direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C7×C3⋊F5, C21⋊3F5, C15⋊1C28, C105⋊3C4, C35⋊2Dic3, C3⋊(C7×F5), C5⋊(C7×Dic3), D5.(S3×C7), (C7×D5).2S3, (D5×C21).3C2, (C3×D5).1C14, SmallGroup(420,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C7×C3⋊F5 |
Generators and relations for C7×C3⋊F5
G = < a,b,c,d | a7=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C7×C3⋊F5
►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 54 94)(2 55 95)(3 56 96)(4 50 97)(5 51 98)(6 52 92)(7 53 93)(8 39 58)(9 40 59)(10 41 60)(11 42 61)(12 36 62)(13 37 63)(14 38 57)(15 85 65)(16 86 66)(17 87 67)(18 88 68)(19 89 69)(20 90 70)(21 91 64)(22 100 43)(23 101 44)(24 102 45)(25 103 46)(26 104 47)(27 105 48)(28 99 49)(29 75 79)(30 76 80)(31 77 81)(32 71 82)(33 72 83)(34 73 84)(35 74 78)
(1 70 79 102 59)(2 64 80 103 60)(3 65 81 104 61)(4 66 82 105 62)(5 67 83 99 63)(6 68 84 100 57)(7 69 78 101 58)(8 53 19 35 44)(9 54 20 29 45)(10 55 21 30 46)(11 56 15 31 47)(12 50 16 32 48)(13 51 17 33 49)(14 52 18 34 43)(22 38 92 88 73)(23 39 93 89 74)(24 40 94 90 75)(25 41 95 91 76)(26 42 96 85 77)(27 36 97 86 71)(28 37 98 87 72)
(8 23 19 74)(9 24 20 75)(10 25 21 76)(11 26 15 77)(12 27 16 71)(13 28 17 72)(14 22 18 73)(29 40 45 90)(30 41 46 91)(31 42 47 85)(32 36 48 86)(33 37 49 87)(34 38 43 88)(35 39 44 89)(50 97)(51 98)(52 92)(53 93)(54 94)(55 95)(56 96)(57 100 68 84)(58 101 69 78)(59 102 70 79)(60 103 64 80)(61 104 65 81)(62 105 66 82)(63 99 67 83)
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), (1,54,94)(2,55,95)(3,56,96)(4,50,97)(5,51,98)(6,52,92)(7,53,93)(8,39,58)(9,40,59)(10,41,60)(11,42,61)(12,36,62)(13,37,63)(14,38,57)(15,85,65)(16,86,66)(17,87,67)(18,88,68)(19,89,69)(20,90,70)(21,91,64)(22,100,43)(23,101,44)(24,102,45)(25,103,46)(26,104,47)(27,105,48)(28,99,49)(29,75,79)(30,76,80)(31,77,81)(32,71,82)(33,72,83)(34,73,84)(35,74,78), (1,70,79,102,59)(2,64,80,103,60)(3,65,81,104,61)(4,66,82,105,62)(5,67,83,99,63)(6,68,84,100,57)(7,69,78,101,58)(8,53,19,35,44)(9,54,20,29,45)(10,55,21,30,46)(11,56,15,31,47)(12,50,16,32,48)(13,51,17,33,49)(14,52,18,34,43)(22,38,92,88,73)(23,39,93,89,74)(24,40,94,90,75)(25,41,95,91,76)(26,42,96,85,77)(27,36,97,86,71)(28,37,98,87,72), (8,23,19,74)(9,24,20,75)(10,25,21,76)(11,26,15,77)(12,27,16,71)(13,28,17,72)(14,22,18,73)(29,40,45,90)(30,41,46,91)(31,42,47,85)(32,36,48,86)(33,37,49,87)(34,38,43,88)(35,39,44,89)(50,97)(51,98)(52,92)(53,93)(54,94)(55,95)(56,96)(57,100,68,84)(58,101,69,78)(59,102,70,79)(60,103,64,80)(61,104,65,81)(62,105,66,82)(63,99,67,83)>;
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), (1,54,94)(2,55,95)(3,56,96)(4,50,97)(5,51,98)(6,52,92)(7,53,93)(8,39,58)(9,40,59)(10,41,60)(11,42,61)(12,36,62)(13,37,63)(14,38,57)(15,85,65)(16,86,66)(17,87,67)(18,88,68)(19,89,69)(20,90,70)(21,91,64)(22,100,43)(23,101,44)(24,102,45)(25,103,46)(26,104,47)(27,105,48)(28,99,49)(29,75,79)(30,76,80)(31,77,81)(32,71,82)(33,72,83)(34,73,84)(35,74,78), (1,70,79,102,59)(2,64,80,103,60)(3,65,81,104,61)(4,66,82,105,62)(5,67,83,99,63)(6,68,84,100,57)(7,69,78,101,58)(8,53,19,35,44)(9,54,20,29,45)(10,55,21,30,46)(11,56,15,31,47)(12,50,16,32,48)(13,51,17,33,49)(14,52,18,34,43)(22,38,92,88,73)(23,39,93,89,74)(24,40,94,90,75)(25,41,95,91,76)(26,42,96,85,77)(27,36,97,86,71)(28,37,98,87,72), (8,23,19,74)(9,24,20,75)(10,25,21,76)(11,26,15,77)(12,27,16,71)(13,28,17,72)(14,22,18,73)(29,40,45,90)(30,41,46,91)(31,42,47,85)(32,36,48,86)(33,37,49,87)(34,38,43,88)(35,39,44,89)(50,97)(51,98)(52,92)(53,93)(54,94)(55,95)(56,96)(57,100,68,84)(58,101,69,78)(59,102,70,79)(60,103,64,80)(61,104,65,81)(62,105,66,82)(63,99,67,83) );
