direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C6×D11, C22⋊C6, C66⋊2C2, C33⋊3C22, C11⋊(C2×C6), SmallGroup(132,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C6×D11 |
Generators and relations for C6×D11
G = < a,b,c | a6=b11=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C6×D11
►On 66 points
Generators in S66
(1 54 32 43 21 65)(2 55 33 44 22 66)(3 45 23 34 12 56)(4 46 24 35 13 57)(5 47 25 36 14 58)(6 48 26 37 15 59)(7 49 27 38 16 60)(8 50 28 39 17 61)(9 51 29 40 18 62)(10 52 30 41 19 63)(11 53 31 42 20 64)
(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)(45 46 47 48 49 50 51 52 53 54 55)(56 57 58 59 60 61 62 63 64 65 66)
(1 42)(2 41)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 44)(11 43)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 55)(20 54)(21 53)(22 52)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 56)(30 66)(31 65)(32 64)(33 63)
G:=sub<Sym(66)| (1,54,32,43,21,65)(2,55,33,44,22,66)(3,45,23,34,12,56)(4,46,24,35,13,57)(5,47,25,36,14,58)(6,48,26,37,15,59)(7,49,27,38,16,60)(8,50,28,39,17,61)(9,51,29,40,18,62)(10,52,30,41,19,63)(11,53,31,42,20,64), (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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,44)(11,43)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,55)(20,54)(21,53)(22,52)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,66)(31,65)(32,64)(33,63)>;
G:=Group( (1,54,32,43,21,65)(2,55,33,44,22,66)(3,45,23,34,12,56)(4,46,24,35,13,57)(5,47,25,36,14,58)(6,48,26,37,15,59)(7,49,27,38,16,60)(8,50,28,39,17,61)(9,51,29,40,18,62)(10,52,30,41,19,63)(11,53,31,42,20,64), (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)(45,46,47,48,49,50,51,52,53,54,55)(56,57,58,59,60,61,62,63,64,65,66), (1,42)(2,41)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,34)(10,44)(11,43)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,55)(20,54)(21,53)(22,52)(23,62)(24,61)(25,60)(26,59)(27,58)(28,57)(29,56)(30,66)(31,65)(32,64)(33,63) );
G=PermutationGroup([[(1,54,32,43,21,65),(2,55,33,44,22,66),(3,45,23,34,12,56),(4,46,24,35,13,57),(5,47,25,36,14,58),(6,48,26,37,15,59),(7,49,27,38,16,60),(8,50,28,39,17,61),(9,51,29,40,18,62),(10,52,30,41,19,63),(11,53,31,42,20,64)], [(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),(45,46,47,48,49,50,51,52,53,54,55),(56,57,58,59,60,61,62,63,64,65,66)], [(1,42),(2,41),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,34),(10,44),(11,43),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,55),(20,54),(21,53),(22,52),(23,62),(24,61),(25,60),(26,59),(27,58),(28,57),(29,56),(30,66),(31,65),(32,64),(33,63)]])
C6×D11 is a maximal subgroup of
C33⋊D4 C3⋊D44
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 11A | ··· | 11E | 22A | ··· | 22E | 33A | ··· | 33J | 66A | ··· | 66J |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 11 | 11 | 1 | 1 | 1 | 1 | 11 | 11 | 11 | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D11 | D22 | C3×D11 | C6×D11 |
| kernel | C6×D11 | C3×D11 | C66 | D22 | D11 | C22 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 5 | 5 | 10 | 10 |
Matrix representation of C6×D11 ►in GL2(𝔽43) generated by
| 7 | 0 |
| 0 | 7 |
| 42 | 24 |
| 28 | 15 |
| 28 | 35 |
| 28 | 15 |
G:=sub<GL(2,GF(43))| [7,0,0,7],[42,28,24,15],[28,28,35,15] >;
C6×D11 in GAP, Magma, Sage, TeX
C_6\times D_{11} % in TeX
G:=Group("C6xD11"); // GroupNames label
G:=SmallGroup(132,7);
// by ID
G=gap.SmallGroup(132,7);
# by ID
G:=PCGroup([4,-2,-2,-3,-11,1923]);
// Polycyclic
G:=Group<a,b,c|a^6=b^11=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export