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G = C8⋊Q8order 64 = 26

The semidirect product of C8 and Q8 acting via Q8/C2=C22

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

Aliases: C8⋊Q8, C42.33C22, C4.8(C2×Q8), (C2×C4).46D4, C4⋊Q8.12C2, C2.9(C4⋊Q8), C4.Q8.3C2, C2.D8.8C2, C8⋊C4.2C2, C4⋊C4.25C22, (C2×C8).22C22, C42.C2.5C2, C2.23(C8⋊C22), (C2×C4).126C23, C22.122(C2×D4), C2.23(C8.C22), SmallGroup(64,182)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C8⋊Q8
C1C2C4C2×C4C2×C8C8⋊C4 — C8⋊Q8
C1C2C2×C4 — C8⋊Q8
C1C22C42 — C8⋊Q8
C1C2C2C2×C4 — C8⋊Q8

Generators and relations for C8⋊Q8
 G = < a,b,c | a8=b4=1, c2=b2, bab-1=a5, cac-1=a3, cbc-1=b-1 >

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

Character table of C8⋊Q8

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B8C8D
 size 1111224488884444
ρ11111111111111111    trivial
ρ2111111-1-11-1-111-11-1    linear of order 2
ρ3111111-1-1-11-11-11-11    linear of order 2
ρ411111111-1-111-1-1-1-1    linear of order 2
ρ5111111-1-11-11-1-11-11    linear of order 2
ρ61111111111-1-1-1-1-1-1    linear of order 2
ρ711111111-1-1-1-11111    linear of order 2
ρ8111111-1-1-111-11-11-1    linear of order 2
ρ92222-2-22-200000000    orthogonal lifted from D4
ρ102222-2-2-2200000000    orthogonal lifted from D4
ρ112-22-2-220000000-202    symplectic lifted from Q8, Schur index 2
ρ122-22-2-22000000020-2    symplectic lifted from Q8, Schur index 2
ρ132-22-22-200000020-20    symplectic lifted from Q8, Schur index 2
ρ142-22-22-2000000-2020    symplectic lifted from Q8, Schur index 2
ρ154-4-44000000000000    orthogonal lifted from C8⋊C22
ρ1644-4-4000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C8⋊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 33 22 28)(2 38 23 25)(3 35 24 30)(4 40 17 27)(5 37 18 32)(6 34 19 29)(7 39 20 26)(8 36 21 31)(9 44 60 54)(10 41 61 51)(11 46 62 56)(12 43 63 53)(13 48 64 50)(14 45 57 55)(15 42 58 52)(16 47 59 49)
(1 50 22 48)(2 53 23 43)(3 56 24 46)(4 51 17 41)(5 54 18 44)(6 49 19 47)(7 52 20 42)(8 55 21 45)(9 37 60 32)(10 40 61 27)(11 35 62 30)(12 38 63 25)(13 33 64 28)(14 36 57 31)(15 39 58 26)(16 34 59 29)

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

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

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

C8⋊Q8 is a maximal subgroup of
C42.6C23  C42.7C23  C42.10C23  M4(2)⋊3Q8  M4(2)⋊4Q8  C42.390C23  C42.391C23  C42.423C23  C42.424C23  C42.425C23  C42.426C23  C42.57C23  C42.58C23  C42.59C23  C42.60C23  C42.61C23  C42.62C23  C42.63C23  C42.64C23  D84Q8  SD16⋊Q8  SD162Q8  SD163Q8  D85Q8
 C42.D2p: C42.252D4  M4(2)⋊5Q8  M4(2)⋊6Q8  C42.255D4  C42.256D4  C42.257D4  C42.258D4  C42.286D4 ...
 C4p⋊Q8.C2: C42.9C23  Q164Q8  Q165Q8  C243Q8  C244Q8  C403Q8  C404Q8  C563Q8 ...
C8⋊Q8 is a maximal quotient of
C42.26Q8  C4.(C4×Q8)  C8⋊(C4⋊C4)
 C42.D2p: C42.124D4  C42.125D4  C8⋊Dic6  C42.68D6  C42.76D6  C8⋊Dic10  C42.68D10  C42.76D10 ...
 C4⋊C4.D2p: C4⋊C4⋊Q8  (C2×C8)⋊Q8  C2.(C8⋊Q8)  (C2×C8).1Q8  C2.(C83Q8)  (C2×C8).24Q8  C243Q8  C244Q8 ...

Matrix representation of C8⋊Q8 in GL6(𝔽17)

010000
1600000
000010
0082130
000100
00015415
,
010000
1600000
000001
001404
0002132
0016000
,
1300000
040000
0013141613
00951316
001191515
001614131

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,2,1,15,0,0,1,13,0,4,0,0,0,0,0,15],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16,0,0,0,4,2,0,0,0,0,0,13,0,0,0,1,4,2,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,9,11,16,0,0,14,5,9,14,0,0,16,13,15,13,0,0,13,16,15,1] >;

C8⋊Q8 in GAP, Magma, Sage, TeX

C_8\rtimes Q_8
% in TeX

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

G:=SmallGroup(64,182);
// by ID

G=gap.SmallGroup(64,182);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,48,121,55,362,332,86,1444,88]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=1,c^2=b^2,b*a*b^-1=a^5,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C8⋊Q8 in TeX
Character table of C8⋊Q8 in TeX

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