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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

C1C2 — C6×Q8
C1C2C6C12C3×Q8 — C6×Q8
C1C2 — C6×Q8
C1C2×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 12A2B2C3A3B4A4B4C4D4E4F6A6B6C6D6E6F12A12B12C12D12E12F12G12H12I12J12K12L
 size 111111222222111111222222222222
ρ1111111111111111111111111111111    trivial
ρ21-1-1111-11-11-1111-1-1-1-11-1-1111-1-1-1-111    linear of order 2
ρ31-1-1111-111-11-111-1-1-1-11111-1-1-111-1-1-1    linear of order 2
ρ411111111-1-1-1-11111111-1-11-1-11-1-11-1-1    linear of order 2
ρ5111111-1-111-1-1111111-111-111-1-1-1-1-1-1    linear of order 2
ρ61-1-11111-1-111-111-1-1-1-1-1-1-1-1111111-1-1    linear of order 2
ρ71-1-11111-11-1-1111-1-1-1-1-111-1-1-11-1-1111    linear of order 2
ρ8111111-1-1-1-111111111-1-1-1-1-1-1-111-111    linear of order 2
ρ91-1-11ζ3ζ32-111-11-1ζ3ζ32ζ65ζ65ζ6ζ6ζ32ζ3ζ32ζ3ζ65ζ6ζ65ζ3ζ32ζ6ζ65ζ6    linear of order 6
ρ101111ζ32ζ3-1-111-1-1ζ32ζ3ζ32ζ32ζ3ζ3ζ65ζ32ζ3ζ6ζ32ζ3ζ6ζ6ζ65ζ65ζ6ζ65    linear of order 6
ρ111111ζ3ζ32111111ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32    linear of order 3
ρ121-1-11ζ32ζ3-11-11-11ζ32ζ3ζ6ζ6ζ65ζ65ζ3ζ6ζ65ζ32ζ32ζ3ζ6ζ6ζ65ζ65ζ32ζ3    linear of order 6
ρ131111ζ3ζ32-1-1-1-111ζ3ζ32ζ3ζ3ζ32ζ32ζ6ζ65ζ6ζ65ζ65ζ6ζ65ζ3ζ32ζ6ζ3ζ32    linear of order 6
ρ141111ζ32ζ3-1-1-1-111ζ32ζ3ζ32ζ32ζ3ζ3ζ65ζ6ζ65ζ6ζ6ζ65ζ6ζ32ζ3ζ65ζ32ζ3    linear of order 6
ρ151-1-11ζ3ζ321-1-111-1ζ3ζ32ζ65ζ65ζ6ζ6ζ6ζ65ζ6ζ65ζ3ζ32ζ3ζ3ζ32ζ32ζ65ζ6    linear of order 6
ρ161-1-11ζ32ζ3-111-11-1ζ32ζ3ζ6ζ6ζ65ζ65ζ3ζ32ζ3ζ32ζ6ζ65ζ6ζ32ζ3ζ65ζ6ζ65    linear of order 6
ρ171-1-11ζ3ζ32-11-11-11ζ3ζ32ζ65ζ65ζ6ζ6ζ32ζ65ζ6ζ3ζ3ζ32ζ65ζ65ζ6ζ6ζ3ζ32    linear of order 6
ρ181111ζ3ζ3211-1-1-1-1ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ65ζ6ζ3ζ65ζ6ζ3ζ65ζ6ζ32ζ65ζ6    linear of order 6
ρ191-1-11ζ32ζ31-1-111-1ζ32ζ3ζ6ζ6ζ65ζ65ζ65ζ6ζ65ζ6ζ32ζ3ζ32ζ32ζ3ζ3ζ6ζ65    linear of order 6
ρ201111ζ32ζ3111111ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3    linear of order 3
ρ211111ζ3ζ32-1-111-1-1ζ3ζ32ζ3ζ3ζ32ζ32ζ6ζ3ζ32ζ65ζ3ζ32ζ65ζ65ζ6ζ6ζ65ζ6    linear of order 6
ρ221-1-11ζ32ζ31-11-1-11ζ32ζ3ζ6ζ6ζ65ζ65ζ65ζ32ζ3ζ6ζ6ζ65ζ32ζ6ζ65ζ3ζ32ζ3    linear of order 6
ρ231-1-11ζ3ζ321-11-1-11ζ3ζ32ζ65ζ65ζ6ζ6ζ6ζ3ζ32ζ65ζ65ζ6ζ3ζ65ζ6ζ32ζ3ζ32    linear of order 6
ρ241111ζ32ζ311-1-1-1-1ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ6ζ65ζ32ζ6ζ65ζ32ζ6ζ65ζ3ζ6ζ65    linear of order 6
ρ2522-2-222000000-2-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ262-22-222000000-2-22-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ272-22-2-1--3-1+-30000001+-31--3-1--31+-3-1+-31--3000000000000    complex lifted from C3×Q8
ρ2822-2-2-1+-3-1--30000001--31+-31--3-1+-31+-3-1--3000000000000    complex lifted from C3×Q8
ρ2922-2-2-1--3-1+-30000001+-31--31+-3-1--31--3-1+-3000000000000    complex lifted from C3×Q8
ρ302-22-2-1+-3-1--30000001--31+-3-1+-31--3-1--31+-3000000000000    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)])

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

9000
01200
0010
0001
,
1000
0100
00012
0010
,
12000
0100
0093
0034
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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