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G = C92⋊11C6  order 486 = 2·35

11st semidirect product of C92 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C92⋊11C6, C9⋊D9⋊8C3, C92⋊8C3⋊C2, C9.3(C9⋊C6), C32⋊C9.17S3, C33.13(C3⋊S3), C3.9(He3.4S3), C3.5(C33.S3), (C3×3- 1+2).5S3, (C3×C9).34(C3×S3), C32.48(C3×C3⋊S3), SmallGroup(486,158)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C92 — C92⋊11C6
Chief series
C1 — C3 — C32 — C3×C9 — C92 — C92⋊8C3 — C92⋊11C6
Lower central
C92 — C92⋊11C6
Upper central
C1

Generators and relations for C92⋊11C6
 G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a2b3, cbc-1=b2 >

Subgroups: 656 in 74 conjugacy classes, 22 normal (10 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, 3- 1+2, C33, C9⋊C6, C9⋊S3, C3×C3⋊S3, C92, C32⋊C9, C9⋊C9, C3×3- 1+2, C32⋊D9, C9⋊D9, C33.S3, C92⋊8C3, C92⋊11C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C9⋊C6, C3×C3⋊S3, C33.S3, He3.4S3, C92⋊11C6

Character table of C92⋊11C6

 class 123A3B3C3D3E3F6A6B9A9B9C9D9E9F9G9H9I9J9K9L9M9N9O9P9Q9R9S9T
 size 18122229981816666666666661818181818181818
ρ1111111111111111111111111111111    trivial
ρ21-1111111-1-111111111111111111111    linear of order 2
ρ31-11111ζ32ζ3ζ6ζ65111111111111ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 6
ρ4111111ζ32ζ3ζ32ζ3111111111111ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ5111111ζ3ζ32ζ3ζ32111111111111ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ61-11111ζ3ζ32ζ65ζ6111111111111ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 6
ρ72022222200-1-1-1-1-1-1-1222-1-122-1-1-1-1-1-1    orthogonal lifted from S3
ρ820222222002-1-1-1-1-1-1-1-1-122-1-12-1-12-1-1    orthogonal lifted from S3
ρ92022222200-1-1-1-1222-1-1-1-1-1-1-1-1-12-1-12    orthogonal lifted from S3
ρ102022222200-1222-1-1-1-1-1-1-1-1-1-1-12-1-12-1    orthogonal lifted from S3
ρ11202222-1+√-3-1-√-300-1-1-1-1-1-1-1222-1-1-1-√-3-1+√-3ζ65ζ65ζ65ζ6ζ6ζ6    complex lifted from C3×S3
ρ12202222-1+√-3-1-√-3002-1-1-1-1-1-1-1-1-122ζ6ζ65-1+√-3ζ65ζ65-1-√-3ζ6ζ6    complex lifted from C3×S3
ρ13202222-1+√-3-1-√-300-1222-1-1-1-1-1-1-1-1ζ6ζ65ζ65-1+√-3ζ65ζ6-1-√-3ζ6    complex lifted from C3×S3
ρ14202222-1-√-3-1+√-300-1-1-1-1-1-1-1222-1-1-1+√-3-1-√-3ζ6ζ6ζ6ζ65ζ65ζ65    complex lifted from C3×S3
ρ15202222-1-√-3-1+√-300-1-1-1-1222-1-1-1-1-1ζ65ζ6ζ6ζ6-1-√-3ζ65ζ65-1+√-3    complex lifted from C3×S3
ρ16202222-1+√-3-1-√-300-1-1-1-1222-1-1-1-1-1ζ6ζ65ζ65ζ65-1+√-3ζ6ζ6-1-√-3    complex lifted from C3×S3
ρ17202222-1-√-3-1+√-300-1222-1-1-1-1-1-1-1-1ζ65ζ6ζ6-1-√-3ζ6ζ65-1+√-3ζ65    complex lifted from C3×S3
ρ18202222-1-√-3-1+√-3002-1-1-1-1-1-1-1-1-122ζ65ζ6-1-√-3ζ6ζ6-1+√-3ζ65ζ65    complex lifted from C3×S3
ρ1960-3-3-36000000000006-3-30000000000    orthogonal lifted from C9⋊C6
ρ2060-3-3-3600000000000-36-30000000000    orthogonal lifted from C9⋊C6
ρ2160-3-3-3600000000000-3-360000000000    orthogonal lifted from C9⋊C6
ρ2260-3-36-3000003ζ98+3ζ93ζ95+3ζ943ζ97+3ζ920000000000000000    orthogonal lifted from He3.4S3
ρ23606-3-3-3000000003ζ98+3ζ93ζ95+3ζ943ζ97+3ζ920000000000000    orthogonal lifted from He3.4S3
ρ2460-3-36-3000003ζ97+3ζ923ζ98+3ζ93ζ95+3ζ940000000000000000    orthogonal lifted from He3.4S3
ρ25606-3-3-3000000003ζ97+3ζ923ζ98+3ζ93ζ95+3ζ940000000000000    orthogonal lifted from He3.4S3
ρ26606-3-3-3000000003ζ95+3ζ943ζ97+3ζ923ζ98+3ζ90000000000000    orthogonal lifted from He3.4S3
ρ2760-3-36-3000003ζ95+3ζ943ζ97+3ζ923ζ98+3ζ90000000000000000    orthogonal lifted from He3.4S3
ρ2860-36-3-300003ζ95+3ζ940000000003ζ97+3ζ923ζ98+3ζ900000000    orthogonal lifted from He3.4S3
ρ2960-36-3-300003ζ97+3ζ920000000003ζ98+3ζ93ζ95+3ζ9400000000    orthogonal lifted from He3.4S3
ρ3060-36-3-300003ζ98+3ζ90000000003ζ95+3ζ943ζ97+3ζ9200000000    orthogonal lifted from He3.4S3

