metabelian, supersoluble, monomial
Aliases: C92⋊3C6, C9⋊D9⋊3C3, (C3×C9)⋊7D9, C9.4(C3×D9), C92⋊3C3⋊3C2, C32⋊C9.21S3, (C32×C9).19S3, C32.11(C9⋊S3), C33.26(C3⋊S3), C3.1(He3.4S3), C3.3(C3×C9⋊S3), (C3×C9).29(C3×S3), C32.32(C3×C3⋊S3), SmallGroup(486,141)
Series: Derived ►Chief ►Lower central ►Upper central
| C92 — C92⋊3C6 |
Generators and relations for C92⋊3C6
G = < a,b,c | a9=b9=c6=1, ab=ba, cac-1=a-1b3, cbc-1=b-1 >
Subgroups: 656 in 81 conjugacy classes, 26 normal (11 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C9, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, C3×C9, C33, C3×D9, C9⋊S3, C3×C3⋊S3, C92, C92, C32⋊C9, C9⋊C9, C32×C9, C32⋊D9, C9⋊D9, C3×C9⋊S3, C92⋊3C3, C92⋊3C6
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3⋊S3, C3×D9, C9⋊S3, C3×C3⋊S3, C3×C9⋊S3, He3.4S3, C92⋊3C6
►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 47 36 43 14 22 74 57 65)(2 48 28 44 15 23 75 58 66)(3 49 29 45 16 24 76 59 67)(4 50 30 37 17 25 77 60 68)(5 51 31 38 18 26 78 61 69)(6 52 32 39 10 27 79 62 70)(7 53 33 40 11 19 80 63 71)(8 54 34 41 12 20 81 55 72)(9 46 35 42 13 21 73 56 64)
(2 73 75 9 44 42)(3 41 45 8 76 81)(4 7)(5 79 78 6 38 39)(10 69 62 26 52 31)(11 25)(12 29 54 24 55 67)(13 66 56 23 46 28)(14 22)(15 35 48 21 58 64)(16 72 59 20 49 34)(17 19)(18 32 51 27 61 70)(30 63)(33 60)(36 57)(37 80)(40 77)(43 74)(47 65)(50 71)(53 68)
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), (1,47,36,43,14,22,74,57,65)(2,48,28,44,15,23,75,58,66)(3,49,29,45,16,24,76,59,67)(4,50,30,37,17,25,77,60,68)(5,51,31,38,18,26,78,61,69)(6,52,32,39,10,27,79,62,70)(7,53,33,40,11,19,80,63,71)(8,54,34,41,12,20,81,55,72)(9,46,35,42,13,21,73,56,64), (2,73,75,9,44,42)(3,41,45,8,76,81)(4,7)(5,79,78,6,38,39)(10,69,62,26,52,31)(11,25)(12,29,54,24,55,67)(13,66,56,23,46,28)(14,22)(15,35,48,21,58,64)(16,72,59,20,49,34)(17,19)(18,32,51,27,61,70)(30,63)(33,60)(36,57)(37,80)(40,77)(43,74)(47,65)(50,71)(53,68)>;
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), (1,47,36,43,14,22,74,57,65)(2,48,28,44,15,23,75,58,66)(3,49,29,45,16,24,76,59,67)(4,50,30,37,17,25,77,60,68)(5,51,31,38,18,26,78,61,69)(6,52,32,39,10,27,79,62,70)(7,53,33,40,11,19,80,63,71)(8,54,34,41,12,20,81,55,72)(9,46,35,42,13,21,73,56,64), (2,73,75,9,44,42)(3,41,45,8,76,81)(4,7)(5,79,78,6,38,39)(10,69,62,26,52,31)(11,25)(12,29,54,24,55,67)(13,66,56,23,46,28)(14,22)(15,35,48,21,58,64)(16,72,59,20,49,34)(17,19)(18,32,51,27,61,70)(30,63)(33,60)(36,57)(37,80)(40,77)(43,74)(47,65)(50,71)(53,68) );
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)], [(1,47,36,43,14,22,74,57,65),(2,48,28,44,15,23,75,58,66),(3,49,29,45,16,24,76,59,67),(4,50,30,37,17,25,77,60,68),(5,51,31,38,18,26,78,61,69),(6,52,32,39,10,27,79,62,70),(7,53,33,40,11,19,80,63,71),(8,54,34,41,12,20,81,55,72),(9,46,35,42,13,21,73,56,64)], [(2,73,75,9,44,42),(3,41,45,8,76,81),(4,7),(5,79,78,6,38,39),(10,69,62,26,52,31),(11,25),(12,29,54,24,55,67),(13,66,56,23,46,28),(14,22),(15,35,48,21,58,64),(16,72,59,20,49,34),(17,19),(18,32,51,27,61,70),(30,63),(33,60),(36,57),(37,80),(40,77),(43,74),(47,65),(50,71),(53,68)]])
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | ··· | 9I | 9J | ··· | 9AP |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 81 | 81 | 2 | ··· | 2 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 |
| type | + | + | + | + | + | + | ||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | D9 | C3×S3 | C3×D9 | He3.4S3 |
| kernel | C92⋊3C6 | C92⋊3C3 | C9⋊D9 | C92 | C32⋊C9 | C32×C9 | C3×C9 | C3×C9 | C9 | C3 |
| # reps | 1 | 1 | 2 | 2 | 3 | 1 | 9 | 8 | 18 | 9 |
Matrix representation of C92⋊3C6 ►in GL8(𝔽19)
| 10 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 17 |
| 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 8 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
G:=sub<GL(8,GF(19))| [10,12,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,7,17,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,7,17,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,7,17],[11,11,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0] >;
C92⋊3C6 in GAP, Magma, Sage, TeX
C_9^2\rtimes_3C_6
% in TeX
G:=Group("C9^2:3C6"); // GroupNames label
G:=SmallGroup(486,141);
// by ID
G=gap.SmallGroup(486,141);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,1520,338,4755,453,3244,11669]);
// 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=b^-1>;
// generators/relations