direct product, metabelian, soluble, monomial, A-group
Aliases: A4×C22, C23⋊C33, C22⋊C66, (C22×C22)⋊C3, (C2×C22)⋊2C6, SmallGroup(264,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C22 — A4×C22 |
Generators and relations for A4×C22
G = < a,b,c,d | a22=b2=c2=d3=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >
Smallest permutation representation of A4×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 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 34)(24 35)(25 36)(26 37)(27 38)(28 39)(29 40)(30 41)(31 42)(32 43)(33 44)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(45 56)(46 57)(47 58)(48 59)(49 60)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)
(1 55 41)(2 56 42)(3 57 43)(4 58 44)(5 59 23)(6 60 24)(7 61 25)(8 62 26)(9 63 27)(10 64 28)(11 65 29)(12 66 30)(13 45 31)(14 46 32)(15 47 33)(16 48 34)(17 49 35)(18 50 36)(19 51 37)(20 52 38)(21 53 39)(22 54 40)
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), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66), (1,55,41)(2,56,42)(3,57,43)(4,58,44)(5,59,23)(6,60,24)(7,61,25)(8,62,26)(9,63,27)(10,64,28)(11,65,29)(12,66,30)(13,45,31)(14,46,32)(15,47,33)(16,48,34)(17,49,35)(18,50,36)(19,51,37)(20,52,38)(21,53,39)(22,54,40)>;
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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,34)(24,35)(25,36)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(32,43)(33,44), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(45,56)(46,57)(47,58)(48,59)(49,60)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66), (1,55,41)(2,56,42)(3,57,43)(4,58,44)(5,59,23)(6,60,24)(7,61,25)(8,62,26)(9,63,27)(10,64,28)(11,65,29)(12,66,30)(13,45,31)(14,46,32)(15,47,33)(16,48,34)(17,49,35)(18,50,36)(19,51,37)(20,52,38)(21,53,39)(22,54,40) );
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),(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(23,34),(24,35),(25,36),(26,37),(27,38),(28,39),(29,40),(30,41),(31,42),(32,43),(33,44)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(45,56),(46,57),(47,58),(48,59),(49,60),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66)], [(1,55,41),(2,56,42),(3,57,43),(4,58,44),(5,59,23),(6,60,24),(7,61,25),(8,62,26),(9,63,27),(10,64,28),(11,65,29),(12,66,30),(13,45,31),(14,46,32),(15,47,33),(16,48,34),(17,49,35),(18,50,36),(19,51,37),(20,52,38),(21,53,39),(22,54,40)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22AD | 33A | ··· | 33T | 66A | ··· | 66T |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C11 | C22 | C33 | C66 | A4 | C2×A4 | C11×A4 | A4×C22 |
| kernel | A4×C22 | C11×A4 | C22×C22 | C2×C22 | C2×A4 | A4 | C23 | C22 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 | 1 | 1 | 10 | 10 |
Matrix representation of A4×C22 ►in GL3(𝔽67) generated by
| 42 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 66 | 0 | 0 |
| 0 | 66 | 0 |
| 0 | 0 | 1 |
| 66 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 66 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(67))| [42,0,0,0,42,0,0,0,42],[66,0,0,0,66,0,0,0,1],[66,0,0,0,1,0,0,0,66],[0,0,1,1,0,0,0,1,0] >;
A4×C22 in GAP, Magma, Sage, TeX
A_4\times C_{22} % in TeX
G:=Group("A4xC22"); // GroupNames label
G:=SmallGroup(264,35);
// by ID
G=gap.SmallGroup(264,35);
# by ID
G:=PCGroup([5,-2,-3,-11,-2,2,1328,2484]);
// Polycyclic
G:=Group<a,b,c,d|a^22=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
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