4Q8
4Q8
4Q8
4Q8
2C2×Q8
2C4⋊C4
2C2×Q8
2C4⋊C4

Character table of C82Q8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 1111222222888822222222
ρ11111111111111111111111    trivial
ρ21111-11-11-1-1-11-11-111-1-11-11    linear of order 2
ρ31111111111-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ41111-11-11-1-111-1-11-1-111-11-1    linear of order 2
ρ511111111111-1-11-1-1-1-1-1-1-1-1    linear of order 2
ρ61111-11-11-1-1-1-1111-1-111-11-1    linear of order 2
ρ71111111111-1-1-1-111111111    linear of order 2
ρ81111-11-11-1-11-11-1-111-1-11-11    linear of order 2
ρ922222-2-2-22-2000000000000    orthogonal lifted from D4
ρ1022-2-200-200200002-222-2-2-22    orthogonal lifted from D8
ρ112222-2-22-2-22000000000000    orthogonal lifted from D4
ρ1222-2-200200-2000022-22-22-2-2    orthogonal lifted from D8
ρ1322-2-200-20020000-22-2-2222-2    orthogonal lifted from D8
ρ1422-2-200200-20000-2-22-22-222    orthogonal lifted from D8
ρ152-22-20-202000000200-2-2020    symplectic lifted from Q8, Schur index 2
ρ162-22-2020-200000002-200-202    symplectic lifted from Q8, Schur index 2
ρ172-22-2020-20000000-220020-2    symplectic lifted from Q8, Schur index 2
ρ182-22-20-202000000-200220-20    symplectic lifted from Q8, Schur index 2
ρ192-2-222000-200000-2222-2-22-2    symplectic lifted from Q16, Schur index 2
ρ202-2-22-2000200000-2-2-22-2222    symplectic lifted from Q16, Schur index 2
ρ212-2-22-2000200000222-22-2-2-2    symplectic lifted from Q16, Schur index 2
ρ222-2-222000-2000002-2-2-222-22    symplectic lifted from Q16, Schur index 2

Smallest permutation representation of C82Q8
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 33 19 26)(2 34 20 27)(3 35 21 28)(4 36 22 29)(5 37 23 30)(6 38 24 31)(7 39 17 32)(8 40 18 25)(9 41 57 52)(10 42 58 53)(11 43 59 54)(12 44 60 55)(13 45 61 56)(14 46 62 49)(15 47 63 50)(16 48 64 51)
(1 51 19 48)(2 50 20 47)(3 49 21 46)(4 56 22 45)(5 55 23 44)(6 54 24 43)(7 53 17 42)(8 52 18 41)(9 40 57 25)(10 39 58 32)(11 38 59 31)(12 37 60 30)(13 36 61 29)(14 35 62 28)(15 34 63 27)(16 33 64 26)

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

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

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

C82Q8 is a maximal subgroup of
C4.10D16  C4.6Q32  D4.7D8  Q84Q16  D44Q16  C42.211C23  D4.1Q16  Q8.1Q16  C8.8SD16  C8.3Q16  C8.7Q16  C42.665C23  D81Q8  Q16⋊Q8  D8⋊Q8  C4.Q32  C4.4D16  C4.SD32  C8.12SD16  C42.364D4  M4(2)⋊4Q8  M4(2)⋊5Q8  C42.366D4  C42.367D4  C42.387C23  C42.259D4  C42.278D4  C42.280D4  C42.282D4  C42.283D4  C42.425C23  D45D8  D46Q16  C42.490C23  C42.491C23  C42.494C23  C42.498C23  Q8×D8  D86Q8  Q8×Q16  Q166Q8  SD16⋊Q8  SD162Q8
 C4p.D8: C84Q16  C8.2D8  C248Q8  C12.17D8  C408Q8  C20.17D8  C568Q8  C28.17D8 ...
 C8p⋊Q8: C162Q8  C163Q8  C16⋊Q8  C242Q8  C402Q8  C562Q8 ...
 C2.(C8pD4): C810SD16  C87Q16  C84SD16  C82Q16  C85D8  C85Q16  C85SD16  C42.666C23 ...
C82Q8 is a maximal quotient of
C42.59Q8  C85(C4⋊C4)  C42.436D4  C4⋊C4⋊Q8  (C2×C8).1Q8
 C8p⋊Q8: C162Q8  C163Q8  C16⋊Q8  C248Q8  C242Q8  C408Q8  C402Q8  C568Q8 ...
 C4p.Q16: C16.5Q8  C12.17D8  C20.17D8  C28.17D8 ...

Matrix representation of C82Q8 in GL4(𝔽17) generated by

1000
0100
00143
001414
,
0100
16000
00016
0010
,
7100
11000
001016
00167
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,14,14,0,0,3,14],[0,16,0,0,1,0,0,0,0,0,0,1,0,0,16,0],[7,1,0,0,1,10,0,0,0,0,10,16,0,0,16,7] >;

C82Q8 in GAP, Magma, Sage, TeX

C_8\rtimes_2Q_8
% in TeX

G:=Group("C8:2Q8");
// GroupNames label

G:=SmallGroup(64,181);
// by ID

G=gap.SmallGroup(64,181);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,55,362,230,1444,88]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C82Q8 in TeX
Character table of C82Q8 in TeX

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