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G = C9⋊S3⋊C9order 486 = 2·35

1st semidirect product of C9⋊S3 and C9 acting via C9/C3=C3

metabelian, supersoluble, monomial

Aliases: C9⋊S31C9, (C3×C9)⋊1D9, (C3×C9)⋊1C18, C3.1(C9×D9), C3.1(C9⋊C18), (C32×C9).1S3, (C32×C9).3C6, C32.8(C9⋊C6), C32.11(S3×C9), C33.73(C3×S3), C3.C921C2, C32.11(C3×D9), C3.1(C32⋊C18), C3.5(C32⋊D9), C32.24(C32⋊C6), (C3×C9⋊S3).1C3, SmallGroup(486,3)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C9⋊S3⋊C9
C1C3C32C3×C9C32×C9C3.C92 — C9⋊S3⋊C9
C3×C9 — C9⋊S3⋊C9
C1C3

Generators and relations for C9⋊S3⋊C9
 G = < a,b,c,d | a9=b3=c2=d9=1, dad-1=ab=ba, cac=a-1, cbc=b-1, bd=db, cd=dc >

Subgroups: 336 in 69 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C3 [×3], C3 [×2], C3 [×4], S3 [×4], C6, C9 [×7], C32 [×3], C32 [×2], C32 [×4], D9, C18, C3×S3 [×4], C3⋊S3, C3×C9 [×2], C3×C9 [×13], C33, C3×D9, S3×C9 [×4], C9⋊S3, C3×C3⋊S3, C32×C9 [×2], C32×C9, C3×C9⋊S3, C9×C3⋊S3, C3.C92, C9⋊S3⋊C9
Quotients: C1, C2, C3, S3, C6, C9, D9, C18, C3×S3, C3×D9, S3×C9, C32⋊C6, C9⋊C6 [×2], C9×D9, C32⋊C18, C32⋊D9, C9⋊C18 [×2], 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 42 31)(2 43 32)(3 44 33)(4 45 34)(5 37 35)(6 38 36)(7 39 28)(8 40 29)(9 41 30)(10 24 54)(11 25 46)(12 26 47)(13 27 48)(14 19 49)(15 20 50)(16 21 51)(17 22 52)(18 23 53)
(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 50)(38 49)(39 48)(40 47)(41 46)(42 54)(43 53)(44 52)(45 51)
(1 3 37 7 9 43 4 6 40)(2 34 36 8 31 33 5 28 30)(10 17 50 13 11 53 16 14 47)(12 24 22 15 27 25 18 21 19)(20 48 46 23 51 49 26 54 52)(29 42 44 35 39 41 32 45 38)

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

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

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

63 conjugacy classes

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

63 irreducible representations

dim1111112222226666
type++++++
imageC1C2C3C6C9C18S3D9C3×S3C3×D9S3×C9C9×D9C32⋊C6C9⋊C6C32⋊C18C9⋊C18
kernelC9⋊S3⋊C9C3.C92C3×C9⋊S3C32×C9C9⋊S3C3×C9C32×C9C3×C9C33C32C32C3C32C32C3C3
# reps11226613266181224

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

50000000
04000000
005100000
000141000
000130000
000004013
00000003
000000115
,
10000000
01000000
00700000
00070000
00007000
000001100
000000110
000000011
,
018000000
180000000
00000100
00000010
00000001
00100000
00010000
00001000
,
10000000
01000000
009015000
00802000
0016110000
000009015
00000802
0000016110

G:=sub<GL(8,GF(19))| [5,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,10,14,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,13,3,15],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11],[0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,8,16,0,0,0,0,0,0,0,1,0,0,0,0,0,15,2,10,0,0,0,0,0,0,0,0,9,8,16,0,0,0,0,0,0,0,1,0,0,0,0,0,15,2,10] >;

C9⋊S3⋊C9 in GAP, Magma, Sage, TeX

C_9\rtimes S_3\rtimes C_9
% in TeX

G:=Group("C9:S3:C9");
// GroupNames label

G:=SmallGroup(486,3);
// by ID

G=gap.SmallGroup(486,3);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,4755,873,453,3244,11669]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=c^2=d^9=1,d*a*d^-1=a*b=b*a,c*a*c=a^-1,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations

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