direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: F5×C21, C5⋊C84, C35⋊7C12, C15⋊2C28, C105⋊4C4, D5.C42, (C7×D5).4C6, (C3×D5).2C14, (D5×C21).4C2, SmallGroup(420,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — F5×C21 |
Generators and relations for F5×C21
G = < a,b,c | a21=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of F5×C21
►On 105 points
Generators in S105
(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)
(1 51 23 83 103)(2 52 24 84 104)(3 53 25 64 105)(4 54 26 65 85)(5 55 27 66 86)(6 56 28 67 87)(7 57 29 68 88)(8 58 30 69 89)(9 59 31 70 90)(10 60 32 71 91)(11 61 33 72 92)(12 62 34 73 93)(13 63 35 74 94)(14 43 36 75 95)(15 44 37 76 96)(16 45 38 77 97)(17 46 39 78 98)(18 47 40 79 99)(19 48 41 80 100)(20 49 42 81 101)(21 50 22 82 102)
(22 102 82 50)(23 103 83 51)(24 104 84 52)(25 105 64 53)(26 85 65 54)(27 86 66 55)(28 87 67 56)(29 88 68 57)(30 89 69 58)(31 90 70 59)(32 91 71 60)(33 92 72 61)(34 93 73 62)(35 94 74 63)(36 95 75 43)(37 96 76 44)(38 97 77 45)(39 98 78 46)(40 99 79 47)(41 100 80 48)(42 101 81 49)
G:=sub<Sym(105)| (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), (1,51,23,83,103)(2,52,24,84,104)(3,53,25,64,105)(4,54,26,65,85)(5,55,27,66,86)(6,56,28,67,87)(7,57,29,68,88)(8,58,30,69,89)(9,59,31,70,90)(10,60,32,71,91)(11,61,33,72,92)(12,62,34,73,93)(13,63,35,74,94)(14,43,36,75,95)(15,44,37,76,96)(16,45,38,77,97)(17,46,39,78,98)(18,47,40,79,99)(19,48,41,80,100)(20,49,42,81,101)(21,50,22,82,102), (22,102,82,50)(23,103,83,51)(24,104,84,52)(25,105,64,53)(26,85,65,54)(27,86,66,55)(28,87,67,56)(29,88,68,57)(30,89,69,58)(31,90,70,59)(32,91,71,60)(33,92,72,61)(34,93,73,62)(35,94,74,63)(36,95,75,43)(37,96,76,44)(38,97,77,45)(39,98,78,46)(40,99,79,47)(41,100,80,48)(42,101,81,49)>;
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), (1,51,23,83,103)(2,52,24,84,104)(3,53,25,64,105)(4,54,26,65,85)(5,55,27,66,86)(6,56,28,67,87)(7,57,29,68,88)(8,58,30,69,89)(9,59,31,70,90)(10,60,32,71,91)(11,61,33,72,92)(12,62,34,73,93)(13,63,35,74,94)(14,43,36,75,95)(15,44,37,76,96)(16,45,38,77,97)(17,46,39,78,98)(18,47,40,79,99)(19,48,41,80,100)(20,49,42,81,101)(21,50,22,82,102), (22,102,82,50)(23,103,83,51)(24,104,84,52)(25,105,64,53)(26,85,65,54)(27,86,66,55)(28,87,67,56)(29,88,68,57)(30,89,69,58)(31,90,70,59)(32,91,71,60)(33,92,72,61)(34,93,73,62)(35,94,74,63)(36,95,75,43)(37,96,76,44)(38,97,77,45)(39,98,78,46)(40,99,79,47)(41,100,80,48)(42,101,81,49) );
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)], [(1,51,23,83,103),(2,52,24,84,104),(3,53,25,64,105),(4,54,26,65,85),(5,55,27,66,86),(6,56,28,67,87),(7,57,29,68,88),(8,58,30,69,89),(9,59,31,70,90),(10,60,32,71,91),(11,61,33,72,92),(12,62,34,73,93),(13,63,35,74,94),(14,43,36,75,95),(15,44,37,76,96),(16,45,38,77,97),(17,46,39,78,98),(18,47,40,79,99),(19,48,41,80,100),(20,49,42,81,101),(21,50,22,82,102)], [(22,102,82,50),(23,103,83,51),(24,104,84,52),(25,105,64,53),(26,85,65,54),(27,86,66,55),(28,87,67,56),(29,88,68,57),(30,89,69,58),(31,90,70,59),(32,91,71,60),(33,92,72,61),(34,93,73,62),(35,94,74,63),(36,95,75,43),(37,96,76,44),(38,97,77,45),(39,98,78,46),(40,99,79,47),(41,100,80,48),(42,101,81,49)]])
105 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 7A | ··· | 7F | 12A | 12B | 12C | 12D | 14A | ··· | 14F | 15A | 15B | 21A | ··· | 21L | 28A | ··· | 28L | 35A | ··· | 35F | 42A | ··· | 42L | 84A | ··· | 84X | 105A | ··· | 105L |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | 6 | 7 | ··· | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 15 | 15 | 21 | ··· | 21 | 28 | ··· | 28 | 35 | ··· | 35 | 42 | ··· | 42 | 84 | ··· | 84 | 105 | ··· | 105 |
| size | 1 | 5 | 1 | 1 | 5 | 5 | 4 | 5 | 5 | 1 | ··· | 1 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 4 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | ··· | 4 | 5 | ··· | 5 | 5 | ··· | 5 | 4 | ··· | 4 |
105 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C7 | C12 | C14 | C21 | C28 | C42 | C84 | F5 | C3×F5 | C7×F5 | F5×C21 |
| kernel | F5×C21 | D5×C21 | C7×F5 | C105 | C7×D5 | C3×F5 | C35 | C3×D5 | F5 | C15 | D5 | C5 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 6 | 12 | 12 | 12 | 24 | 1 | 2 | 6 | 12 |
Matrix representation of F5×C21 ►in GL4(𝔽421) generated by
| 229 | 0 | 0 | 0 |
| 0 | 229 | 0 | 0 |
| 0 | 0 | 229 | 0 |
| 0 | 0 | 0 | 229 |
| 420 | 420 | 420 | 420 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 420 | 420 | 420 | 420 |
G:=sub<GL(4,GF(421))| [229,0,0,0,0,229,0,0,0,0,229,0,0,0,0,229],[420,1,0,0,420,0,1,0,420,0,0,1,420,0,0,0],[1,0,0,420,0,0,1,420,0,0,0,420,0,1,0,420] >;
F5×C21 in GAP, Magma, Sage, TeX
F_5\times C_{21} % in TeX
G:=Group("F5xC21"); // GroupNames label
G:=SmallGroup(420,20);
// by ID
G=gap.SmallGroup(420,20);
# by ID
G:=PCGroup([5,-2,-3,-7,-2,-5,210,4204,219]);
// Polycyclic
G:=Group<a,b,c|a^21=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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