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G = Q8⋊C8order 64 = 26

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

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

Aliases: Q8⋊C8, C4.8Q16, C4.14SD16, C4.2M4(2), C42.63C22, C2.2C4≀C2, C4⋊C4.4C4, C4⋊C8.1C2, C4.2(C2×C8), (C4×C8).1C2, (C2×Q8).4C4, (C4×Q8).1C2, (C2×C4).137D4, C2.6(C22⋊C8), C2.1(Q8⋊C4), C22.22(C22⋊C4), (C2×C4).38(C2×C4), SmallGroup(64,7)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — Q8⋊C8
C1C2C22C2×C4C42C4×Q8 — Q8⋊C8
C1C2C4 — Q8⋊C8
C1C2×C4C42 — Q8⋊C8
C1C22C22C42 — Q8⋊C8

Generators and relations for Q8⋊C8
 G = < a,b,c | a4=c8=1, b2=a2, bab-1=cac-1=a-1, cbc-1=a-1b >

2C4
2C4
2C4
4C4
2C8
2C8
2Q8
2C2×C4
2C2×C4
4C8
2C2×C8
2C2×C8
2C4⋊C4
2C42

Character table of Q8⋊C8

 class 12A2B2C4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D8E8F8G8H8I8J8K8L
 size 1111111122224444222222224444
ρ11111111111111111111111111111    trivial
ρ2111111111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ3111111111111-1-1-1-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ41111111111111111-1-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-1-1-1-111-1-11-11-iiiii-i-i-iii-i-i    linear of order 4
ρ61111-1-1-1-1-111-1-11-11i-i-i-i-iiii-i-iii    linear of order 4
ρ71111-1-1-1-1-111-11-11-1i-i-i-i-iiiiii-i-i    linear of order 4
ρ81111-1-1-1-1-111-11-11-1-iiiii-i-i-i-i-iii    linear of order 4
ρ91-11-1-ii-iii1-1-i-1i1-iζ87ζ85ζ8ζ85ζ8ζ87ζ83ζ83ζ87ζ83ζ8ζ85    linear of order 8
ρ101-11-1-ii-iii1-1-i-1i1-iζ83ζ8ζ85ζ8ζ85ζ83ζ87ζ87ζ83ζ87ζ85ζ8    linear of order 8
ρ111-11-1-ii-iii1-1-i1-i-1iζ83ζ8ζ85ζ8ζ85ζ83ζ87ζ87ζ87ζ83ζ8ζ85    linear of order 8
ρ121-11-1-ii-iii1-1-i1-i-1iζ87ζ85ζ8ζ85ζ8ζ87ζ83ζ83ζ83ζ87ζ85ζ8    linear of order 8
ρ131-11-1i-ii-i-i1-1i1i-1-iζ85ζ87ζ83ζ87ζ83ζ85ζ8ζ8ζ8ζ85ζ87ζ83    linear of order 8
ρ141-11-1i-ii-i-i1-1i-1-i1iζ85ζ87ζ83ζ87ζ83ζ85ζ8ζ8ζ85ζ8ζ83ζ87    linear of order 8
ρ151-11-1i-ii-i-i1-1i-1-i1iζ8ζ83ζ87ζ83ζ87ζ8ζ85ζ85ζ8ζ85ζ87ζ83    linear of order 8
ρ161-11-1i-ii-i-i1-1i1i-1-iζ8ζ83ζ87ζ83ζ87ζ8ζ85ζ85ζ85ζ8ζ83ζ87    linear of order 8
ρ1722222222-2-2-2-20000000000000000    orthogonal lifted from D4
ρ182222-2-2-2-22-2-220000000000000000    orthogonal lifted from D4
ρ192-2-222-2-2200000000-2-222-22-220000    symplectic lifted from Q16, Schur index 2
ρ202-2-222-2-220000000022-2-22-22-20000    symplectic lifted from Q16, Schur index 2
ρ2122-2-2-2i-2i2i2i000000001-i-1-i-1-i1+i1+i-1+i-1+i1-i0000    complex lifted from C4≀C2
ρ2222-2-2-2i-2i2i2i00000000-1+i1+i1+i-1-i-1-i1-i1-i-1+i0000    complex lifted from C4≀C2
ρ232-22-22i-2i2i-2i2i-22-2i0000000000000000    complex lifted from M4(2)
ρ242-22-2-2i2i-2i2i-2i-222i0000000000000000    complex lifted from M4(2)
ρ2522-2-22i2i-2i-2i000000001+i-1+i-1+i1-i1-i-1-i-1-i1+i0000    complex lifted from C4≀C2
ρ262-2-22-222-200000000-2--2-2-2--2--2-2--20000    complex lifted from SD16
ρ2722-2-22i2i-2i-2i00000000-1-i1-i1-i-1+i-1+i1+i1+i-1-i0000    complex lifted from C4≀C2
ρ282-2-22-222-200000000--2-2--2--2-2-2--2-20000    complex lifted from SD16

