direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: F5×C22, C10⋊C44, D5⋊C44, C110⋊2C4, D10.C22, C5⋊(C2×C44), C55⋊3(C2×C4), D5.(C2×C22), (D5×C11)⋊3C4, (D5×C22).3C2, (D5×C11).3C22, SmallGroup(440,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — F5×C22 |
Generators and relations for F5×C22
G = < a,b,c | a22=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of F5×C22
►On 110 points
Generators in S110
(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)(67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110)
(1 81 58 90 35)(2 82 59 91 36)(3 83 60 92 37)(4 84 61 93 38)(5 85 62 94 39)(6 86 63 95 40)(7 87 64 96 41)(8 88 65 97 42)(9 67 66 98 43)(10 68 45 99 44)(11 69 46 100 23)(12 70 47 101 24)(13 71 48 102 25)(14 72 49 103 26)(15 73 50 104 27)(16 74 51 105 28)(17 75 52 106 29)(18 76 53 107 30)(19 77 54 108 31)(20 78 55 109 32)(21 79 56 110 33)(22 80 57 89 34)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(23 89 69 57)(24 90 70 58)(25 91 71 59)(26 92 72 60)(27 93 73 61)(28 94 74 62)(29 95 75 63)(30 96 76 64)(31 97 77 65)(32 98 78 66)(33 99 79 45)(34 100 80 46)(35 101 81 47)(36 102 82 48)(37 103 83 49)(38 104 84 50)(39 105 85 51)(40 106 86 52)(41 107 87 53)(42 108 88 54)(43 109 67 55)(44 110 68 56)
G:=sub<Sym(110)| (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)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110), (1,81,58,90,35)(2,82,59,91,36)(3,83,60,92,37)(4,84,61,93,38)(5,85,62,94,39)(6,86,63,95,40)(7,87,64,96,41)(8,88,65,97,42)(9,67,66,98,43)(10,68,45,99,44)(11,69,46,100,23)(12,70,47,101,24)(13,71,48,102,25)(14,72,49,103,26)(15,73,50,104,27)(16,74,51,105,28)(17,75,52,106,29)(18,76,53,107,30)(19,77,54,108,31)(20,78,55,109,32)(21,79,56,110,33)(22,80,57,89,34), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,89,69,57)(24,90,70,58)(25,91,71,59)(26,92,72,60)(27,93,73,61)(28,94,74,62)(29,95,75,63)(30,96,76,64)(31,97,77,65)(32,98,78,66)(33,99,79,45)(34,100,80,46)(35,101,81,47)(36,102,82,48)(37,103,83,49)(38,104,84,50)(39,105,85,51)(40,106,86,52)(41,107,87,53)(42,108,88,54)(43,109,67,55)(44,110,68,56)>;
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)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110), (1,81,58,90,35)(2,82,59,91,36)(3,83,60,92,37)(4,84,61,93,38)(5,85,62,94,39)(6,86,63,95,40)(7,87,64,96,41)(8,88,65,97,42)(9,67,66,98,43)(10,68,45,99,44)(11,69,46,100,23)(12,70,47,101,24)(13,71,48,102,25)(14,72,49,103,26)(15,73,50,104,27)(16,74,51,105,28)(17,75,52,106,29)(18,76,53,107,30)(19,77,54,108,31)(20,78,55,109,32)(21,79,56,110,33)(22,80,57,89,34), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(23,89,69,57)(24,90,70,58)(25,91,71,59)(26,92,72,60)(27,93,73,61)(28,94,74,62)(29,95,75,63)(30,96,76,64)(31,97,77,65)(32,98,78,66)(33,99,79,45)(34,100,80,46)(35,101,81,47)(36,102,82,48)(37,103,83,49)(38,104,84,50)(39,105,85,51)(40,106,86,52)(41,107,87,53)(42,108,88,54)(43,109,67,55)(44,110,68,56) );
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),(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110)], [(1,81,58,90,35),(2,82,59,91,36),(3,83,60,92,37),(4,84,61,93,38),(5,85,62,94,39),(6,86,63,95,40),(7,87,64,96,41),(8,88,65,97,42),(9,67,66,98,43),(10,68,45,99,44),(11,69,46,100,23),(12,70,47,101,24),(13,71,48,102,25),(14,72,49,103,26),(15,73,50,104,27),(16,74,51,105,28),(17,75,52,106,29),(18,76,53,107,30),(19,77,54,108,31),(20,78,55,109,32),(21,79,56,110,33),(22,80,57,89,34)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(23,89,69,57),(24,90,70,58),(25,91,71,59),(26,92,72,60),(27,93,73,61),(28,94,74,62),(29,95,75,63),(30,96,76,64),(31,97,77,65),(32,98,78,66),(33,99,79,45),(34,100,80,46),(35,101,81,47),(36,102,82,48),(37,103,83,49),(38,104,84,50),(39,105,85,51),(40,106,86,52),(41,107,87,53),(42,108,88,54),(43,109,67,55),(44,110,68,56)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 10 | 11A | ··· | 11J | 22A | ··· | 22J | 22K | ··· | 22AD | 44A | ··· | 44AN | 55A | ··· | 55J | 110A | ··· | 110J |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 5 | ··· | 5 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C11 | C22 | C22 | C44 | C44 | F5 | C2×F5 | C11×F5 | F5×C22 |
| kernel | F5×C22 | C11×F5 | D5×C22 | D5×C11 | C110 | C2×F5 | F5 | D10 | D5 | C10 | C22 | C11 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 10 | 20 | 10 | 20 | 20 | 1 | 1 | 10 | 10 |
Matrix representation of F5×C22 ►in GL5(𝔽661)
| 660 | 0 | 0 | 0 | 0 |
| 0 | 81 | 0 | 0 | 0 |
| 0 | 0 | 81 | 0 | 0 |
| 0 | 0 | 0 | 81 | 0 |
| 0 | 0 | 0 | 0 | 81 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 660 | 660 | 660 | 660 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 106 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 660 | 660 | 660 | 660 |
G:=sub<GL(5,GF(661))| [660,0,0,0,0,0,81,0,0,0,0,0,81,0,0,0,0,0,81,0,0,0,0,0,81],[1,0,0,0,0,0,660,1,0,0,0,660,0,1,0,0,660,0,0,1,0,660,0,0,0],[106,0,0,0,0,0,1,0,0,660,0,0,0,1,660,0,0,0,0,660,0,0,1,0,660] >;
F5×C22 in GAP, Magma, Sage, TeX
F_5\times C_{22} % in TeX
G:=Group("F5xC22"); // GroupNames label
G:=SmallGroup(440,45);
// by ID
G=gap.SmallGroup(440,45);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-5,220,4404,219]);
// Polycyclic
G:=Group<a,b,c|a^22=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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