Smallest permutation representation of C92⋊11C6
►On 81 points
Generators in S81
(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)(55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81)

(1 67 36 17 23 50 74 57 38)(2 68 28 18 24 51 75 58 39)(3 69 29 10 25 52 76 59 40)(4 70 30 11 26 53 77 60 41)(5 71 31 12 27 54 78 61 42)(6 72 32 13 19 46 79 62 43)(7 64 33 14 20 47 80 63 44)(8 65 34 15 21 48 81 55 45)(9 66 35 16 22 49 73 56 37)

(2 79 81 9 12 10)(3 18 13 8 73 78)(4 7)(5 76 75 6 15 16)(11 80)(14 77)(17 74)(19 48 22 54 25 51)(20 30 64 53 63 41)(21 37 61 52 68 32)(23 36 67 50 57 38)(24 43 55 49 71 29)(26 33 70 47 60 44)(27 40 58 46 65 35)(28 62 34 56 31 59)(39 72 45 66 42 69)
 
G:=sub<Sym(81)| (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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,67,36,17,23,50,74,57,38)(2,68,28,18,24,51,75,58,39)(3,69,29,10,25,52,76,59,40)(4,70,30,11,26,53,77,60,41)(5,71,31,12,27,54,78,61,42)(6,72,32,13,19,46,79,62,43)(7,64,33,14,20,47,80,63,44)(8,65,34,15,21,48,81,55,45)(9,66,35,16,22,49,73,56,37), (2,79,81,9,12,10)(3,18,13,8,73,78)(4,7)(5,76,75,6,15,16)(11,80)(14,77)(17,74)(19,48,22,54,25,51)(20,30,64,53,63,41)(21,37,61,52,68,32)(23,36,67,50,57,38)(24,43,55,49,71,29)(26,33,70,47,60,44)(27,40,58,46,65,35)(28,62,34,56,31,59)(39,72,45,66,42,69)>;
 
