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G = C3⋊Q16order 48 = 24·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C12 — C3⋊Q16
C1C3C6C12Dic6 — C3⋊Q16
C3C6C12 — C3⋊Q16
C1C2C4Q8

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 >

2C4
6C4
3C8
3Q8
2Dic3
2C12
3Q16

Character table of C3⋊Q16

 class 1234A4B4C68A8B12A12B12C
 size 1122412266444
ρ1111111111111    trivial
ρ21111-111-1-1-11-1    linear of order 2
ρ31111-1-1111-11-1    linear of order 2
ρ411111-11-1-1111    linear of order 2
ρ522-1220-100-1-1-1    orthogonal lifted from S3
ρ6222-2002000-20    orthogonal lifted from D4
ρ722-12-20-1001-11    orthogonal lifted from D6
ρ82-22000-22-2000    symplectic lifted from Q16, Schur index 2
ρ92-22000-2-22000    symplectic lifted from Q16, Schur index 2
ρ1022-1-200-100--31-3    complex lifted from C3⋊D4
ρ1122-1-200-100-31--3    complex lifted from C3⋊D4
ρ124-4-2000200000    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

1023
4243
1420
0443
,
2233
2402
0330
1411
,
3240
1213
2041
1101
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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