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G = C3×C3⋊C16order 144 = 24·32

Direct product of C3 and C3⋊C16

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×C3⋊C16, C3⋊C48, C6.C24, C24.5C6, C323C16, C12.2C12, C24.10S3, C12.10Dic3, C6.4(C3⋊C8), C8.2(C3×S3), (C3×C6).3C8, (C3×C12).6C4, (C3×C24).4C2, C4.2(C3×Dic3), C2.(C3×C3⋊C8), SmallGroup(144,28)

Series: Derived Chief Lower central Upper central

C1C3 — C3×C3⋊C16
C1C3C6C12C24C3×C24 — C3×C3⋊C16
C3 — C3×C3⋊C16
C1C24

Generators and relations for C3×C3⋊C16
 G = < a,b,c | a3=b3=c16=1, ab=ba, ac=ca, cbc-1=b-1 >

2C3
2C6
2C12
3C16
2C24
3C48

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

C3×C3⋊C16 is a maximal subgroup of
C24.60D6  C24.61D6  C24.62D6  C3⋊D48  C323SD32  C24.49D6  C323Q32  S3×C48  He33C16  C9⋊C48  He34C16
C3×C3⋊C16 is a maximal quotient of
He33C16  C9⋊C48

72 conjugacy classes

class 1  2 3A3B3C3D3E4A4B6A6B6C6D6E8A8B8C8D12A12B12C12D12E···12J16A···16H24A···24H24I···24T48A···48P
order1233333446666688881212121212···1216···1624···2424···2448···48
size11112221111222111111112···23···31···12···23···3

72 irreducible representations

dim111111111122222222
type+++-
imageC1C2C3C4C6C8C12C16C24C48S3Dic3C3×S3C3⋊C8C3×Dic3C3⋊C16C3×C3⋊C8C3×C3⋊C16
kernelC3×C3⋊C16C3×C24C3⋊C16C3×C12C24C3×C6C12C32C6C3C24C12C8C6C4C3C2C1
# reps1122244881611222448

Matrix representation of C3×C3⋊C16 in GL3(𝔽97) generated by

3500
0610
0061
,
100
0610
0035
,
8500
001
010
G:=sub<GL(3,GF(97))| [35,0,0,0,61,0,0,0,61],[1,0,0,0,61,0,0,0,35],[85,0,0,0,0,1,0,1,0] >;

C3×C3⋊C16 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes C_{16}
% in TeX

G:=Group("C3xC3:C16");
// GroupNames label

G:=SmallGroup(144,28);
// by ID

G=gap.SmallGroup(144,28);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-2,-3,36,50,69,3461]);
// Polycyclic

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

Export

Subgroup lattice of C3×C3⋊C16 in TeX

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