direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D11×C22, C22⋊C22, C112⋊2C22, C11⋊(C2×C22), (C11×C22)⋊1C2, SmallGroup(484,10)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — D11×C22 |
Generators and relations for D11×C22
G = < a,b,c | a22=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D11×C22
►On 44 points
Generators in S44
(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)
(1 5 9 13 17 21 3 7 11 15 19)(2 6 10 14 18 22 4 8 12 16 20)(23 41 37 33 29 25 43 39 35 31 27)(24 42 38 34 30 26 44 40 36 32 28)
(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 23)(19 24)(20 25)(21 26)(22 27)
G:=sub<Sym(44)| (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), (1,5,9,13,17,21,3,7,11,15,19)(2,6,10,14,18,22,4,8,12,16,20)(23,41,37,33,29,25,43,39,35,31,27)(24,42,38,34,30,26,44,40,36,32,28), (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)(18,23)(19,24)(20,25)(21,26)(22,27)>;
G:=Group( (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), (1,5,9,13,17,21,3,7,11,15,19)(2,6,10,14,18,22,4,8,12,16,20)(23,41,37,33,29,25,43,39,35,31,27)(24,42,38,34,30,26,44,40,36,32,28), (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)(18,23)(19,24)(20,25)(21,26)(22,27) );
G=PermutationGroup([[(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)], [(1,5,9,13,17,21,3,7,11,15,19),(2,6,10,14,18,22,4,8,12,16,20),(23,41,37,33,29,25,43,39,35,31,27),(24,42,38,34,30,26,44,40,36,32,28)], [(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),(18,23),(19,24),(20,25),(21,26),(22,27)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 11A | ··· | 11J | 11K | ··· | 11BM | 22A | ··· | 22J | 22K | ··· | 22BM | 22BN | ··· | 22CG |
| order | 1 | 2 | 2 | 2 | 11 | ··· | 11 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 22 | ··· | 22 |
| size | 1 | 1 | 11 | 11 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 11 | ··· | 11 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | D11 | D22 | C11×D11 | D11×C22 |
| kernel | D11×C22 | C11×D11 | C11×C22 | D22 | D11 | C22 | C22 | C11 | C2 | C1 |
| # reps | 1 | 2 | 1 | 10 | 20 | 10 | 5 | 5 | 50 | 50 |
Matrix representation of D11×C22 ►in GL2(𝔽23) generated by
| 19 | 0 |
| 0 | 19 |
| 3 | 0 |
| 0 | 8 |
| 0 | 8 |
| 3 | 0 |
G:=sub<GL(2,GF(23))| [19,0,0,19],[3,0,0,8],[0,3,8,0] >;
D11×C22 in GAP, Magma, Sage, TeX
D_{11}\times C_{22} % in TeX
G:=Group("D11xC22"); // GroupNames label
G:=SmallGroup(484,10);
// by ID
G=gap.SmallGroup(484,10);
# by ID
G:=PCGroup([4,-2,-2,-11,-11,7043]);
// Polycyclic
G:=Group<a,b,c|a^22=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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