metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5⋊Dic21, C21⋊1F5, C105⋊1C4, D5.D21, C15⋊1Dic7, C35⋊1Dic3, C3⋊(C7⋊F5), C7⋊(C3⋊F5), (C3×D5).1D7, (C7×D5).1S3, (D5×C21).1C2, SmallGroup(420,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C105 — C5⋊Dic21 |
Generators and relations for C5⋊Dic21
G = < a,b,c | a5=b42=1, c2=b21, bab-1=a-1, cac-1=a3, cbc-1=b-1 >
Smallest permutation representation of C5⋊Dic21
►On 105 points
Generators in S105
(1 96 61 40 75)(2 76 41 62 97)(3 98 63 42 77)(4 78 43 22 99)(5 100 23 44 79)(6 80 45 24 101)(7 102 25 46 81)(8 82 47 26 103)(9 104 27 48 83)(10 84 49 28 105)(11 64 29 50 85)(12 86 51 30 65)(13 66 31 52 87)(14 88 53 32 67)(15 68 33 54 89)(16 90 55 34 69)(17 70 35 56 91)(18 92 57 36 71)(19 72 37 58 93)(20 94 59 38 73)(21 74 39 60 95)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 21)(17 20)(18 19)(22 86 43 65)(23 85 44 64)(24 84 45 105)(25 83 46 104)(26 82 47 103)(27 81 48 102)(28 80 49 101)(29 79 50 100)(30 78 51 99)(31 77 52 98)(32 76 53 97)(33 75 54 96)(34 74 55 95)(35 73 56 94)(36 72 57 93)(37 71 58 92)(38 70 59 91)(39 69 60 90)(40 68 61 89)(41 67 62 88)(42 66 63 87)
G:=sub<Sym(105)| (1,96,61,40,75)(2,76,41,62,97)(3,98,63,42,77)(4,78,43,22,99)(5,100,23,44,79)(6,80,45,24,101)(7,102,25,46,81)(8,82,47,26,103)(9,104,27,48,83)(10,84,49,28,105)(11,64,29,50,85)(12,86,51,30,65)(13,66,31,52,87)(14,88,53,32,67)(15,68,33,54,89)(16,90,55,34,69)(17,70,35,56,91)(18,92,57,36,71)(19,72,37,58,93)(20,94,59,38,73)(21,74,39,60,95), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,21)(17,20)(18,19)(22,86,43,65)(23,85,44,64)(24,84,45,105)(25,83,46,104)(26,82,47,103)(27,81,48,102)(28,80,49,101)(29,79,50,100)(30,78,51,99)(31,77,52,98)(32,76,53,97)(33,75,54,96)(34,74,55,95)(35,73,56,94)(36,72,57,93)(37,71,58,92)(38,70,59,91)(39,69,60,90)(40,68,61,89)(41,67,62,88)(42,66,63,87)>;
G:=Group( (1,96,61,40,75)(2,76,41,62,97)(3,98,63,42,77)(4,78,43,22,99)(5,100,23,44,79)(6,80,45,24,101)(7,102,25,46,81)(8,82,47,26,103)(9,104,27,48,83)(10,84,49,28,105)(11,64,29,50,85)(12,86,51,30,65)(13,66,31,52,87)(14,88,53,32,67)(15,68,33,54,89)(16,90,55,34,69)(17,70,35,56,91)(18,92,57,36,71)(19,72,37,58,93)(20,94,59,38,73)(21,74,39,60,95), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,21)(17,20)(18,19)(22,86,43,65)(23,85,44,64)(24,84,45,105)(25,83,46,104)(26,82,47,103)(27,81,48,102)(28,80,49,101)(29,79,50,100)(30,78,51,99)(31,77,52,98)(32,76,53,97)(33,75,54,96)(34,74,55,95)(35,73,56,94)(36,72,57,93)(37,71,58,92)(38,70,59,91)(39,69,60,90)(40,68,61,89)(41,67,62,88)(42,66,63,87) );
G=PermutationGroup([[(1,96,61,40,75),(2,76,41,62,97),(3,98,63,42,77),(4,78,43,22,99),(5,100,23,44,79),(6,80,45,24,101),(7,102,25,46,81),(8,82,47,26,103),(9,104,27,48,83),(10,84,49,28,105),(11,64,29,50,85),(12,86,51,30,65),(13,66,31,52,87),(14,88,53,32,67),(15,68,33,54,89),(16,90,55,34,69),(17,70,35,56,91),(18,92,57,36,71),(19,72,37,58,93),(20,94,59,38,73),(21,74,39,60,95)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,21),(17,20),(18,19),(22,86,43,65),(23,85,44,64),(24,84,45,105),(25,83,46,104),(26,82,47,103),(27,81,48,102),(28,80,49,101),(29,79,50,100),(30,78,51,99),(31,77,52,98),(32,76,53,97),(33,75,54,96),(34,74,55,95),(35,73,56,94),(36,72,57,93),(37,71,58,92),(38,70,59,91),(39,69,60,90),(40,68,61,89),(41,67,62,88),(42,66,63,87)]])
45 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5 | 6 | 7A | 7B | 7C | 14A | 14B | 14C | 15A | 15B | 21A | ··· | 21F | 35A | ··· | 35F | 42A | ··· | 42F | 105A | ··· | 105L |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | 7 | 7 | 14 | 14 | 14 | 15 | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 42 | ··· | 42 | 105 | ··· | 105 |
| size | 1 | 5 | 2 | 105 | 105 | 4 | 10 | 2 | 2 | 2 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | + | - | + | ||||
| image | C1 | C2 | C4 | S3 | Dic3 | D7 | Dic7 | D21 | Dic21 | F5 | C3⋊F5 | C7⋊F5 | C5⋊Dic21 |
| kernel | C5⋊Dic21 | D5×C21 | C105 | C7×D5 | C35 | C3×D5 | C15 | D5 | C5 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 6 | 6 | 1 | 2 | 6 | 12 |
Matrix representation of C5⋊Dic21 ►in GL4(𝔽421) generated by
| 1 | 0 | 0 | 287 |
| 0 | 1 | 134 | 340 |
| 135 | 236 | 50 | 263 |
| 287 | 108 | 158 | 368 |
| 111 | 371 | 0 | 0 |
| 100 | 357 | 0 | 0 |
| 274 | 38 | 216 | 50 |
| 136 | 209 | 371 | 158 |
| 340 | 18 | 0 | 0 |
| 384 | 81 | 0 | 0 |
| 406 | 260 | 370 | 158 |
| 313 | 167 | 234 | 51 |
G:=sub<GL(4,GF(421))| [1,0,135,287,0,1,236,108,0,134,50,158,287,340,263,368],[111,100,274,136,371,357,38,209,0,0,216,371,0,0,50,158],[340,384,406,313,18,81,260,167,0,0,370,234,0,0,158,51] >;
C5⋊Dic21 in GAP, Magma, Sage, TeX
C_5\rtimes {\rm Dic}_{21} % in TeX
G:=Group("C5:Dic21"); // GroupNames label
G:=SmallGroup(420,23);
// by ID
G=gap.SmallGroup(420,23);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,10,122,483,488,9004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^42=1,c^2=b^21,b*a*b^-1=a^-1,c*a*c^-1=a^3,c*b*c^-1=b^-1>;
// generators/relations
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