direct product, abelian, monomial, 2-elementary
Aliases: C2×C22, SmallGroup(44,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C22 |
| C1 — C2×C22 |
| C1 — C2×C22 |
Generators and relations for C2×C22
G = < a,b | a2=b22=1, ab=ba >
Smallest permutation representation of C2×C22
►Regular action on 44 points
Generators in S44
(1 43)(2 44)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)
(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)
G:=sub<Sym(44)| (1,43)(2,44)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42), (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)>;
G:=Group( (1,43)(2,44)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42), (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) );
G=PermutationGroup([[(1,43),(2,44),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42)], [(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)]])
C2×C22 is a maximal subgroup of
C11⋊D4
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 11A | ··· | 11J | 22A | ··· | 22AD |
| order | 1 | 2 | 2 | 2 | 11 | ··· | 11 | 22 | ··· | 22 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C11 | C22 |
| kernel | C2×C22 | C22 | C22 | C2 |
| # reps | 1 | 3 | 10 | 30 |
Matrix representation of C2×C22 ►in GL2(𝔽23) generated by
| 22 | 0 |
| 0 | 1 |
| 10 | 0 |
| 0 | 7 |
G:=sub<GL(2,GF(23))| [22,0,0,1],[10,0,0,7] >;
C2×C22 in GAP, Magma, Sage, TeX
C_2\times C_{22} % in TeX
G:=Group("C2xC22"); // GroupNames label
G:=SmallGroup(44,4);
// by ID
G=gap.SmallGroup(44,4);
# by ID
G:=PCGroup([3,-2,-2,-11]);
// Polycyclic
G:=Group<a,b|a^2=b^22=1,a*b=b*a>;
// generators/relations
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