direct product, metacyclic, nilpotent (class 2), monomial, 3-elementary
Aliases: C2×C27⋊C3, C54⋊C3, C18.C9, C9.C18, C27⋊2C6, C32.C18, C18.2C32, (C3×C6).C9, C9.1(C3×C6), (C3×C9).4C6, C6.3(C3×C9), (C3×C18).3C3, C3.3(C3×C18), SmallGroup(162,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C27⋊C3
G = < a,b,c | a2=b27=c3=1, ab=ba, ac=ca, cbc-1=b10 >
Smallest permutation representation of C2×C27⋊C3
►On 54 points
Generators in S54
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 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)
(2 20 11)(3 12 21)(5 23 14)(6 15 24)(8 26 17)(9 18 27)(29 47 38)(30 39 48)(32 50 41)(33 42 51)(35 53 44)(36 45 54)
G:=sub<Sym(54)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54), (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54)>;
G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54), (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), (2,20,11)(3,12,21)(5,23,14)(6,15,24)(8,26,17)(9,18,27)(29,47,38)(30,39,48)(32,50,41)(33,42,51)(35,53,44)(36,45,54) );
G=PermutationGroup([[(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54)], [(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)], [(2,20,11),(3,12,21),(5,23,14),(6,15,24),(8,26,17),(9,18,27),(29,47,38),(30,39,48),(32,50,41),(33,42,51),(35,53,44),(36,45,54)]])
C2×C27⋊C3 is a maximal subgroup of
C27⋊C12
66 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 9A | ··· | 9F | 9G | 9H | 9I | 9J | 18A | ··· | 18F | 18G | 18H | 18I | 18J | 27A | ··· | 27R | 54A | ··· | 54R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 18 | 18 | 18 | 18 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 1 | 1 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | C9 | C9 | C18 | C18 | C27⋊C3 | C2×C27⋊C3 |
| kernel | C2×C27⋊C3 | C27⋊C3 | C54 | C3×C18 | C27 | C3×C9 | C18 | C3×C6 | C9 | C32 | C2 | C1 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 12 | 6 | 12 | 6 | 6 | 6 |
Matrix representation of C2×C27⋊C3 ►in GL3(𝔽109) generated by
| 108 | 0 | 0 |
| 0 | 108 | 0 |
| 0 | 0 | 108 |
| 0 | 1 | 0 |
| 0 | 0 | 63 |
| 27 | 0 | 0 |
| 1 | 0 | 0 |
| 0 | 63 | 0 |
| 0 | 0 | 45 |
G:=sub<GL(3,GF(109))| [108,0,0,0,108,0,0,0,108],[0,0,27,1,0,0,0,63,0],[1,0,0,0,63,0,0,0,45] >;
C2×C27⋊C3 in GAP, Magma, Sage, TeX
C_2\times C_{27}\rtimes C_3 % in TeX
G:=Group("C2xC27:C3"); // GroupNames label
G:=SmallGroup(162,27);
// by ID
G=gap.SmallGroup(162,27);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,96,457,78]);
// Polycyclic
G:=Group<a,b,c|a^2=b^27=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^10>;
// generators/relations
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