Copied to
clipboard

G = C9⋊(S3×C9)  order 486 = 2·35

The semidirect product of C9 and S3×C9 acting via S3×C9/C32=C6

metabelian, supersoluble, monomial

Aliases: C9⋊(S3×C9), C3⋊(C9⋊C18), C9⋊S3⋊4C9, C9⋊C9⋊5S3, (C3×C9)⋊5C18, (C32×C9).6S3, (C32×C9).9C6, C33.80(C3×S3), C32.19(S3×C9), C32.11(C9⋊C6), C3.7(C33.S3), (C3×C9⋊C9)⋊2C2, C3.3(C9×C3⋊S3), (C3×C9⋊S3).4C3, (C3×C9).2(C3⋊S3), (C3×C9).38(C3×S3), C32.29(C3×C3⋊S3), SmallGroup(486,138)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C9 — C9⋊(S3×C9)
Chief series
C1 — C3 — C32 — C3×C9 — C32×C9 — C3×C9⋊C9 — C9⋊(S3×C9)
Lower central
C3×C9 — C9⋊(S3×C9)
Upper central
C1 — C3

Generators and relations for C9⋊(S3×C9)
 G = < a,b,c,d | a9=b9=c3=d2=1, bab-1=a7, ac=ca, dad=a-1, bc=cb, bd=db, dcd=c-1 >

Subgroups: 354 in 90 conjugacy classes, 29 normal (14 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C18, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, C33, C3×D9, S3×C9, C9⋊S3, C3×C3⋊S3, C9⋊C9, C9⋊C9, C32×C9, C32×C9, C9⋊C18, C3×C9⋊S3, C9×C3⋊S3, C3×C9⋊C9, C9⋊(S3×C9)
Quotients: C1, C2, C3, S3, C6, C9, C18, C3×S3, C3⋊S3, S3×C9, C9⋊C6, C3×C3⋊S3, C9⋊C18, C9×C3⋊S3, C33.S3, C9⋊(S3×C9)

Smallest permutation representation of C9⋊(S3×C9)
►On 54 points
Generators in S54
(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)

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

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

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

63 conjugacy classes

class 1  2 3A3B3C···3N6A6B9A···9F9G···9AM18A···18F
order12333···3669···99···918···18
size127112···227273···36···627···27

63 irreducible representations

dim11111122222266
type+++++
imageC1C2C3C6C9C18S3S3C3×S3C3×S3S3×C9S3×C9C9⋊C6C9⋊C18
kernelC9⋊(S3×C9)C3×C9⋊C9C3×C9⋊S3C32×C9C9⋊S3C3×C9C9⋊C9C32×C9C3×C9C33C9C32C32C3
# reps112266316218636

Matrix representation of C9⋊(S3×C9) ►in GL8(𝔽19)

10000000
01000000
000110000
000011000
00100000
00000001
00000700
00000070
,
50000000
05000000
00010000
000011000
001100000
00000010
000000011
000001100
,
70000000
011000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
10000000
00000100
00000010
00000001
00100000
00010000
00001000

G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,1,0,0],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,0],[7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C9⋊(S3×C9) in GAP, Magma, Sage, TeX 

C_9\rtimes (S_3\times C_9)
 
% in TeX
 
G:=Group("C9:(S3xC9)");
 
// GroupNames label
 
G:=SmallGroup(486,138);
 
// by ID
 
G=gap.SmallGroup(486,138);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,4755,2169,453,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,b*a*b^-1=a^7,a*c=c*a,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