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G = C3×C3⋊C16  order 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, C32⋊3C16, 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

Derived series
C1 — C3 — C3×C3⋊C16
Chief series
C1 — C3 — C6 — C12 — C24 — C3×C24 — C3×C3⋊C16
Lower central
C3 — C3×C3⋊C16
Upper central
C1 — C24

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

C1
C2
C3
C3
2C3
C4
C6
C6
2C6
C32
C8
C12
C12
2C12
C3×C6
3C16
C24
C24
2C24
C3×C12
C3⋊C16
3C48
C3×C24
C3×C3⋊C16

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

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

C3×C3⋊C16 is a maximal subgroup of
 C24.60D6  C24.61D6  C24.62D6  C3⋊D48  C32⋊3SD32  C24.49D6  C32⋊3Q32  S3×C48  He3⋊3C16  C9⋊C48  He3⋊4C16
C3×C3⋊C16 is a maximal quotient of
 He3⋊3C16  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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