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## G = C6×Q8order 48 = 24·3

### Direct product of C6 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C6×Q8, C6.12C23, C12.20C22, C4.4(C2×C6), (C2×C4).3C6, (C2×C12).9C2, C22.4(C2×C6), C2.2(C22×C6), (C2×C6).15C22, SmallGroup(48,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C6×Q8
 Chief series C1 — C2 — C6 — C12 — C3×Q8 — C6×Q8
 Lower central C1 — C2 — C6×Q8
 Upper central C1 — C2×C6 — C6×Q8

Generators and relations for C6×Q8
G = < a,b,c | a6=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C6×Q8

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L size 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 -1 -1 1 1 1 -1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ6 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ7 1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 1 linear of order 2 ρ8 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 -1 1 1 linear of order 2 ρ9 1 -1 -1 1 ζ3 ζ32 -1 1 1 -1 1 -1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ32 ζ3 ζ32 ζ3 ζ65 ζ6 ζ65 ζ3 ζ32 ζ6 ζ65 ζ6 linear of order 6 ρ10 1 1 1 1 ζ32 ζ3 -1 -1 1 1 -1 -1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ65 ζ32 ζ3 ζ6 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ6 ζ65 linear of order 6 ρ11 1 1 1 1 ζ3 ζ32 1 1 1 1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ12 1 -1 -1 1 ζ32 ζ3 -1 1 -1 1 -1 1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ3 ζ6 ζ65 ζ32 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ32 ζ3 linear of order 6 ρ13 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 1 1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ6 ζ65 ζ6 ζ65 ζ65 ζ6 ζ65 ζ3 ζ32 ζ6 ζ3 ζ32 linear of order 6 ρ14 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ65 ζ6 ζ65 ζ6 ζ6 ζ65 ζ6 ζ32 ζ3 ζ65 ζ32 ζ3 linear of order 6 ρ15 1 -1 -1 1 ζ3 ζ32 1 -1 -1 1 1 -1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ6 ζ65 ζ6 ζ65 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 linear of order 6 ρ16 1 -1 -1 1 ζ32 ζ3 -1 1 1 -1 1 -1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ3 ζ32 ζ3 ζ32 ζ6 ζ65 ζ6 ζ32 ζ3 ζ65 ζ6 ζ65 linear of order 6 ρ17 1 -1 -1 1 ζ3 ζ32 -1 1 -1 1 -1 1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ32 ζ65 ζ6 ζ3 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ3 ζ32 linear of order 6 ρ18 1 1 1 1 ζ3 ζ32 1 1 -1 -1 -1 -1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ32 ζ65 ζ6 ζ3 ζ65 ζ6 ζ3 ζ65 ζ6 ζ32 ζ65 ζ6 linear of order 6 ρ19 1 -1 -1 1 ζ32 ζ3 1 -1 -1 1 1 -1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ65 ζ6 ζ65 ζ6 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 linear of order 6 ρ20 1 1 1 1 ζ32 ζ3 1 1 1 1 1 1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ21 1 1 1 1 ζ3 ζ32 -1 -1 1 1 -1 -1 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ6 ζ3 ζ32 ζ65 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ65 ζ6 linear of order 6 ρ22 1 -1 -1 1 ζ32 ζ3 1 -1 1 -1 -1 1 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ65 ζ32 ζ3 ζ6 ζ6 ζ65 ζ32 ζ6 ζ65 ζ3 ζ32 ζ3 linear of order 6 ρ23 1 -1 -1 1 ζ3 ζ32 1 -1 1 -1 -1 1 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ6 ζ3 ζ32 ζ65 ζ65 ζ6 ζ3 ζ65 ζ6 ζ32 ζ3 ζ32 linear of order 6 ρ24 1 1 1 1 ζ32 ζ3 1 1 -1 -1 -1 -1 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ3 ζ6 ζ65 ζ32 ζ6 ζ65 ζ32 ζ6 ζ65 ζ3 ζ6 ζ65 linear of order 6 ρ25 2 2 -2 -2 2 2 0 0 0 0 0 0 -2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ26 2 -2 2 -2 2 2 0 0 0 0 0 0 -2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ27 2 -2 2 -2 -1-√-3 -1+√-3 0 0 0 0 0 0 1+√-3 1-√-3 -1-√-3 1+√-3 -1+√-3 1-√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×Q8 ρ28 2 2 -2 -2 -1+√-3 -1-√-3 0 0 0 0 0 0 1-√-3 1+√-3 1-√-3 -1+√-3 1+√-3 -1-√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×Q8 ρ29 2 2 -2 -2 -1-√-3 -1+√-3 0 0 0 0 0 0 1+√-3 1-√-3 1+√-3 -1-√-3 1-√-3 -1+√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×Q8 ρ30 2 -2 2 -2 -1+√-3 -1-√-3 0 0 0 0 0 0 1-√-3 1+√-3 -1+√-3 1-√-3 -1-√-3 1+√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C3×Q8

