non-abelian, supersoluble, monomial
Aliases: C32⋊2D27, C33.2D9, (C3×C27)⋊4S3, (C3×C9).2D9, C32⋊C27⋊3C2, C3.3(C27⋊S3), (C32×C9).10S3, C32.15(C9⋊S3), C9.1(He3⋊C2), C3.2(C32⋊2D9), (C3×C9).18(C3⋊S3), SmallGroup(486,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C27 — C32⋊2D27 |
Generators and relations for C32⋊2D27
G = < a,b,c,d | a3=b3=c27=d2=1, ab=ba, cac-1=ab-1, dad=a-1, bc=cb, bd=db, dcd=c-1 >
Subgroups: 538 in 58 conjugacy classes, 18 normal (10 characteristic)
C1, C2, C3, C3, S3, C6, C9, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C27, C3×C9, C3×C9, C3×C9, C33, D27, C3×D9, C9⋊S3, C3×C3⋊S3, C3×C27, C32×C9, C3×D27, C3×C9⋊S3, C32⋊C27, C32⋊2D27
Quotients: C1, C2, S3, D9, C3⋊S3, D27, C9⋊S3, He3⋊C2, C32⋊2D9, C27⋊S3, C32⋊2D27
►On 54 points
Generators in S54
(1 19 10)(3 12 21)(4 22 13)(6 15 24)(7 25 16)(9 18 27)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(36 45 54)
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 37 46)(29 38 47)(30 39 48)(31 40 49)(32 41 50)(33 42 51)(34 43 52)(35 44 53)(36 45 54)
(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)
(1 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 54)(25 53)(26 52)(27 51)
G:=sub<Sym(54)| (1,19,10)(3,12,21)(4,22,13)(6,15,24)(7,25,16)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54), (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), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,54)(25,53)(26,52)(27,51)>;
G:=Group( (1,19,10)(3,12,21)(4,22,13)(6,15,24)(7,25,16)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,37,46)(29,38,47)(30,39,48)(31,40,49)(32,41,50)(33,42,51)(34,43,52)(35,44,53)(36,45,54), (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), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,54)(25,53)(26,52)(27,51) );
G=PermutationGroup([[(1,19,10),(3,12,21),(4,22,13),(6,15,24),(7,25,16),(9,18,27),(29,47,38),(30,39,48),(32,50,41),(33,42,51),(35,53,44),(36,45,54)], [(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,25,16),(8,26,17),(9,27,18),(28,37,46),(29,38,47),(30,39,48),(31,40,49),(32,41,50),(33,42,51),(34,43,52),(35,44,53),(36,45,54)], [(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)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,54),(25,53),(26,52),(27,51)]])
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | ··· | 9I | 9J | ··· | 9O | 27A | ··· | 27AA |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 27 | ··· | 27 |
| size | 1 | 81 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 81 | 81 | 2 | ··· | 2 | 6 | ··· | 6 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 |
| type | + | + | + | + | + | + | + | |||
| image | C1 | C2 | S3 | S3 | D9 | D9 | D27 | He3⋊C2 | C32⋊2D9 | C32⋊2D27 |
| kernel | C32⋊2D27 | C32⋊C27 | C3×C27 | C32×C9 | C3×C9 | C33 | C32 | C9 | C3 | C1 |
| # reps | 1 | 1 | 3 | 1 | 6 | 3 | 27 | 4 | 2 | 6 |
Matrix representation of C32⋊2D27 ►in GL5(𝔽109)
| 45 | 0 | 0 | 0 | 0 |
| 6 | 63 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 45 | 0 | 0 |
| 0 | 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 0 | 45 |
| 7 | 0 | 0 | 0 | 0 |
| 60 | 78 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 45 | 0 |
| 0 | 0 | 0 | 0 | 63 |
| 108 | 106 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 |
| 0 | 0 | 0 | 64 | 0 |
G:=sub<GL(5,GF(109))| [45,6,0,0,0,0,63,0,0,0,0,0,0,1,0,0,0,0,0,64,0,0,46,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,45,0,0,0,0,0,45,0,0,0,0,0,45],[7,60,0,0,0,0,78,0,0,0,0,0,1,0,0,0,0,0,45,0,0,0,0,0,63],[108,0,0,0,0,106,1,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,46,0] >;
C32⋊2D27 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2D_{27} % in TeX
G:=Group("C3^2:2D27"); // GroupNames label
G:=SmallGroup(486,51);
// by ID
G=gap.SmallGroup(486,51);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,1195,218,548,4755,453,11669]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^27=d^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations