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G = C92⋊4C6  order 486 = 2·35

4th semidirect product of C92 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C92⋊4C6, C9⋊D9⋊4C3, C92⋊4C3⋊C2, C32⋊C9.15S3, C33.11(C3⋊S3), C3.7(He3.4S3), (C3×C9).32(C3×S3), C32.45(C3×C3⋊S3), SmallGroup(486,155)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C92 — C92⋊4C6
Chief series
C1 — C3 — C32 — C3×C9 — C92 — C92⋊4C3 — C92⋊4C6
Lower central
C92 — C92⋊4C6
Upper central
C1

Generators and relations for C92⋊4C6
 G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a-1b3, cbc-1=a3b2 >

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⋊4C3, C92⋊4C6
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C3×C3⋊S3, He3.4S3, C92⋊4C6

Character table of C92⋊4C6

 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
ρ1960-36-3-3000000000003ζ97+3ζ923ζ95+3ζ943ζ98+3ζ90000000000    orthogonal lifted from He3.4S3
ρ2060-3-36-3000003ζ98+3ζ93ζ95+3ζ943ζ97+3ζ920000000000000000    orthogonal lifted from He3.4S3
ρ2160-36-3-3000000000003ζ95+3ζ943ζ98+3ζ93ζ97+3ζ920000000000    orthogonal lifted from He3.4S3
ρ22606-3-3-3000000003ζ98+3ζ93ζ95+3ζ943ζ97+3ζ920000000000000    orthogonal lifted from He3.4S3
ρ2360-3-36-3000003ζ97+3ζ923ζ98+3ζ93ζ95+3ζ940000000000000000    orthogonal lifted from He3.4S3
ρ24606-3-3-3000000003ζ97+3ζ923ζ98+3ζ93ζ95+3ζ940000000000000    orthogonal lifted from He3.4S3
ρ2560-3-3-3600003ζ98+3ζ90000000003ζ97+3ζ923ζ95+3ζ9400000000    orthogonal lifted from He3.4S3
ρ2660-3-3-3600003ζ95+3ζ940000000003ζ98+3ζ93ζ97+3ζ9200000000    orthogonal lifted from He3.4S3
ρ27606-3-3-3000000003ζ95+3ζ943ζ97+3ζ923ζ98+3ζ90000000000000    orthogonal lifted from He3.4S3
ρ2860-3-36-3000003ζ95+3ζ943ζ97+3ζ923ζ98+3ζ90000000000000000    orthogonal lifted from He3.4S3
ρ2960-3-3-3600003ζ97+3ζ920000000003ζ95+3ζ943ζ98+3ζ900000000    orthogonal lifted from He3.4S3
ρ3060-36-3-3000000000003ζ98+3ζ93ζ97+3ζ923ζ95+3ζ940000000000    orthogonal lifted from He3.4S3

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

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

Matrix representation of C92⋊4C6 ►in GL12(𝔽19)

1020000000000
12120000000000
11115700000000
32121700000000
1420057000000
1611001217000000
000000372100
000000371200
000000310271818
00000012172710
00000091317500
000000101317500
,
1203000000000
702100000000
807010000000
1603001000000
603000000000
217000000000
0000001770000
0000001250000
000000535700
000000311121700
0000001480057
000000816001217
,
100000000000
1180000000000
120001818000000
500001000000
000100000000
001000000000
00000014170000
0000001250000
0000001713001412
00000051000175
00000091417500
0000001597200

G:=sub<GL(12,GF(19))| [10,12,11,3,14,16,0,0,0,0,0,0,2,12,11,2,2,11,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,0,0,7,17,0,0,0,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,0,0,7,17,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,12,9,10,0,0,0,0,0,0,7,7,10,17,13,13,0,0,0,0,0,0,2,1,2,2,17,17,0,0,0,0,0,0,1,2,7,7,5,5,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,18,0,0,0],[12,7,8,16,6,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,3,2,7,3,3,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,17,12,5,3,14,8,0,0,0,0,0,0,7,5,3,11,8,16,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,0,0,7,17,0,0,0,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,0,0,0,0,7,17],[1,1,12,5,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,14,12,17,5,9,15,0,0,0,0,0,0,17,5,13,10,14,9,0,0,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,14,17,0,0,0,0,0,0,0,0,0,0,12,5,0,0] >;
 

C92⋊4C6 in GAP, Magma, Sage, TeX 

C_9^2\rtimes_4C_6
 
% in TeX
 
G:=Group("C9^2:4C6");
 
// GroupNames label
 
G:=SmallGroup(486,155);
 
// by ID
 
G=gap.SmallGroup(486,155);
 
# by ID
 
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,1520,338,4755,2817,453,3244,11669]);
 
// Polycyclic
 
G:=Group<a,b,c|a^9=b^9=c^6=1,a*b=b*a,c*a*c^-1=a^-1*b^3,c*b*c^-1=a^3*b^2>;
 
// generators/relations
 

Export

Character table of C92⋊4C6 in TeX

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