direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C22, C6⋊C22, C66⋊3C2, C33⋊4C22, C3⋊(C2×C22), SmallGroup(132,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C22 |
Generators and relations for S3×C22
G = < a,b,c | a22=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C22
►On 66 points
Generators in S66
(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 47 24)(2 48 25)(3 49 26)(4 50 27)(5 51 28)(6 52 29)(7 53 30)(8 54 31)(9 55 32)(10 56 33)(11 57 34)(12 58 35)(13 59 36)(14 60 37)(15 61 38)(16 62 39)(17 63 40)(18 64 41)(19 65 42)(20 66 43)(21 45 44)(22 46 23)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)
G:=sub<Sym(66)| (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), (1,47,24)(2,48,25)(3,49,26)(4,50,27)(5,51,28)(6,52,29)(7,53,30)(8,54,31)(9,55,32)(10,56,33)(11,57,34)(12,58,35)(13,59,36)(14,60,37)(15,61,38)(16,62,39)(17,63,40)(18,64,41)(19,65,42)(20,66,43)(21,45,44)(22,46,23), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)>;
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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,47,24)(2,48,25)(3,49,26)(4,50,27)(5,51,28)(6,52,29)(7,53,30)(8,54,31)(9,55,32)(10,56,33)(11,57,34)(12,58,35)(13,59,36)(14,60,37)(15,61,38)(16,62,39)(17,63,40)(18,64,41)(19,65,42)(20,66,43)(21,45,44)(22,46,23), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56) );
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),(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)], [(1,47,24),(2,48,25),(3,49,26),(4,50,27),(5,51,28),(6,52,29),(7,53,30),(8,54,31),(9,55,32),(10,56,33),(11,57,34),(12,58,35),(13,59,36),(14,60,37),(15,61,38),(16,62,39),(17,63,40),(18,64,41),(19,65,42),(20,66,43),(21,45,44),(22,46,23)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(23,57),(24,58),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,55),(44,56)]])
S3×C22 is a maximal subgroup of
C33⋊D4 C11⋊D12
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 6 | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22AD | 33A | ··· | 33J | 66A | ··· | 66J |
| order | 1 | 2 | 2 | 2 | 3 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C11 | C22 | C22 | S3 | D6 | S3×C11 | S3×C22 |
| kernel | S3×C22 | S3×C11 | C66 | D6 | S3 | C6 | C22 | C11 | C2 | C1 |
| # reps | 1 | 2 | 1 | 10 | 20 | 10 | 1 | 1 | 10 | 10 |
Matrix representation of S3×C22 ►in GL2(𝔽23) generated by
| 7 | 0 |
| 0 | 7 |
| 22 | 10 |
| 16 | 0 |
| 22 | 10 |
| 0 | 1 |
G:=sub<GL(2,GF(23))| [7,0,0,7],[22,16,10,0],[22,0,10,1] >;
S3×C22 in GAP, Magma, Sage, TeX
S_3\times C_{22} % in TeX
G:=Group("S3xC22"); // GroupNames label
G:=SmallGroup(132,8);
// by ID
G=gap.SmallGroup(132,8);
# by ID
G:=PCGroup([4,-2,-2,-11,-3,1411]);
// Polycyclic
G:=Group<a,b,c|a^22=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export