Smallest permutation representation of Q8⋊C8
Regular action on 64 points
Generators in S64
(1 31 55 17)(2 18 56 32)(3 25 49 19)(4 20 50 26)(5 27 51 21)(6 22 52 28)(7 29 53 23)(8 24 54 30)(9 36 62 46)(10 47 63 37)(11 38 64 48)(12 41 57 39)(13 40 58 42)(14 43 59 33)(15 34 60 44)(16 45 61 35)
(1 59 55 14)(2 34 56 44)(3 61 49 16)(4 36 50 46)(5 63 51 10)(6 38 52 48)(7 57 53 12)(8 40 54 42)(9 26 62 20)(11 28 64 22)(13 30 58 24)(15 32 60 18)(17 33 31 43)(19 35 25 45)(21 37 27 47)(23 39 29 41)
(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)

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

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

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

Q8⋊C8 is a maximal subgroup of
C42.455D4  C42.397D4  C42.399D4  C42.46D4  C42.373D4  C42.47D4  C42.401D4  Q8⋊M4(2)  C42.374D4  D44M4(2)  Q85M4(2)  C42.316D4  C42.305D4  C42.52D4  C42.54D4  C8×SD16  C8×Q16  SD16⋊C8  Q165C8  C812SD16  C815SD16  C89Q16  D4.M4(2)  Q8.M4(2)  Q82M4(2)  C89SD16  C8⋊M4(2)  C42.181C23  Q8⋊D8  D4⋊SD16  Q8⋊SD16  Q86SD16  Q83D8  C42.189C23  C42.191C23  Q82SD16  D4⋊Q16  Q8⋊Q16  Q8.Q16  D4.3Q16  C42.199C23  C42.201C23  Q8.D8  Q83SD16  D4.5SD16  D43Q16  Q83Q16  Q84Q16  D44Q16  C42.211C23  Q84SD16  C42.213C23  Q8.SD16  C2.U2(𝔽3)
 C4p.M4(2): C86Q16  C8.M4(2)  C4.8Dic12  Dic62C8  C12.26Q16  Dic103C8  Dic104C8  C20.26Q16 ...
Q8⋊C8 is a maximal quotient of
C4⋊C4⋊C8  (C2×Q8)⋊C8  C42.46Q8  Dic101C8  Dic5.12Q16
 C4p.Q16: Q8⋊C16  C8.17Q16  C4.8Dic12  Dic62C8  C12.26Q16  Dic103C8  Dic104C8  C20.26Q16 ...

Matrix representation of Q8⋊C8 in GL3(𝔽17) generated by

100
001
0160
,
1600
0167
071
,
1500
0411
01113
G:=sub<GL(3,GF(17))| [1,0,0,0,0,16,0,1,0],[16,0,0,0,16,7,0,7,1],[15,0,0,0,4,11,0,11,13] >;

Q8⋊C8 in GAP, Magma, Sage, TeX

Q_8\rtimes C_8
% in TeX

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

G:=SmallGroup(64,7);
// by ID

G=gap.SmallGroup(64,7);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,2,48,73,103,362,188,86,117]);
// Polycyclic

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

Export

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

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