direct product, abelian, monomial, 2-elementary
Aliases: C2×C18, SmallGroup(36,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C18 |
| C1 — C2×C18 |
| C1 — C2×C18 |
Generators and relations for C2×C18
G = < a,b | a2=b18=1, ab=ba >
Smallest permutation representation of C2×C18
►Regular action on 36 points
Generators in S36
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)
(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)
G:=sub<Sym(36)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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)>;
G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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) );
G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26)], [(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)]])
C2×C18 is a maximal subgroup of
C9⋊D4 C9.A4 C9⋊A4
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | ··· | 6F | 9A | ··· | 9F | 18A | ··· | 18R |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C3 | C6 | C9 | C18 |
| kernel | C2×C18 | C18 | C2×C6 | C6 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 6 | 18 |
Matrix representation of C2×C18 ►in GL2(𝔽19) generated by
| 1 | 0 |
| 0 | 18 |
| 15 | 0 |
| 0 | 11 |
G:=sub<GL(2,GF(19))| [1,0,0,18],[15,0,0,11] >;
C2×C18 in GAP, Magma, Sage, TeX
C_2\times C_{18} % in TeX
G:=Group("C2xC18"); // GroupNames label
G:=SmallGroup(36,5);
// by ID
G=gap.SmallGroup(36,5);
# by ID
G:=PCGroup([4,-2,-2,-3,-3,46]);
// Polycyclic
G:=Group<a,b|a^2=b^18=1,a*b=b*a>;
// generators/relations
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