p-group, metabelian, nilpotent (class 3), monomial
Aliases: 3- 1+2⋊C9, C32.22He3, C33.22C32, C32.23- 1+2, C3.3C3≀C3, C32⋊C9.7C3, C32.2(C3×C9), (C32×C9).5C3, C3.8(C32⋊C9), C3.3(He3.C3), C3.2(C3.He3), (C3×3- 1+2).3C3, SmallGroup(243,18)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for 3- 1+2⋊C9
G = < a,b,c | a9=b3=c9=1, bab-1=a4, cac-1=ab-1, cbc-1=a6b >
Smallest permutation representation of 3- 1+2⋊C9
►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 38 33)(2 45 28)(3 43 32)(4 41 36)(5 39 31)(6 37 35)(7 44 30)(8 42 34)(9 40 29)(10 73 25)(11 80 20)(12 78 24)(13 76 19)(14 74 23)(15 81 27)(16 79 22)(17 77 26)(18 75 21)(46 58 66)(47 56 70)(48 63 65)(49 61 69)(50 59 64)(51 57 68)(52 55 72)(53 62 67)(54 60 71)
(1 78 51 38 27 63 33 18 71)(2 25 72 39 16 52 34 76 55)(3 11 59 40 80 67 35 20 47)(4 81 54 41 21 57 36 12 65)(5 19 66 42 10 46 28 79 58)(6 14 62 43 74 70 29 23 50)(7 75 48 44 24 60 30 15 68)(8 22 69 45 13 49 31 73 61)(9 17 56 37 77 64 32 26 53)
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), (1,38,33)(2,45,28)(3,43,32)(4,41,36)(5,39,31)(6,37,35)(7,44,30)(8,42,34)(9,40,29)(10,73,25)(11,80,20)(12,78,24)(13,76,19)(14,74,23)(15,81,27)(16,79,22)(17,77,26)(18,75,21)(46,58,66)(47,56,70)(48,63,65)(49,61,69)(50,59,64)(51,57,68)(52,55,72)(53,62,67)(54,60,71), (1,78,51,38,27,63,33,18,71)(2,25,72,39,16,52,34,76,55)(3,11,59,40,80,67,35,20,47)(4,81,54,41,21,57,36,12,65)(5,19,66,42,10,46,28,79,58)(6,14,62,43,74,70,29,23,50)(7,75,48,44,24,60,30,15,68)(8,22,69,45,13,49,31,73,61)(9,17,56,37,77,64,32,26,53)>;
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), (1,38,33)(2,45,28)(3,43,32)(4,41,36)(5,39,31)(6,37,35)(7,44,30)(8,42,34)(9,40,29)(10,73,25)(11,80,20)(12,78,24)(13,76,19)(14,74,23)(15,81,27)(16,79,22)(17,77,26)(18,75,21)(46,58,66)(47,56,70)(48,63,65)(49,61,69)(50,59,64)(51,57,68)(52,55,72)(53,62,67)(54,60,71), (1,78,51,38,27,63,33,18,71)(2,25,72,39,16,52,34,76,55)(3,11,59,40,80,67,35,20,47)(4,81,54,41,21,57,36,12,65)(5,19,66,42,10,46,28,79,58)(6,14,62,43,74,70,29,23,50)(7,75,48,44,24,60,30,15,68)(8,22,69,45,13,49,31,73,61)(9,17,56,37,77,64,32,26,53) );
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)], [(1,38,33),(2,45,28),(3,43,32),(4,41,36),(5,39,31),(6,37,35),(7,44,30),(8,42,34),(9,40,29),(10,73,25),(11,80,20),(12,78,24),(13,76,19),(14,74,23),(15,81,27),(16,79,22),(17,77,26),(18,75,21),(46,58,66),(47,56,70),(48,63,65),(49,61,69),(50,59,64),(51,57,68),(52,55,72),(53,62,67),(54,60,71)], [(1,78,51,38,27,63,33,18,71),(2,25,72,39,16,52,34,76,55),(3,11,59,40,80,67,35,20,47),(4,81,54,41,21,57,36,12,65),(5,19,66,42,10,46,28,79,58),(6,14,62,43,74,70,29,23,50),(7,75,48,44,24,60,30,15,68),(8,22,69,45,13,49,31,73,61),(9,17,56,37,77,64,32,26,53)]])
3- 1+2⋊C9 is a maximal subgroup of
3- 1+2⋊D9
51 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3N | 9A | ··· | 9R | 9S | ··· | 9AJ |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 |
| type | + | |||||||||
| image | C1 | C3 | C3 | C3 | C9 | He3 | 3- 1+2 | C3≀C3 | He3.C3 | C3.He3 |
| kernel | 3- 1+2⋊C9 | C32⋊C9 | C32×C9 | C3×3- 1+2 | 3- 1+2 | C32 | C32 | C3 | C3 | C3 |
| # reps | 1 | 4 | 2 | 2 | 18 | 2 | 4 | 6 | 6 | 6 |
Matrix representation of 3- 1+2⋊C9 ►in GL6(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 18 | 18 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 10 | 14 | 16 | 0 | 0 | 0 |
| 0 | 12 | 14 | 0 | 0 | 0 |
| 17 | 17 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(19))| [1,0,18,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,7,0],[7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11],[10,0,17,0,0,0,14,12,17,0,0,0,16,14,16,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;
3- 1+2⋊C9 in GAP, Magma, Sage, TeX
3_-^{1+2}\rtimes C_9 % in TeX
G:=Group("ES-(3,1):C9"); // GroupNames label
G:=SmallGroup(243,18);
// by ID
G=gap.SmallGroup(243,18);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,276,1352,457]);
// Polycyclic
G:=Group<a,b,c|a^9=b^3=c^9=1,b*a*b^-1=a^4,c*a*c^-1=a*b^-1,c*b*c^-1=a^6*b>;
// generators/relations
Export