direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C3×C27⋊C3, C27⋊C32, C9.3C33, C33.3C9, (C3×C27)⋊4C3, (C3×C9).6C9, C9.4(C3×C9), C3.6(C32×C9), C32.17(C3×C9), (C32×C9).13C3, (C3×C9).27C32, SmallGroup(243,49)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C3×C27⋊C3
G = < a,b,c | a3=b27=c3=1, ab=ba, ac=ca, cbc-1=b10 >
Subgroups: 72 in 60 conjugacy classes, 54 normal (8 characteristic)
C1, C3, C3, C3, C9, C9, C32, C32, C32, C27, C3×C9, C3×C9, C33, C3×C27, C27⋊C3, C32×C9, C3×C27⋊C3
Quotients: C1, C3, C9, C32, C3×C9, C33, C27⋊C3, C32×C9, C3×C27⋊C3
►On 81 points
Generators in S81
(1 54 71)(2 28 72)(3 29 73)(4 30 74)(5 31 75)(6 32 76)(7 33 77)(8 34 78)(9 35 79)(10 36 80)(11 37 81)(12 38 55)(13 39 56)(14 40 57)(15 41 58)(16 42 59)(17 43 60)(18 44 61)(19 45 62)(20 46 63)(21 47 64)(22 48 65)(23 49 66)(24 50 67)(25 51 68)(26 52 69)(27 53 70)
(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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(28 46 37)(29 38 47)(31 49 40)(32 41 50)(34 52 43)(35 44 53)(55 64 73)(57 75 66)(58 67 76)(60 78 69)(61 70 79)(63 81 72)
G:=sub<Sym(81)| (1,54,71)(2,28,72)(3,29,73)(4,30,74)(5,31,75)(6,32,76)(7,33,77)(8,34,78)(9,35,79)(10,36,80)(11,37,81)(12,38,55)(13,39,56)(14,40,57)(15,41,58)(16,42,59)(17,43,60)(18,44,61)(19,45,62)(20,46,63)(21,47,64)(22,48,65)(23,49,66)(24,50,67)(25,51,68)(26,52,69)(27,53,70), (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(28,46,37)(29,38,47)(31,49,40)(32,41,50)(34,52,43)(35,44,53)(55,64,73)(57,75,66)(58,67,76)(60,78,69)(61,70,79)(63,81,72)>;
G:=Group( (1,54,71)(2,28,72)(3,29,73)(4,30,74)(5,31,75)(6,32,76)(7,33,77)(8,34,78)(9,35,79)(10,36,80)(11,37,81)(12,38,55)(13,39,56)(14,40,57)(15,41,58)(16,42,59)(17,43,60)(18,44,61)(19,45,62)(20,46,63)(21,47,64)(22,48,65)(23,49,66)(24,50,67)(25,51,68)(26,52,69)(27,53,70), (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(28,46,37)(29,38,47)(31,49,40)(32,41,50)(34,52,43)(35,44,53)(55,64,73)(57,75,66)(58,67,76)(60,78,69)(61,70,79)(63,81,72) );
G=PermutationGroup([[(1,54,71),(2,28,72),(3,29,73),(4,30,74),(5,31,75),(6,32,76),(7,33,77),(8,34,78),(9,35,79),(10,36,80),(11,37,81),(12,38,55),(13,39,56),(14,40,57),(15,41,58),(16,42,59),(17,43,60),(18,44,61),(19,45,62),(20,46,63),(21,47,64),(22,48,65),(23,49,66),(24,50,67),(25,51,68),(26,52,69),(27,53,70)], [(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)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27),(28,46,37),(29,38,47),(31,49,40),(32,41,50),(34,52,43),(35,44,53),(55,64,73),(57,75,66),(58,67,76),(60,78,69),(61,70,79),(63,81,72)]])
C3×C27⋊C3 is a maximal subgroup of
C33.5D9
99 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3N | 9A | ··· | 9R | 9S | ··· | 9AD | 27A | ··· | 27BB |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 3 | ··· | 3 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | C3 | C9 | C9 | C27⋊C3 |
| kernel | C3×C27⋊C3 | C3×C27 | C27⋊C3 | C32×C9 | C3×C9 | C33 | C3 |
| # reps | 1 | 6 | 18 | 2 | 48 | 6 | 18 |
Matrix representation of C3×C27⋊C3 ►in GL4(𝔽109) generated by
| 45 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 63 | 0 |
| 0 | 0 | 0 | 63 |
| 27 | 0 | 0 | 0 |
| 0 | 45 | 44 | 0 |
| 0 | 63 | 64 | 45 |
| 0 | 42 | 46 | 0 |
| 63 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 45 | 45 | 0 |
| 0 | 64 | 0 | 63 |
G:=sub<GL(4,GF(109))| [45,0,0,0,0,63,0,0,0,0,63,0,0,0,0,63],[27,0,0,0,0,45,63,42,0,44,64,46,0,0,45,0],[63,0,0,0,0,1,45,64,0,0,45,0,0,0,0,63] >;
C3×C27⋊C3 in GAP, Magma, Sage, TeX
C_3\times C_{27}\rtimes C_3 % in TeX
G:=Group("C3xC27:C3"); // GroupNames label
G:=SmallGroup(243,49);
// by ID
G=gap.SmallGroup(243,49);
# by ID
G:=PCGroup([5,-3,3,3,-3,-3,135,841,78]);
// Polycyclic
G:=Group<a,b,c|a^3=b^27=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations