p-group, metacyclic, nilpotent (class 2), monomial
Aliases: C9⋊C9, C32.7C32, C3.23- 1+2, C3.2(C3×C9), (C3×C9).1C3, SmallGroup(81,4)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C9⋊C9
G = < a,b | a9=b9=1, bab-1=a7 >
Smallest permutation representation of C9⋊C9
►Regular action 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 78 51 33 22 66 39 13 57)(2 73 49 34 26 64 40 17 55)(3 77 47 35 21 71 41 12 62)(4 81 54 36 25 69 42 16 60)(5 76 52 28 20 67 43 11 58)(6 80 50 29 24 65 44 15 56)(7 75 48 30 19 72 45 10 63)(8 79 46 31 23 70 37 14 61)(9 74 53 32 27 68 38 18 59)
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), (1,78,51,33,22,66,39,13,57)(2,73,49,34,26,64,40,17,55)(3,77,47,35,21,71,41,12,62)(4,81,54,36,25,69,42,16,60)(5,76,52,28,20,67,43,11,58)(6,80,50,29,24,65,44,15,56)(7,75,48,30,19,72,45,10,63)(8,79,46,31,23,70,37,14,61)(9,74,53,32,27,68,38,18,59)>;
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), (1,78,51,33,22,66,39,13,57)(2,73,49,34,26,64,40,17,55)(3,77,47,35,21,71,41,12,62)(4,81,54,36,25,69,42,16,60)(5,76,52,28,20,67,43,11,58)(6,80,50,29,24,65,44,15,56)(7,75,48,30,19,72,45,10,63)(8,79,46,31,23,70,37,14,61)(9,74,53,32,27,68,38,18,59) );
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)], [(1,78,51,33,22,66,39,13,57),(2,73,49,34,26,64,40,17,55),(3,77,47,35,21,71,41,12,62),(4,81,54,36,25,69,42,16,60),(5,76,52,28,20,67,43,11,58),(6,80,50,29,24,65,44,15,56),(7,75,48,30,19,72,45,10,63),(8,79,46,31,23,70,37,14,61),(9,74,53,32,27,68,38,18,59)]])
C9⋊C9 is a maximal subgroup of
C9⋊C18 C27⋊C9 C32.He3 C32.5He3 C32.6He3 C92⋊3C3 C9×3- 1+2 C9⋊3- 1+2 C33.31C32 C92⋊7C3 C92⋊4C3 C92⋊5C3 C92⋊8C3 C92⋊9C3 C62.11C32 C62.12C32
C9⋊C9 is a maximal quotient of
C3.C92 C9⋊C27 C27⋊C9 C62.11C32 C62.12C32
33 conjugacy classes
| class | 1 | 3A | ··· | 3H | 9A | ··· | 9X |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 |
33 irreducible representations
| dim | 1 | 1 | 1 | 3 |
| type | + | |||
| image | C1 | C3 | C9 | 3- 1+2 |
| kernel | C9⋊C9 | C3×C9 | C9 | C3 |
| # reps | 1 | 8 | 18 | 6 |
Matrix representation of C9⋊C9 ►in GL4(𝔽19) generated by
| 7 | 0 | 0 | 0 |
| 0 | 11 | 9 | 0 |
| 0 | 0 | 8 | 1 |
| 0 | 7 | 12 | 0 |
| 16 | 0 | 0 | 0 |
| 0 | 13 | 12 | 4 |
| 0 | 1 | 13 | 14 |
| 0 | 13 | 4 | 12 |
G:=sub<GL(4,GF(19))| [7,0,0,0,0,11,0,7,0,9,8,12,0,0,1,0],[16,0,0,0,0,13,1,13,0,12,13,4,0,4,14,12] >;
C9⋊C9 in GAP, Magma, Sage, TeX
C_9\rtimes C_9
% in TeX
G:=Group("C9:C9"); // GroupNames label
G:=SmallGroup(81,4);
// by ID
G=gap.SmallGroup(81,4);
# by ID
G:=PCGroup([4,-3,3,-3,3,108,97,29]);
// Polycyclic
G:=Group<a,b|a^9=b^9=1,b*a*b^-1=a^7>;
// generators/relations
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