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G = C92⋊10C6  order 486 = 2·35

10th semidirect product of C92 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C92⋊10C6, C9⋊D9⋊7C3, C9⋊1(C9⋊C6), C92⋊7C3⋊3C2, C32⋊C9.14S3, C33.10(C3⋊S3), C3.6(He3.4S3), C3.4(C33.S3), (C3×3- 1+2).4S3, (C3×C9).40(C3×S3), C32.44(C3×C3⋊S3), SmallGroup(486,154)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C92 — C92⋊10C6
Chief series
C1 — C3 — C32 — C3×C9 — C92 — C92⋊7C3 — C92⋊10C6
Lower central
C92 — C92⋊10C6
Upper central
C1

Generators and relations for C92⋊10C6
 G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a2, cbc-1=b5 >

Subgroups: 692 in 84 conjugacy classes, 25 normal (7 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, 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⋊7C3, C92⋊10C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C9⋊C6, C3×C3⋊S3, C33.S3, He3.4S3, C92⋊10C6

Character table of C92⋊10C6

 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
ρ20606-3-3-30000-30000000006-300000000    orthogonal lifted from C9⋊C6
ρ21606-3-3-30000-3000000000-3600000000    orthogonal lifted from C9⋊C6
ρ2260-3-3-3600000000000-36-30000000000    orthogonal lifted from C9⋊C6
ρ2360-3-3-3600000000000-3-360000000000    orthogonal lifted from C9⋊C6
ρ24606-3-3-300006000000000-3-300000000    orthogonal lifted from C9⋊C6
ρ2560-36-3-3000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90000000000000000    orthogonal lifted from He3.4S3
ρ2660-3-36-3000000003ζ98+3ζ93ζ95+3ζ943ζ97+3ζ920000000000000    orthogonal lifted from He3.4S3
ρ2760-36-3-3000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940000000000000000    orthogonal lifted from He3.4S3
ρ2860-36-3-3000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920000000000000000    orthogonal lifted from He3.4S3
ρ2960-3-36-3000000003ζ95+3ζ943ζ97+3ζ923ζ98+3ζ90000000000000    orthogonal lifted from He3.4S3
ρ3060-3-36-3000000003ζ97+3ζ923ζ98+3ζ93ζ95+3ζ940000000000000    orthogonal lifted from He3.4S3

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

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

Matrix representation of C92⋊10C6 ►in GL12(ℤ)

-21-1-200000000
001-100000000
101101000000
2-211-1-1000000
-10-1000000000
00-1000000000
0000000-10000
0000001-10000
000000-11-1-100
000000101000
0000000-100-1-1
000000-110010
,
-110000000000
-100000000000
1-10100000000
-10-1-100000000
010001000000
1-100-1-1000000
0000002-12100
0000002-11200
0000000-1-10-1-1
000000-11-1010
000000-100-100
000000000-100
,
010000000000
100000000000
000010000000
1-200-1-1000000
-21-1-100000000
000100000000
000000-110000
000000010000
0000001-100-10
000000010011
0000001-11100
000000-100-100

G:=sub<GL(12,Integers())| [-2,0,1,2,-1,0,0,0,0,0,0,0,1,0,0,-2,0,0,0,0,0,0,0,0,-1,1,1,1,-1,-1,0,0,0,0,0,0,-2,-1,1,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,1,0,-1,0,0,0,0,0,0,-1,-1,1,0,-1,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0],[-1,-1,1,-1,0,1,0,0,0,0,0,0,1,0,-1,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,2,2,0,-1,-1,0,0,0,0,0,0,0,-1,-1,-1,1,0,0,0,0,0,0,0,0,2,1,-1,-1,0,0,0,0,0,0,0,0,1,2,0,0,-1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0],[0,1,0,1,-2,0,0,0,0,0,0,0,1,0,0,-2,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,-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,-1,0,1,0,1,-1,0,0,0,0,0,0,1,1,-1,1,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
 

C92⋊10C6 in GAP, Magma, Sage, TeX 

C_9^2\rtimes_{10}C_6
 
% in TeX
 
G:=Group("C9^2:10C6");
 
// GroupNames label
 
G:=SmallGroup(486,154);
 
// by ID
 
G=gap.SmallGroup(486,154);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,548,338,8643,2169,237,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^5>;
 
// generators/relations
 

Export

Character table of C92⋊10C6 in TeX

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