Copied to
clipboard

G = C83Q8order 64 = 26

2nd semidirect product of C8 and Q8 acting via Q8/C4=C2

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

Aliases: C83Q8, C4.5SD16, C42.83C22, C4.5(C2×Q8), (C4×C8).12C2, (C2×C4).79D4, C4⋊Q8.10C2, C2.6(C4⋊Q8), C4.Q8.6C2, C4⋊C4.22C22, (C2×C8).93C22, C2.17(C2×SD16), (C2×C4).123C23, C22.119(C2×D4), SmallGroup(64,179)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C83Q8
C1C2C4C2×C4C2×C8C4×C8 — C83Q8
C1C2C2×C4 — C83Q8
C1C22C42 — C83Q8
C1C2C2C2×C4 — C83Q8

Generators and relations for C83Q8
 G = < a,b,c | a8=b4=1, c2=b2, ab=ba, cac-1=a3, cbc-1=b-1 >

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

Character table of C83Q8

 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
ρ102222-2-22-2-22000000000000    orthogonal lifted from D4
ρ112-22-20-202000000200-2-2020    symplectic lifted from Q8, Schur index 2
ρ122-22-2020-200000002-200-202    symplectic lifted from Q8, Schur index 2
ρ132-22-2020-20000000-220020-2    symplectic lifted from Q8, Schur index 2
ρ142-22-20-202000000-200220-20    symplectic lifted from Q8, Schur index 2
ρ1522-2-200-20020000-2--2-2-2--2--2--2-2    complex lifted from SD16
ρ162-2-222000-200000--2-2-2-2--2--2-2--2    complex lifted from SD16
ρ1722-2-200-20020000--2-2--2--2-2-2-2--2    complex lifted from SD16
ρ182-2-222000-200000-2--2--2--2-2-2--2-2    complex lifted from SD16
ρ192-2-22-2000200000-2-2-2--2-2--2--2--2    complex lifted from SD16
ρ2022-2-200200-20000-2-2--2-2--2-2--2--2    complex lifted from SD16
ρ212-2-22-2000200000--2--2--2-2--2-2-2-2    complex lifted from SD16
ρ2222-2-200200-20000--2--2-2--2-2--2-2-2    complex lifted from SD16

Smallest permutation representation of C83Q8
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 37 23 26)(2 38 24 27)(3 39 17 28)(4 40 18 29)(5 33 19 30)(6 34 20 31)(7 35 21 32)(8 36 22 25)(9 46 62 51)(10 47 63 52)(11 48 64 53)(12 41 57 54)(13 42 58 55)(14 43 59 56)(15 44 60 49)(16 45 61 50)
(1 51 23 46)(2 54 24 41)(3 49 17 44)(4 52 18 47)(5 55 19 42)(6 50 20 45)(7 53 21 48)(8 56 22 43)(9 37 62 26)(10 40 63 29)(11 35 64 32)(12 38 57 27)(13 33 58 30)(14 36 59 25)(15 39 60 28)(16 34 61 31)

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

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

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

C83Q8 is a maximal subgroup of
C88SD16  C85SD16  C86SD16  C42.665C23  C42.667C23  C83Q16  C42.364D4  M4(2)⋊5Q8  C42.365D4  C42.261D4  C42.262D4  C42.279D4  C42.280D4  C42.281D4  C42.283D4
 C4p.SD16: C85Q16  C8212C2  C8.9SD16  D83Q8  C249Q8  C12.SD16  C409Q8  C20.SD16 ...
 C4⋊C4.D2p: Q84SD16  C42.213C23  Q8.SD16  D44SD16  C811SD16  C88Q16  C83SD16  C8⋊Q16 ...
C83Q8 is a maximal quotient of
C42.58Q8  C87(C4⋊C4)
 C42.D2p: C42.436D4  C249Q8  C12.SD16  C409Q8  C20.SD16  C569Q8  C28.SD16 ...
 C4⋊C4.D2p: (C2×C8)⋊Q8  C2.(C83Q8)  C245Q8  C405Q8  C565Q8 ...

Matrix representation of C83Q8 in GL4(𝔽17) generated by

12500
121200
00125
001212
,
0100
16000
0010
0001
,
13000
0400
00411
001113
G:=sub<GL(4,GF(17))| [12,12,0,0,5,12,0,0,0,0,12,12,0,0,5,12],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,4,11,0,0,11,13] >;

C83Q8 in GAP, Magma, Sage, TeX

C_8\rtimes_3Q_8
% in TeX

G:=Group("C8:3Q8");
// GroupNames label

G:=SmallGroup(64,179);
// by ID

G=gap.SmallGroup(64,179);
# by ID

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

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

Export

Subgroup lattice of C83Q8 in TeX
Character table of C83Q8 in TeX

׿
×
𝔽