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G = C92⋊12C6  order 486 = 2·35

12nd semidirect product of C92 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C92⋊12C6, C9⋊D9⋊9C3, C9⋊2(C9⋊C6), C92⋊9C3⋊2C2, C33.14(C3⋊S3), C3.6(C33.S3), (C3×3- 1+2).6S3, (C3×C9).41(C3×S3), C32.49(C3×C3⋊S3), SmallGroup(486,159)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C92 — C92⋊12C6
Chief series
C1 — C3 — C32 — C3×C9 — C92 — C92⋊9C3 — C92⋊12C6
Lower central
C92 — C92⋊12C6
Upper central
C1

Generators and relations for C92⋊12C6
 G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a2, cbc-1=b2 >

Subgroups: 764 in 104 conjugacy classes, 31 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, C9⋊C9, C3×3- 1+2, C9⋊D9, C33.S3, C92⋊9C3, C92⋊12C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C9⋊C6, C3×C3⋊S3, C33.S3, C92⋊12C6

Character table of C92⋊12C6

 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
ρ2160-36-3-300000-3-360000000000000000    orthogonal lifted from C9⋊C6
ρ22606-3-3-30000-3000000000-3600000000    orthogonal lifted from C9⋊C6
ρ2360-3-3-3600000000000-36-30000000000    orthogonal lifted from C9⋊C6
ρ2460-3-36-3000000006-3-30000000000000    orthogonal lifted from C9⋊C6
ρ2560-3-36-300000000-3-360000000000000    orthogonal lifted from C9⋊C6
ρ2660-3-36-300000000-36-30000000000000    orthogonal lifted from C9⋊C6
ρ2760-3-3-3600000000000-3-360000000000    orthogonal lifted from C9⋊C6
ρ2860-36-3-3000006-3-30000000000000000    orthogonal lifted from C9⋊C6
ρ29606-3-3-300006000000000-3-300000000    orthogonal lifted from C9⋊C6
ρ3060-36-3-300000-36-30000000000000000    orthogonal lifted from C9⋊C6

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

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

Matrix representation of C92⋊12C6 ►in GL12(ℤ)

111121000000
111112000000
-110000000000
-100000000000
0-10-1-1-1000000
0-1-10-1-1000000
000000-30-1100
00000000-1000
0000001100-1-2
000000-51-421-1
000000410-10-2
000000-3-1-1101
,
00-1100000000
00-1000000000
-1-1-1-1-1-2000000
00001-1000000
001001000000
101001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
010000000000
100000000000
-1-1-1-1-2-1000000
0000-11000000
000110000000
001010000000
000000100000
0000001-10000
000000-4-111-11
0000002-10012
000000-5-11102
000000500-10-1

G:=sub<GL(12,Integers())| [1,1,-1,-1,0,0,0,0,0,0,0,0,1,1,1,0,-1,-1,0,0,0,0,0,0,1,1,0,0,0,-1,0,0,0,0,0,0,1,1,0,0,-1,0,0,0,0,0,0,0,2,1,0,0,-1,-1,0,0,0,0,0,0,1,2,0,0,-1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-3,0,1,-5,4,-3,0,0,0,0,0,0,0,0,1,1,1,-1,0,0,0,0,0,0,-1,-1,0,-4,0,-1,0,0,0,0,0,0,1,0,0,2,-1,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-2,-1,-2,1],[0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,-1,-1,0,1,1,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-2,-1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,1,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-2,-1,1,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,-4,2,-5,5,0,0,0,0,0,0,0,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,-1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,1,2,2,-1] >;
 

C92⋊12C6 in GAP, Magma, Sage, TeX 

C_9^2\rtimes_{12}C_6
 
% in TeX
 
G:=Group("C9^2:12C6");
 
// GroupNames label
 
G:=SmallGroup(486,159);
 
// by ID
 
G=gap.SmallGroup(486,159);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,548,338,4755,2169,453,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^2>;
 
// generators/relations
 

Export

Character table of C92⋊12C6 in TeX

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