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)], [(1,54,94),(2,55,95),(3,56,96),(4,50,97),(5,51,98),(6,52,92),(7,53,93),(8,39,58),(9,40,59),(10,41,60),(11,42,61),(12,36,62),(13,37,63),(14,38,57),(15,85,65),(16,86,66),(17,87,67),(18,88,68),(19,89,69),(20,90,70),(21,91,64),(22,100,43),(23,101,44),(24,102,45),(25,103,46),(26,104,47),(27,105,48),(28,99,49),(29,75,79),(30,76,80),(31,77,81),(32,71,82),(33,72,83),(34,73,84),(35,74,78)], [(1,70,79,102,59),(2,64,80,103,60),(3,65,81,104,61),(4,66,82,105,62),(5,67,83,99,63),(6,68,84,100,57),(7,69,78,101,58),(8,53,19,35,44),(9,54,20,29,45),(10,55,21,30,46),(11,56,15,31,47),(12,50,16,32,48),(13,51,17,33,49),(14,52,18,34,43),(22,38,92,88,73),(23,39,93,89,74),(24,40,94,90,75),(25,41,95,91,76),(26,42,96,85,77),(27,36,97,86,71),(28,37,98,87,72)], [(8,23,19,74),(9,24,20,75),(10,25,21,76),(11,26,15,77),(12,27,16,71),(13,28,17,72),(14,22,18,73),(29,40,45,90),(30,41,46,91),(31,42,47,85),(32,36,48,86),(33,37,49,87),(34,38,43,88),(35,39,44,89),(50,97),(51,98),(52,92),(53,93),(54,94),(55,95),(56,96),(57,100,68,84),(58,101,69,78),(59,102,70,79),(60,103,64,80),(61,104,65,81),(62,105,66,82),(63,99,67,83)]])
63 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5 | 6 | 7A | ··· | 7F | 14A | ··· | 14F | 15A | 15B | 21A | ··· | 21F | 28A | ··· | 28L | 35A | ··· | 35F | 42A | ··· | 42F | 105A | ··· | 105L |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 6 | 7 | ··· | 7 | 14 | ··· | 14 | 15 | 15 | 21 | ··· | 21 | 28 | ··· | 28 | 35 | ··· | 35 | 42 | ··· | 42 | 105 | ··· | 105 |
| size | 1 | 5 | 2 | 15 | 15 | 4 | 10 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 2 | ··· | 2 | 15 | ··· | 15 | 4 | ··· | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C4 | C7 | C14 | C28 | S3 | Dic3 | S3×C7 | C7×Dic3 | F5 | C3⋊F5 | C7×F5 | C7×C3⋊F5 |
| kernel | C7×C3⋊F5 | D5×C21 | C105 | C3⋊F5 | C3×D5 | C15 | C7×D5 | C35 | D5 | C5 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 6 | 6 | 12 | 1 | 1 | 6 | 6 | 1 | 2 | 6 | 12 |
Matrix representation of C7×C3⋊F5 ►in GL6(𝔽421)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 75 | 0 | 0 | 0 |
| 0 | 0 | 0 | 75 | 0 | 0 |
| 0 | 0 | 0 | 0 | 75 | 0 |
| 0 | 0 | 0 | 0 | 0 | 75 |
| 100 | 169 | 0 | 0 | 0 | 0 |
| 5 | 320 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 420 | 420 | 420 | 420 |
| 18 | 220 | 0 | 0 | 0 | 0 |
| 276 | 403 | 0 | 0 | 0 | 0 |
| 0 | 0 | 189 | 0 | 377 | 377 |
| 0 | 0 | 377 | 377 | 0 | 189 |
| 0 | 0 | 44 | 233 | 44 | 0 |
| 0 | 0 | 232 | 188 | 188 | 232 |
G:=sub<GL(6,GF(421))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,75,0,0,0,0,0,0,75,0,0,0,0,0,0,75,0,0,0,0,0,0,75],[100,5,0,0,0,0,169,320,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,420,0,0,1,0,0,420,0,0,0,1,0,420,0,0,0,0,1,420],[18,276,0,0,0,0,220,403,0,0,0,0,0,0,189,377,44,232,0,0,0,377,233,188,0,0,377,0,44,188,0,0,377,189,0,232] >;
C7×C3⋊F5 in GAP, Magma, Sage, TeX
C_7\times C_3\rtimes F_5
% in TeX
G:=Group("C7xC3:F5"); // GroupNames label
G:=SmallGroup(420,22);
// by ID
G=gap.SmallGroup(420,22);
# by ID
G:=PCGroup([5,-2,-7,-2,-3,-5,70,1123,6304,614]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^3=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
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