Smallest permutation representation of C6×Q8
Regular action on 48 points
Generators in S48
(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)
(1 29 16 19)(2 30 17 20)(3 25 18 21)(4 26 13 22)(5 27 14 23)(6 28 15 24)(7 37 48 33)(8 38 43 34)(9 39 44 35)(10 40 45 36)(11 41 46 31)(12 42 47 32)
(1 41 16 31)(2 42 17 32)(3 37 18 33)(4 38 13 34)(5 39 14 35)(6 40 15 36)(7 21 48 25)(8 22 43 26)(9 23 44 27)(10 24 45 28)(11 19 46 29)(12 20 47 30)

G:=sub<Sym(48)| (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), (1,29,16,19)(2,30,17,20)(3,25,18,21)(4,26,13,22)(5,27,14,23)(6,28,15,24)(7,37,48,33)(8,38,43,34)(9,39,44,35)(10,40,45,36)(11,41,46,31)(12,42,47,32), (1,41,16,31)(2,42,17,32)(3,37,18,33)(4,38,13,34)(5,39,14,35)(6,40,15,36)(7,21,48,25)(8,22,43,26)(9,23,44,27)(10,24,45,28)(11,19,46,29)(12,20,47,30)>;

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), (1,29,16,19)(2,30,17,20)(3,25,18,21)(4,26,13,22)(5,27,14,23)(6,28,15,24)(7,37,48,33)(8,38,43,34)(9,39,44,35)(10,40,45,36)(11,41,46,31)(12,42,47,32), (1,41,16,31)(2,42,17,32)(3,37,18,33)(4,38,13,34)(5,39,14,35)(6,40,15,36)(7,21,48,25)(8,22,43,26)(9,23,44,27)(10,24,45,28)(11,19,46,29)(12,20,47,30) );

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)], [(1,29,16,19),(2,30,17,20),(3,25,18,21),(4,26,13,22),(5,27,14,23),(6,28,15,24),(7,37,48,33),(8,38,43,34),(9,39,44,35),(10,40,45,36),(11,41,46,31),(12,42,47,32)], [(1,41,16,31),(2,42,17,32),(3,37,18,33),(4,38,13,34),(5,39,14,35),(6,40,15,36),(7,21,48,25),(8,22,43,26),(9,23,44,27),(10,24,45,28),(11,19,46,29),(12,20,47,30)]])

C6×Q8 is a maximal subgroup of   Q82Dic3  C12.10D4  Q8.11D6  Dic3⋊Q8  D63Q8  C12.23D4  Q8.15D6

Matrix representation of C6×Q8 in GL4(𝔽13) generated by

 9 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 12 0 0 1 0
,
 12 0 0 0 0 1 0 0 0 0 9 3 0 0 3 4
G:=sub<GL(4,GF(13))| [9,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,12,0],[12,0,0,0,0,1,0,0,0,0,9,3,0,0,3,4] >;

C6×Q8 in GAP, Magma, Sage, TeX

C_6\times Q_8
% in TeX

G:=Group("C6xQ8");
// GroupNames label

G:=SmallGroup(48,46);
// by ID

G=gap.SmallGroup(48,46);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-2,120,261,126]);
// Polycyclic

G:=Group<a,b,c|a^6=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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