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)(55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81), (1,67,36,17,23,50,74,57,38)(2,68,28,18,24,51,75,58,39)(3,69,29,10,25,52,76,59,40)(4,70,30,11,26,53,77,60,41)(5,71,31,12,27,54,78,61,42)(6,72,32,13,19,46,79,62,43)(7,64,33,14,20,47,80,63,44)(8,65,34,15,21,48,81,55,45)(9,66,35,16,22,49,73,56,37), (2,79,81,9,12,10)(3,18,13,8,73,78)(4,7)(5,76,75,6,15,16)(11,80)(14,77)(17,74)(19,48,22,54,25,51)(20,30,64,53,63,41)(21,37,61,52,68,32)(23,36,67,50,57,38)(24,43,55,49,71,29)(26,33,70,47,60,44)(27,40,58,46,65,35)(28,62,34,56,31,59)(39,72,45,66,42,69) );
 
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),(55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81)], [(1,67,36,17,23,50,74,57,38),(2,68,28,18,24,51,75,58,39),(3,69,29,10,25,52,76,59,40),(4,70,30,11,26,53,77,60,41),(5,71,31,12,27,54,78,61,42),(6,72,32,13,19,46,79,62,43),(7,64,33,14,20,47,80,63,44),(8,65,34,15,21,48,81,55,45),(9,66,35,16,22,49,73,56,37)], [(2,79,81,9,12,10),(3,18,13,8,73,78),(4,7),(5,76,75,6,15,16),(11,80),(14,77),(17,74),(19,48,22,54,25,51),(20,30,64,53,63,41),(21,37,61,52,68,32),(23,36,67,50,57,38),(24,43,55,49,71,29),(26,33,70,47,60,44),(27,40,58,46,65,35),(28,62,34,56,31,59),(39,72,45,66,42,69)]])
 

Matrix representation of C92⋊11C6 ►in GL12(𝔽19)

0000142000000
00001712000000
750000000000
1420000000000
007500000000
0014200000000
0000001220000
00000017140000
00000017014200
00000002171200
00000017000142
00000002001712
,
000100000000
00181800000000
000001000000
00001818000000
100000000000
010000000000
00000011181700
0000000011800
0000000001801
000000110181818
0000000001800
0000001001800
,
1800000000000
110000000000
0000180000000
000011000000
001100000000
0001800000000
0000005120000
00000017140000
00000050001214
00000017120027
00000017122700
00000001451700

G:=sub<GL(12,GF(19))| [0,0,7,14,0,0,0,0,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,0,0,0,0,7,14,0,0,0,0,0,0,0,0,0,0,5,2,0,0,0,0,0,0,14,17,0,0,0,0,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,17,17,0,17,0,0,0,0,0,0,0,2,14,0,2,0,2,0,0,0,0,0,0,0,0,14,17,0,0,0,0,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,0,0,0,0,14,17,0,0,0,0,0,0,0,0,0,0,2,12],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,17,18,18,18,18,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,1,18,0,0],[18,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,17,5,17,17,0,0,0,0,0,0,0,12,14,0,12,12,14,0,0,0,0,0,0,0,0,0,0,2,5,0,0,0,0,0,0,0,0,0,0,7,17,0,0,0,0,0,0,0,0,12,2,0,0,0,0,0,0,0,0,0,0,14,7,0,0] >;
 

C92⋊11C6 in GAP, Magma, Sage, TeX 

C_9^2\rtimes_{11}C_6
 
% in TeX
 
G:=Group("C9^2:11C6");
 
// GroupNames label
 
G:=SmallGroup(486,158);
 
// by ID
 
G=gap.SmallGroup(486,158);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,2006,338,4755,2169,453,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^2*b^3,c*b*c^-1=b^2>;
 
// generators/relations
 

Export

Character table of C92⋊11C6 in TeX

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