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G = C62⋊C9  order 324 = 22·34

1st semidirect product of C62 and C9 acting via C9/C3=C3

metabelian, soluble, monomial

Aliases: C62⋊1C9, C33.1A4, C62.17C32, C32⋊(C3.A4), (C3×C62).1C3, (C2×C6).11He3, C3.2(C32⋊A4), C22⋊2(C32⋊C9), C32.19(C3×A4), C3.2(C32.A4), (C2×C6).43- 1+2, (C3×C3.A4)⋊5C3, (C2×C6).9(C3×C9), C3.5(C3×C3.A4), SmallGroup(324,59)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C62⋊C9
Chief series
C1 — C22 — C2×C6 — C62 — C3×C3.A4 — C62⋊C9
Lower central
C22 — C2×C6 — C62⋊C9
Upper central
C1 — C32 — C33

Generators and relations for C62⋊C9
 G = < a,b,c | a6=b6=c9=1, ab=ba, cac-1=ab-1, cbc-1=a3b4 >

Subgroups: 205 in 74 conjugacy classes, 24 normal (11 characteristic)
C1, C2, C3, C3, C3, C22, C6, C9, C32, C32, C32, C2×C6, C2×C6, C2×C6, C3×C6, C3×C9, C33, C3.A4, C62, C62, C62, C32×C6, C32⋊C9, C3×C3.A4, C3×C62, C62⋊C9
Quotients: C1, C3, C9, C32, A4, C3×C9, He3, 3- 1+2, C3.A4, C3×A4, C32⋊C9, C3×C3.A4, C32.A4, C32⋊A4, C62⋊C9

Smallest permutation representation of C62⋊C9
►On 54 points
Generators in S54
(1 30 7 36 4 33)(2 52 19 28 39 16)(3 26 43)(5 46 22 31 42 10)(6 20 37)(8 49 25 34 45 13)(9 23 40)(11 53 35)(12 27 18 24 15 21)(14 47 29)(17 50 32)(38 54 44 51 41 48)

(1 21 44)(2 10 45 28 22 49)(3 11 37 29 23 50)(4 24 38)(5 13 39 31 25 52)(6 14 40 32 26 53)(7 27 41)(8 16 42 34 19 46)(9 17 43 35 20 47)(12 51 30)(15 54 33)(18 48 36)

(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 49 50 51 52 53 54)
 
G:=sub<Sym(54)| (1,30,7,36,4,33)(2,52,19,28,39,16)(3,26,43)(5,46,22,31,42,10)(6,20,37)(8,49,25,34,45,13)(9,23,40)(11,53,35)(12,27,18,24,15,21)(14,47,29)(17,50,32)(38,54,44,51,41,48), (1,21,44)(2,10,45,28,22,49)(3,11,37,29,23,50)(4,24,38)(5,13,39,31,25,52)(6,14,40,32,26,53)(7,27,41)(8,16,42,34,19,46)(9,17,43,35,20,47)(12,51,30)(15,54,33)(18,48,36), (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)(46,47,48,49,50,51,52,53,54)>;
 
G:=Group( (1,30,7,36,4,33)(2,52,19,28,39,16)(3,26,43)(5,46,22,31,42,10)(6,20,37)(8,49,25,34,45,13)(9,23,40)(11,53,35)(12,27,18,24,15,21)(14,47,29)(17,50,32)(38,54,44,51,41,48), (1,21,44)(2,10,45,28,22,49)(3,11,37,29,23,50)(4,24,38)(5,13,39,31,25,52)(6,14,40,32,26,53)(7,27,41)(8,16,42,34,19,46)(9,17,43,35,20,47)(12,51,30)(15,54,33)(18,48,36), (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)(46,47,48,49,50,51,52,53,54) );
 
G=PermutationGroup([[(1,30,7,36,4,33),(2,52,19,28,39,16),(3,26,43),(5,46,22,31,42,10),(6,20,37),(8,49,25,34,45,13),(9,23,40),(11,53,35),(12,27,18,24,15,21),(14,47,29),(17,50,32),(38,54,44,51,41,48)], [(1,21,44),(2,10,45,28,22,49),(3,11,37,29,23,50),(4,24,38),(5,13,39,31,25,52),(6,14,40,32,26,53),(7,27,41),(8,16,42,34,19,46),(9,17,43,35,20,47),(12,51,30),(15,54,33),(18,48,36)], [(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),(46,47,48,49,50,51,52,53,54)]])
 

60 conjugacy classes

class 1  2 3A···3H3I···3N6A···6Z9A···9R
order123···33···36···69···9
size131···13···33···312···12

60 irreducible representations

dim11113333333
type++
imageC1C3C3C9A4He33- 1+2C3.A4C3×A4C32.A4C32⋊A4
kernelC62⋊C9C3×C3.A4C3×C62C62C33C2×C6C2×C6C32C32C3C3
# reps1621812462126

Matrix representation of C62⋊C9 ►in GL4(𝔽19) generated by

11000
0800
0010
00012
,
1000
0700
00120
00012
,
6000
0010
0001
01100
G:=sub<GL(4,GF(19))| [11,0,0,0,0,8,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,7,0,0,0,0,12,0,0,0,0,12],[6,0,0,0,0,0,0,11,0,1,0,0,0,0,1,0] >;
 

C62⋊C9 in GAP, Magma, Sage, TeX 

C_6^2\rtimes C_9
 
% in TeX
 
G:=Group("C6^2:C9");
 
// GroupNames label
 
G:=SmallGroup(324,59);
 
// by ID
 
G=gap.SmallGroup(324,59);
 
# by ID
 
G:=PCGroup([6,-3,-3,-3,-3,-2,2,54,361,4864,8753]);
 
// Polycyclic
 
G:=Group<a,b,c|a^6=b^6=c^9=1,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^3*b^4>;
 
// generators/relations
 

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