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G = C92⋊5C6  order 486 = 2·35

5th semidirect product of C92 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C92⋊5C6, C9⋊D9⋊5C3, C92⋊5C3⋊2C2, C32⋊C9.16S3, C33.12(C3⋊S3), C3.8(He3.4S3), (C3×C9).33(C3×S3), C32.47(C3×C3⋊S3), SmallGroup(486,157)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C92 — C92⋊5C6
Chief series
C1 — C3 — C32 — C3×C9 — C92 — C92⋊5C3 — C92⋊5C6
Lower central
C92 — C92⋊5C6
Upper central
C1

Generators and relations for C92⋊5C6
 G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a-1b6, cbc-1=a3b-1 >

Subgroups: 620 in 64 conjugacy classes, 19 normal (7 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C33, C9⋊S3, C3×C3⋊S3, C92, C32⋊C9, C9⋊C9, C32⋊D9, C9⋊D9, C92⋊5C3, C92⋊5C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C3×C3⋊S3, He3.4S3, C92⋊5C6

Character table of C92⋊5C6

 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
ρ19606-3-3-3000000000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920000000000    orthogonal lifted from He3.4S3
ρ2060-3-36-3000003ζ98+3ζ93ζ95+3ζ943ζ97+3ζ920000000000000000    orthogonal lifted from He3.4S3
ρ2160-36-3-3000000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920000000000000    orthogonal lifted from He3.4S3
ρ22606-3-3-3000000000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90000000000    orthogonal lifted from He3.4S3
ρ2360-3-36-3000003ζ97+3ζ923ζ98+3ζ93ζ95+3ζ940000000000000000    orthogonal lifted from He3.4S3
ρ2460-3-3-3600003ζ98+3ζ90000000003ζ97+3ζ923ζ95+3ζ9400000000    orthogonal lifted from He3.4S3
ρ2560-3-3-3600003ζ95+3ζ940000000003ζ98+3ζ93ζ97+3ζ9200000000    orthogonal lifted from He3.4S3
ρ26606-3-3-3000000000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940000000000    orthogonal lifted from He3.4S3
ρ2760-3-36-3000003ζ95+3ζ943ζ97+3ζ923ζ98+3ζ90000000000000000    orthogonal lifted from He3.4S3
ρ2860-3-3-3600003ζ97+3ζ920000000003ζ95+3ζ943ζ98+3ζ900000000    orthogonal lifted from He3.4S3
ρ2960-36-3-3000000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90000000000000    orthogonal lifted from He3.4S3
ρ3060-36-3-3000000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940000000000000    orthogonal lifted from He3.4S3

Smallest permutation representation of C92⋊5C6
►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 20 36 70 10 53 74 57 37)(2 21 28 71 11 54 75 58 38)(3 22 29 72 12 46 76 59 39)(4 23 30 64 13 47 77 60 40)(5 24 31 65 14 48 78 61 41)(6 25 32 66 15 49 79 62 42)(7 26 33 67 16 50 80 63 43)(8 27 34 68 17 51 81 55 44)(9 19 35 69 18 52 73 56 45)

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

Matrix representation of C92⋊5C6 ►in GL12(𝔽19)

1422100000000
171218100000000
0051710000000
002701000000
01817500000000
11141200000000
00000016130000
0000002100000
00000055141700
000000151821200
0000001513001417
00000051800212
,
12170000000000
2140000000000
00121700000000
0021400000000
00001217000000
0000214000000
0000001203000
000000701100
000000807010
0000005010001
000000603000
0000008110000
,
0180000000000
1800000000000
720001000000
14120010000000
00181800000000
000100000000
0000001830000
000000010000
0000000700118
00000001000018
00000014318000
00000021018100

G:=sub<GL(12,GF(19))| [14,17,0,0,0,1,0,0,0,0,0,0,2,12,0,0,18,1,0,0,0,0,0,0,2,18,5,2,17,14,0,0,0,0,0,0,1,1,17,7,5,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16,2,5,15,15,5,0,0,0,0,0,0,13,10,5,18,13,18,0,0,0,0,0,0,0,0,14,2,0,0,0,0,0,0,0,0,0,0,17,12,0,0,0,0,0,0,0,0,0,0,0,0,14,2,0,0,0,0,0,0,0,0,0,0,17,12],[12,2,0,0,0,0,0,0,0,0,0,0,17,14,0,0,0,0,0,0,0,0,0,0,0,0,12,2,0,0,0,0,0,0,0,0,0,0,17,14,0,0,0,0,0,0,0,0,0,0,0,0,12,2,0,0,0,0,0,0,0,0,0,0,17,14,0,0,0,0,0,0,0,0,0,0,0,0,12,7,8,5,6,8,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,7,10,3,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,18,7,14,0,0,0,0,0,0,0,0,18,0,2,12,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0,14,2,0,0,0,0,0,0,3,1,7,10,3,10,0,0,0,0,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,18,18,0,0] >;
 

C92⋊5C6 in GAP, Magma, Sage, TeX 

C_9^2\rtimes_5C_6
 
% in TeX
 
G:=Group("C9^2:5C6");
 
// GroupNames label
 
G:=SmallGroup(486,157);
 
// by ID
 
G=gap.SmallGroup(486,157);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,2978,338,4755,873,453,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^-1*b^6,c*b*c^-1=a^3*b^-1>;
 
// generators/relations
 

Export

Character table of C92⋊5C6 in TeX

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