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G = C3⋊C16order 48 = 24·3

The semidirect product of C3 and C16 acting via C16/C8=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3⋊C16, C6.C8, C8.2S3, C24.3C2, C12.2C4, C4.2Dic3, C2.(C3⋊C8), SmallGroup(48,1)

Series: Derived Chief Lower central Upper central

C1C3 — C3⋊C16
C1C3C6C12C24 — C3⋊C16
C3 — C3⋊C16
C1C8

Generators and relations for C3⋊C16
 G = < a,b | a3=b16=1, bab-1=a-1 >

3C16

Character table of C3⋊C16

 class 1234A4B68A8B8C8D12A12B16A16B16C16D16E16F16G16H24A24B24C24D
 size 112112111122333333332222
ρ1111111111111111111111111    trivial
ρ2111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ3111111-1-1-1-111-iiii-i-i-ii-1-1-1-1    linear of order 4
ρ4111111-1-1-1-111i-i-i-iiii-i-1-1-1-1    linear of order 4
ρ5111-1-11i-i-ii-1-1ζ8ζ87ζ83ζ83ζ85ζ85ζ8ζ87-ii-ii    linear of order 8
ρ6111-1-11-iii-i-1-1ζ87ζ8ζ85ζ85ζ83ζ83ζ87ζ8i-ii-i    linear of order 8
ρ7111-1-11i-i-ii-1-1ζ85ζ83ζ87ζ87ζ8ζ8ζ85ζ83-ii-ii    linear of order 8
ρ8111-1-11-iii-i-1-1ζ83ζ85ζ8ζ8ζ87ζ87ζ83ζ85i-ii-i    linear of order 8
ρ91-11-ii-1ζ166ζ1610ζ162ζ1614-iiζ167ζ16ζ1613ζ165ζ1611ζ163ζ1615ζ169ζ1610ζ166ζ162ζ1614    linear of order 16
ρ101-11i-i-1ζ162ζ1614ζ166ζ1610i-iζ165ζ163ζ167ζ1615ζ16ζ169ζ1613ζ1611ζ1614ζ162ζ166ζ1610    linear of order 16
ρ111-11i-i-1ζ1610ζ166ζ1614ζ162i-iζ16ζ167ζ1611ζ163ζ1613ζ165ζ169ζ1615ζ166ζ1610ζ1614ζ162    linear of order 16
ρ121-11-ii-1ζ1614ζ162ζ1610ζ166-iiζ163ζ165ζ16ζ169ζ167ζ1615ζ1611ζ1613ζ162ζ1614ζ1610ζ166    linear of order 16
ρ131-11i-i-1ζ162ζ1614ζ166ζ1610i-iζ1613ζ1611ζ1615ζ167ζ169ζ16ζ165ζ163ζ1614ζ162ζ166ζ1610    linear of order 16
ρ141-11i-i-1ζ1610ζ166ζ1614ζ162i-iζ169ζ1615ζ163ζ1611ζ165ζ1613ζ16ζ167ζ166ζ1610ζ1614ζ162    linear of order 16
ρ151-11-ii-1ζ1614ζ162ζ1610ζ166-iiζ1611ζ1613ζ169ζ16ζ1615ζ167ζ163ζ165ζ162ζ1614ζ1610ζ166    linear of order 16
ρ161-11-ii-1ζ166ζ1610ζ162ζ1614-iiζ1615ζ169ζ165ζ1613ζ163ζ1611ζ167ζ16ζ1610ζ166ζ162ζ1614    linear of order 16
ρ1722-122-12222-1-100000000-1-1-1-1    orthogonal lifted from S3
ρ1822-122-1-2-2-2-2-1-1000000001111    symplectic lifted from Dic3, Schur index 2
ρ1922-1-2-2-12i-2i-2i2i1100000000i-ii-i    complex lifted from C3⋊C8
ρ2022-1-2-2-1-2i2i2i-2i1100000000-ii-ii    complex lifted from C3⋊C8
ρ212-2-12i-2i18583878-ii00000000ζ87ζ8ζ83ζ85    complex faithful, Schur index 2
ρ222-2-12i-2i18878385-ii00000000ζ83ζ85ζ87ζ8    complex faithful, Schur index 2
ρ232-2-1-2i2i18788583i-i00000000ζ85ζ83ζ8ζ87    complex faithful, Schur index 2
ρ242-2-1-2i2i18385887i-i00000000ζ8ζ87ζ85ζ83    complex faithful, Schur index 2

Smallest permutation representation of C3⋊C16
Regular action on 48 points
Generators in S48
(1 21 41)(2 42 22)(3 23 43)(4 44 24)(5 25 45)(6 46 26)(7 27 47)(8 48 28)(9 29 33)(10 34 30)(11 31 35)(12 36 32)(13 17 37)(14 38 18)(15 19 39)(16 40 20)
(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)

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

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

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

Matrix representation of C3⋊C16 in GL2(𝔽17) generated by

166
140
,
615
011
G:=sub<GL(2,GF(17))| [16,14,6,0],[6,0,15,11] >;

C3⋊C16 in GAP, Magma, Sage, TeX

C_3\rtimes C_{16}
% in TeX

G:=Group("C3:C16");
// GroupNames label

G:=SmallGroup(48,1);
// by ID

G=gap.SmallGroup(48,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,10,26,42,804]);
// Polycyclic

G:=Group<a,b|a^3=b^16=1,b*a*b^-1=a^-1>;
// generators/relations

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