direct product, p-group, abelian, monomial
Aliases: C32×C9, SmallGroup(81,11)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C32×C9 |
| C1 — C32×C9 |
| C1 — C32×C9 |
Generators and relations for C32×C9
G = < a,b,c | a3=b3=c9=1, ab=ba, ac=ca, bc=cb >
Subgroups: 50, all normal (4 characteristic)
C1, C3, C3, C9, C32, C3×C9, C33, C32×C9
Quotients: C1, C3, C9, C32, C3×C9, C33, C32×C9
►Regular action on 81 points
Generators in S81
(1 38 17)(2 39 18)(3 40 10)(4 41 11)(5 42 12)(6 43 13)(7 44 14)(8 45 15)(9 37 16)(19 58 30)(20 59 31)(21 60 32)(22 61 33)(23 62 34)(24 63 35)(25 55 36)(26 56 28)(27 57 29)(46 64 79)(47 65 80)(48 66 81)(49 67 73)(50 68 74)(51 69 75)(52 70 76)(53 71 77)(54 72 78)
(1 31 47)(2 32 48)(3 33 49)(4 34 50)(5 35 51)(6 36 52)(7 28 53)(8 29 54)(9 30 46)(10 61 73)(11 62 74)(12 63 75)(13 55 76)(14 56 77)(15 57 78)(16 58 79)(17 59 80)(18 60 81)(19 64 37)(20 65 38)(21 66 39)(22 67 40)(23 68 41)(24 69 42)(25 70 43)(26 71 44)(27 72 45)
(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)
G:=sub<Sym(81)| (1,38,17)(2,39,18)(3,40,10)(4,41,11)(5,42,12)(6,43,13)(7,44,14)(8,45,15)(9,37,16)(19,58,30)(20,59,31)(21,60,32)(22,61,33)(23,62,34)(24,63,35)(25,55,36)(26,56,28)(27,57,29)(46,64,79)(47,65,80)(48,66,81)(49,67,73)(50,68,74)(51,69,75)(52,70,76)(53,71,77)(54,72,78), (1,31,47)(2,32,48)(3,33,49)(4,34,50)(5,35,51)(6,36,52)(7,28,53)(8,29,54)(9,30,46)(10,61,73)(11,62,74)(12,63,75)(13,55,76)(14,56,77)(15,57,78)(16,58,79)(17,59,80)(18,60,81)(19,64,37)(20,65,38)(21,66,39)(22,67,40)(23,68,41)(24,69,42)(25,70,43)(26,71,44)(27,72,45), (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)>;
G:=Group( (1,38,17)(2,39,18)(3,40,10)(4,41,11)(5,42,12)(6,43,13)(7,44,14)(8,45,15)(9,37,16)(19,58,30)(20,59,31)(21,60,32)(22,61,33)(23,62,34)(24,63,35)(25,55,36)(26,56,28)(27,57,29)(46,64,79)(47,65,80)(48,66,81)(49,67,73)(50,68,74)(51,69,75)(52,70,76)(53,71,77)(54,72,78), (1,31,47)(2,32,48)(3,33,49)(4,34,50)(5,35,51)(6,36,52)(7,28,53)(8,29,54)(9,30,46)(10,61,73)(11,62,74)(12,63,75)(13,55,76)(14,56,77)(15,57,78)(16,58,79)(17,59,80)(18,60,81)(19,64,37)(20,65,38)(21,66,39)(22,67,40)(23,68,41)(24,69,42)(25,70,43)(26,71,44)(27,72,45), (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) );
G=PermutationGroup([[(1,38,17),(2,39,18),(3,40,10),(4,41,11),(5,42,12),(6,43,13),(7,44,14),(8,45,15),(9,37,16),(19,58,30),(20,59,31),(21,60,32),(22,61,33),(23,62,34),(24,63,35),(25,55,36),(26,56,28),(27,57,29),(46,64,79),(47,65,80),(48,66,81),(49,67,73),(50,68,74),(51,69,75),(52,70,76),(53,71,77),(54,72,78)], [(1,31,47),(2,32,48),(3,33,49),(4,34,50),(5,35,51),(6,36,52),(7,28,53),(8,29,54),(9,30,46),(10,61,73),(11,62,74),(12,63,75),(13,55,76),(14,56,77),(15,57,78),(16,58,79),(17,59,80),(18,60,81),(19,64,37),(20,65,38),(21,66,39),(22,67,40),(23,68,41),(24,69,42),(25,70,43),(26,71,44),(27,72,45)], [(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)]])
C32×C9 is a maximal subgroup of
C32⋊4D9 C3.C92 C32⋊C27 C32.19He3 C32.20He3 C9.4He3 He3⋊C9 3- 1+2⋊C9 C92⋊3C3 C9⋊He3 C32.23C33 C9⋊3- 1+2 C33.31C32 C9.He3
C32×C9 is a maximal quotient of
C92⋊3C3 C27○He3
81 conjugacy classes
| class | 1 | 3A | ··· | 3Z | 9A | ··· | 9BB |
| order | 1 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C3 | C9 |
| kernel | C32×C9 | C3×C9 | C33 | C32 |
| # reps | 1 | 24 | 2 | 54 |
Matrix representation of C32×C9 ►in GL3(𝔽19) generated by
| 1 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 7 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 16 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 16 |
G:=sub<GL(3,GF(19))| [1,0,0,0,7,0,0,0,7],[7,0,0,0,11,0,0,0,11],[16,0,0,0,9,0,0,0,16] >;
C32×C9 in GAP, Magma, Sage, TeX
C_3^2\times C_9
% in TeX
G:=Group("C3^2xC9"); // GroupNames label
G:=SmallGroup(81,11);
// by ID
G=gap.SmallGroup(81,11);
# by ID
G:=PCGroup([4,-3,3,3,-3,108]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^9=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations