direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C11, C4⋊C22, C44⋊3C2, C22⋊C22, C22.6C22, (C2×C22)⋊1C2, C2.1(C2×C22), SmallGroup(88,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C11
G = < a,b,c | a11=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C11
►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 38 27 20)(2 39 28 21)(3 40 29 22)(4 41 30 12)(5 42 31 13)(6 43 32 14)(7 44 33 15)(8 34 23 16)(9 35 24 17)(10 36 25 18)(11 37 26 19)
(12 41)(13 42)(14 43)(15 44)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)
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), (1,38,27,20)(2,39,28,21)(3,40,29,22)(4,41,30,12)(5,42,31,13)(6,43,32,14)(7,44,33,15)(8,34,23,16)(9,35,24,17)(10,36,25,18)(11,37,26,19), (12,41)(13,42)(14,43)(15,44)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)>;
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), (1,38,27,20)(2,39,28,21)(3,40,29,22)(4,41,30,12)(5,42,31,13)(6,43,32,14)(7,44,33,15)(8,34,23,16)(9,35,24,17)(10,36,25,18)(11,37,26,19), (12,41)(13,42)(14,43)(15,44)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40) );
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)], [(1,38,27,20),(2,39,28,21),(3,40,29,22),(4,41,30,12),(5,42,31,13),(6,43,32,14),(7,44,33,15),(8,34,23,16),(9,35,24,17),(10,36,25,18),(11,37,26,19)], [(12,41),(13,42),(14,43),(15,44),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40)]])
D4×C11 is a maximal subgroup of
D4⋊D11 D4.D11 D4⋊2D11
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22AD | 44A | ··· | 44J |
| order | 1 | 2 | 2 | 2 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | D4 | D4×C11 |
| kernel | D4×C11 | C44 | C2×C22 | D4 | C4 | C22 | C11 | C1 |
| # reps | 1 | 1 | 2 | 10 | 10 | 20 | 1 | 10 |
Matrix representation of D4×C11 ►in GL2(𝔽23) generated by
| 18 | 0 |
| 0 | 18 |
| 0 | 8 |
| 20 | 0 |
| 0 | 15 |
| 20 | 0 |
G:=sub<GL(2,GF(23))| [18,0,0,18],[0,20,8,0],[0,20,15,0] >;
D4×C11 in GAP, Magma, Sage, TeX
D_4\times C_{11} % in TeX
G:=Group("D4xC11"); // GroupNames label
G:=SmallGroup(88,9);
// by ID
G=gap.SmallGroup(88,9);
# by ID
G:=PCGroup([4,-2,-2,-11,-2,369]);
// Polycyclic
G:=Group<a,b,c|a^11=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export