direct product, metabelian, supersoluble, monomial
Aliases: S3×C27⋊C3, C33.2C18, (S3×C27)⋊C3, (S3×C9).C9, C27⋊2(C3×S3), (C3×C27)⋊7C6, C9.2(S3×C9), (C3×C9).2C18, (S3×C32).C9, C9.7(S3×C32), (S3×C9).3C32, (C32×C9).12C6, C32.10(S3×C9), C32.17(C3×C18), C3⋊(C2×C27⋊C3), (C3×C27⋊C3)⋊3C2, (S3×C3×C9).3C3, C3.10(S3×C3×C9), (C3×S3).5(C3×C9), (C3×C9).46(C3×S3), (C3×C9).38(C3×C6), SmallGroup(486,114)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C27⋊C3
G = < a,b,c,d | a3=b2=c27=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c10 >
Subgroups: 128 in 66 conjugacy classes, 33 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C9, C32, C32, C18, C3×S3, C3×S3, C3×C6, C27, C27, C3×C9, C3×C9, C3×C9, C33, C54, S3×C9, S3×C9, C3×C18, S3×C32, C3×C27, C27⋊C3, C27⋊C3, C32×C9, S3×C27, C2×C27⋊C3, S3×C3×C9, C3×C27⋊C3, S3×C27⋊C3
Quotients: C1, C2, C3, S3, C6, C9, C32, C18, C3×S3, C3×C6, C3×C9, S3×C9, C3×C18, S3×C32, C27⋊C3, C2×C27⋊C3, S3×C3×C9, S3×C27⋊C3
►On 54 points
(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 28)(21 29)(22 30)(23 31)(24 32)(25 33)(26 34)(27 35)
(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)
(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)
G:=sub<Sym(54)| (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,28)(21,29)(22,30)(23,31)(24,32)(25,33)(26,34)(27,35), (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), (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)>;
G:=Group( (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,28)(21,29)(22,30)(23,31)(24,32)(25,33)(26,34)(27,35), (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), (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) );
G=PermutationGroup([[(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,25),(8,17,26),(9,18,27),(28,46,37),(29,47,38),(30,48,39),(31,49,40),(32,50,41),(33,51,42),(34,52,43),(35,53,44),(36,54,45)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,28),(21,29),(22,30),(23,31),(24,32),(25,33),(26,34),(27,35)], [(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)], [(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)]])
99 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 6A | 6B | 6C | 6D | 9A | ··· | 9F | 9G | ··· | 9L | 9M | 9N | 9O | 9P | 9Q | 9R | 9S | 9T | 18A | ··· | 18F | 18G | 18H | 18I | 18J | 27A | ··· | 27R | 27S | ··· | 27AJ | 54A | ··· | 54R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 18 | 18 | 18 | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 3 | 3 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 3 | ··· | 3 | 9 | 9 | 9 | 9 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
99 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C9 | C18 | C18 | S3 | C3×S3 | C3×S3 | S3×C9 | S3×C9 | C27⋊C3 | C2×C27⋊C3 | S3×C27⋊C3 |
| kernel | S3×C27⋊C3 | C3×C27⋊C3 | S3×C27 | S3×C3×C9 | C3×C27 | C32×C9 | S3×C9 | S3×C32 | C3×C9 | C33 | C27⋊C3 | C27 | C3×C9 | C9 | C32 | S3 | C3 | C1 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 12 | 6 | 12 | 6 | 1 | 6 | 2 | 12 | 6 | 6 | 6 | 6 |
Matrix representation of S3×C27⋊C3 ►in GL5(𝔽109)
| 108 | 1 | 0 | 0 | 0 |
| 108 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 108 | 0 | 0 | 0 |
| 108 | 0 | 0 | 0 | 0 |
| 0 | 0 | 108 | 0 | 0 |
| 0 | 0 | 0 | 108 | 0 |
| 0 | 0 | 0 | 0 | 108 |
| 63 | 0 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 36 |
| 0 | 0 | 4 | 0 | 93 |
| 0 | 0 | 11 | 0 | 103 |
| 45 | 0 | 0 | 0 | 0 |
| 0 | 45 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 24 |
| 0 | 0 | 0 | 63 | 43 |
| 0 | 0 | 0 | 0 | 45 |
G:=sub<GL(5,GF(109))| [108,108,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,108,0,0,0,108,0,0,0,0,0,0,108,0,0,0,0,0,108,0,0,0,0,0,108],[63,0,0,0,0,0,63,0,0,0,0,0,6,4,11,0,0,1,0,0,0,0,36,93,103],[45,0,0,0,0,0,45,0,0,0,0,0,1,0,0,0,0,0,63,0,0,0,24,43,45] >;
S3×C27⋊C3 in GAP, Magma, Sage, TeX
S_3\times C_{27}\rtimes C_3 % in TeX
G:=Group("S3xC27:C3"); // GroupNames label
G:=SmallGroup(486,114);
// by ID
G=gap.SmallGroup(486,114);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,68,93,11669]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^27=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^10>;
// generators/relations