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G = C3×Q16order 48 = 24·3

Direct product of C3 and Q16

direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×Q16, C8.C6, C24.2C2, C6.16D4, Q8.2C6, C12.19C22, C4.3(C2×C6), C2.5(C3×D4), (C3×Q8).2C2, SmallGroup(48,27)

Series: Derived Chief Lower central Upper central

C1C4 — C3×Q16
C1C2C4C12C3×Q8 — C3×Q16
C1C2C4 — C3×Q16
C1C6C12 — C3×Q16

Generators and relations for C3×Q16
 G = < a,b,c | a3=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

2C4
2C4
2C12
2C12

Character table of C3×Q16

 class 123A3B4A4B4C6A6B8A8B12A12B12C12D12E12F24A24B24C24D
 size 111124411222244442222
ρ1111111111111111111111    trivial
ρ211111-1-1111111-1-1-1-11111    linear of order 2
ρ311111-1111-1-1111-11-1-1-1-1-1    linear of order 2
ρ4111111-111-1-111-11-11-1-1-1-1    linear of order 2
ρ511ζ32ζ31-11ζ32ζ3-1-1ζ32ζ3ζ32ζ65ζ3ζ6ζ6ζ6ζ65ζ65    linear of order 6
ρ611ζ32ζ311-1ζ32ζ3-1-1ζ32ζ3ζ6ζ3ζ65ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ711ζ3ζ32111ζ3ζ3211ζ3ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ811ζ3ζ321-1-1ζ3ζ3211ζ3ζ32ζ65ζ6ζ6ζ65ζ3ζ3ζ32ζ32    linear of order 6
ρ911ζ32ζ31-1-1ζ32ζ311ζ32ζ3ζ6ζ65ζ65ζ6ζ32ζ32ζ3ζ3    linear of order 6
ρ1011ζ3ζ3211-1ζ3ζ32-1-1ζ3ζ32ζ65ζ32ζ6ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ1111ζ32ζ3111ζ32ζ311ζ32ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ1211ζ3ζ321-11ζ3ζ32-1-1ζ3ζ32ζ3ζ6ζ32ζ65ζ65ζ65ζ6ζ6    linear of order 6
ρ132222-2002200-2-200000000    orthogonal lifted from D4
ρ142-222000-2-2-220000002-22-2    symplectic lifted from Q16, Schur index 2
ρ152-222000-2-22-2000000-22-22    symplectic lifted from Q16, Schur index 2
ρ1622-1+-3-1--3-200-1+-3-1--3001--31+-300000000    complex lifted from C3×D4
ρ1722-1--3-1+-3-200-1--3-1+-3001+-31--300000000    complex lifted from C3×D4
ρ182-2-1--3-1+-30001+-31--3-2200000083ζ328ζ3287ζ3285ζ3283ζ38ζ387ζ385ζ3    complex faithful
ρ192-2-1+-3-1--30001--31+-32-200000087ζ385ζ383ζ38ζ387ζ3285ζ3283ζ328ζ32    complex faithful
ρ202-2-1--3-1+-30001+-31--32-200000087ζ3285ζ3283ζ328ζ3287ζ385ζ383ζ38ζ3    complex faithful
ρ212-2-1+-3-1--30001--31+-3-2200000083ζ38ζ387ζ385ζ383ζ328ζ3287ζ3285ζ32    complex faithful

Smallest permutation representation of C3×Q16
Regular action on 48 points
Generators in S48
(1 34 41)(2 35 42)(3 36 43)(4 37 44)(5 38 45)(6 39 46)(7 40 47)(8 33 48)(9 19 32)(10 20 25)(11 21 26)(12 22 27)(13 23 28)(14 24 29)(15 17 30)(16 18 31)
(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 21 5 17)(2 20 6 24)(3 19 7 23)(4 18 8 22)(9 47 13 43)(10 46 14 42)(11 45 15 41)(12 44 16 48)(25 39 29 35)(26 38 30 34)(27 37 31 33)(28 36 32 40)

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

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

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

Matrix representation of C3×Q16 in GL2(𝔽7) generated by

40
04
,
06
13
,
65
11
G:=sub<GL(2,GF(7))| [4,0,0,4],[0,1,6,3],[6,1,5,1] >;

C3×Q16 in GAP, Magma, Sage, TeX

C_3\times Q_{16}
% in TeX

G:=Group("C3xQ16");
// GroupNames label

G:=SmallGroup(48,27);
// by ID

G=gap.SmallGroup(48,27);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-2,120,141,126,723,368,58]);
// Polycyclic

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

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