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## G = C3⋊Q16order 48 = 24·3

### The semidirect product of C3 and Q16 acting via Q16/Q8=C2

Aliases: C32Q16, C4.4D6, C6.10D4, Q8.2S3, C12.4C22, Dic6.2C2, C3⋊C8.C2, (C3×Q8).1C2, C2.7(C3⋊D4), SmallGroup(48,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3⋊Q16
 Chief series C1 — C3 — C6 — C12 — Dic6 — C3⋊Q16
 Lower central C3 — C6 — C12 — C3⋊Q16
 Upper central C1 — C2 — C4 — Q8

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

Character table of C3⋊Q16

 class 1 2 3 4A 4B 4C 6 8A 8B 12A 12B 12C size 1 1 2 2 4 12 2 6 6 4 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 -1 2 2 0 -1 0 0 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 2 -2 0 0 2 0 0 0 -2 0 orthogonal lifted from D4 ρ7 2 2 -1 2 -2 0 -1 0 0 1 -1 1 orthogonal lifted from D6 ρ8 2 -2 2 0 0 0 -2 √2 -√2 0 0 0 symplectic lifted from Q16, Schur index 2 ρ9 2 -2 2 0 0 0 -2 -√2 √2 0 0 0 symplectic lifted from Q16, Schur index 2 ρ10 2 2 -1 -2 0 0 -1 0 0 -√-3 1 √-3 complex lifted from C3⋊D4 ρ11 2 2 -1 -2 0 0 -1 0 0 √-3 1 -√-3 complex lifted from C3⋊D4 ρ12 4 -4 -2 0 0 0 2 0 0 0 0 0 symplectic faithful, Schur index 2

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

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

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

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

Matrix representation of C3⋊Q16 in GL4(𝔽5) generated by

 1 0 2 3 4 2 4 3 1 4 2 0 0 4 4 3
,
 2 2 3 3 2 4 0 2 0 3 3 0 1 4 1 1
,
 3 2 4 0 1 2 1 3 2 0 4 1 1 1 0 1
G:=sub<GL(4,GF(5))| [1,4,1,0,0,2,4,4,2,4,2,4,3,3,0,3],[2,2,0,1,2,4,3,4,3,0,3,1,3,2,0,1],[3,1,2,1,2,2,0,1,4,1,4,0,0,3,1,1] >;

C3⋊Q16 in GAP, Magma, Sage, TeX

C_3\rtimes Q_{16}
% in TeX

G:=Group("C3:Q16");
// GroupNames label

G:=SmallGroup(48,18);
// by ID

G=gap.SmallGroup(48,18);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,40,61,46,182,97,42,804]);
// Polycyclic

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

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