direct product, abelian, monomial, 2-elementary
Aliases: C2×C20, SmallGroup(40,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C20 |
| C1 — C2×C20 |
| C1 — C2×C20 |
Generators and relations for C2×C20
G = < a,b | a2=b20=1, ab=ba >
Smallest permutation representation of C2×C20
►Regular action on 40 points
Generators in S40
(1 39)(2 40)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)
(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)
G:=sub<Sym(40)| (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38), (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)>;
G:=Group( (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38), (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) );
G=PermutationGroup([[(1,39),(2,40),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38)], [(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)]])
C2×C20 is a maximal subgroup of
C4.Dic5 C10.D4 C4⋊Dic5 D10⋊C4 C4○D20
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 |
| kernel | C2×C20 | C20 | C2×C10 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 |
Matrix representation of C2×C20 ►in GL2(𝔽41) generated by
| 1 | 0 |
| 0 | 40 |
| 21 | 0 |
| 0 | 5 |
G:=sub<GL(2,GF(41))| [1,0,0,40],[21,0,0,5] >;
C2×C20 in GAP, Magma, Sage, TeX
C_2\times C_{20} % in TeX
G:=Group("C2xC20"); // GroupNames label
G:=SmallGroup(40,9);
// by ID
G=gap.SmallGroup(40,9);
# by ID
G:=PCGroup([4,-2,-2,-5,-2,80]);
// Polycyclic
G:=Group<a,b|a^2=b^20=1,a*b=b*a>;
// generators/relations
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