metabelian, supersoluble, monomial, A-group
Aliases: C9⋊D9, C92⋊2C2, (C3×C9).6S3, C3.1(C9⋊S3), C32.7(C3⋊S3), SmallGroup(162,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C92 — C9⋊D9 |
Generators and relations for C9⋊D9
G = < a,b,c | a9=b9=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Smallest permutation representation of C9⋊D9
►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 74 29 17 42 46 27 58 67)(2 75 30 18 43 47 19 59 68)(3 76 31 10 44 48 20 60 69)(4 77 32 11 45 49 21 61 70)(5 78 33 12 37 50 22 62 71)(6 79 34 13 38 51 23 63 72)(7 80 35 14 39 52 24 55 64)(8 81 36 15 40 53 25 56 65)(9 73 28 16 41 54 26 57 66)
(1 67)(2 66)(3 65)(4 64)(5 72)(6 71)(7 70)(8 69)(9 68)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 54)(19 28)(20 36)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(37 38)(39 45)(40 44)(41 43)(55 77)(56 76)(57 75)(58 74)(59 73)(60 81)(61 80)(62 79)(63 78)
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), (1,74,29,17,42,46,27,58,67)(2,75,30,18,43,47,19,59,68)(3,76,31,10,44,48,20,60,69)(4,77,32,11,45,49,21,61,70)(5,78,33,12,37,50,22,62,71)(6,79,34,13,38,51,23,63,72)(7,80,35,14,39,52,24,55,64)(8,81,36,15,40,53,25,56,65)(9,73,28,16,41,54,26,57,66), (1,67)(2,66)(3,65)(4,64)(5,72)(6,71)(7,70)(8,69)(9,68)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,54)(19,28)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(37,38)(39,45)(40,44)(41,43)(55,77)(56,76)(57,75)(58,74)(59,73)(60,81)(61,80)(62,79)(63,78)>;
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), (1,74,29,17,42,46,27,58,67)(2,75,30,18,43,47,19,59,68)(3,76,31,10,44,48,20,60,69)(4,77,32,11,45,49,21,61,70)(5,78,33,12,37,50,22,62,71)(6,79,34,13,38,51,23,63,72)(7,80,35,14,39,52,24,55,64)(8,81,36,15,40,53,25,56,65)(9,73,28,16,41,54,26,57,66), (1,67)(2,66)(3,65)(4,64)(5,72)(6,71)(7,70)(8,69)(9,68)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,54)(19,28)(20,36)(21,35)(22,34)(23,33)(24,32)(25,31)(26,30)(27,29)(37,38)(39,45)(40,44)(41,43)(55,77)(56,76)(57,75)(58,74)(59,73)(60,81)(61,80)(62,79)(63,78) );
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)], [(1,74,29,17,42,46,27,58,67),(2,75,30,18,43,47,19,59,68),(3,76,31,10,44,48,20,60,69),(4,77,32,11,45,49,21,61,70),(5,78,33,12,37,50,22,62,71),(6,79,34,13,38,51,23,63,72),(7,80,35,14,39,52,24,55,64),(8,81,36,15,40,53,25,56,65),(9,73,28,16,41,54,26,57,66)], [(1,67),(2,66),(3,65),(4,64),(5,72),(6,71),(7,70),(8,69),(9,68),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,54),(19,28),(20,36),(21,35),(22,34),(23,33),(24,32),(25,31),(26,30),(27,29),(37,38),(39,45),(40,44),(41,43),(55,77),(56,76),(57,75),(58,74),(59,73),(60,81),(61,80),(62,79),(63,78)]])
C9⋊D9 is a maximal subgroup of
C92⋊C4 D92 C92⋊C6 C92⋊2C6 C9⋊D27 C92⋊3C6 C92⋊9C6 C92⋊10C6 C92⋊4C6 C92⋊5C6 C92⋊11C6 C92⋊12C6 C92⋊8S3
C9⋊D9 is a maximal quotient of
C9⋊Dic9 C3.2(C9⋊D9) C9⋊D27 C92⋊8S3
42 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 9A | ··· | 9AJ |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 9 | ··· | 9 |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 2 | 2 |
| type | + | + | + | + |
| image | C1 | C2 | S3 | D9 |
| kernel | C9⋊D9 | C92 | C3×C9 | C9 |
| # reps | 1 | 1 | 4 | 36 |
Matrix representation of C9⋊D9 ►in GL4(𝔽19) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 7 |
| 0 | 0 | 12 | 17 |
| 5 | 7 | 0 | 0 |
| 12 | 17 | 0 | 0 |
| 0 | 0 | 12 | 17 |
| 0 | 0 | 2 | 14 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 2 |
| 0 | 0 | 14 | 12 |
G:=sub<GL(4,GF(19))| [1,0,0,0,0,1,0,0,0,0,5,12,0,0,7,17],[5,12,0,0,7,17,0,0,0,0,12,2,0,0,17,14],[0,1,0,0,1,0,0,0,0,0,7,14,0,0,2,12] >;
C9⋊D9 in GAP, Magma, Sage, TeX
C_9\rtimes D_9
% in TeX
G:=Group("C9:D9"); // GroupNames label
G:=SmallGroup(162,16);
// by ID
G=gap.SmallGroup(162,16);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,221,186,992,282,723,2704]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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