direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C7⋊F5, C21⋊2F5, C105⋊2C4, C35⋊6C12, C15⋊2Dic7, C5⋊(C3×Dic7), C7⋊3(C3×F5), D5.(C3×D7), (C3×D5).2D7, (C7×D5).3C6, (D5×C21).2C2, SmallGroup(420,21)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C3×C7⋊F5 |
Generators and relations for C3×C7⋊F5
G = < a,b,c,d | a3=b7=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C3×C7⋊F5
►On 105 points
Generators in S105
(1 71 36)(2 72 37)(3 73 38)(4 74 39)(5 75 40)(6 76 41)(7 77 42)(8 78 43)(9 79 44)(10 80 45)(11 81 46)(12 82 47)(13 83 48)(14 84 49)(15 85 50)(16 86 51)(17 87 52)(18 88 53)(19 89 54)(20 90 55)(21 91 56)(22 92 57)(23 93 58)(24 94 59)(25 95 60)(26 96 61)(27 97 62)(28 98 63)(29 99 64)(30 100 65)(31 101 66)(32 102 67)(33 103 68)(34 104 69)(35 105 70)
(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 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)(36 64 57 50 43)(37 65 58 51 44)(38 66 59 52 45)(39 67 60 53 46)(40 68 61 54 47)(41 69 62 55 48)(42 70 63 56 49)(71 99 92 85 78)(72 100 93 86 79)(73 101 94 87 80)(74 102 95 88 81)(75 103 96 89 82)(76 104 97 90 83)(77 105 98 91 84)
(2 7)(3 6)(4 5)(8 15 29 22)(9 21 30 28)(10 20 31 27)(11 19 32 26)(12 18 33 25)(13 17 34 24)(14 16 35 23)(37 42)(38 41)(39 40)(43 50 64 57)(44 56 65 63)(45 55 66 62)(46 54 67 61)(47 53 68 60)(48 52 69 59)(49 51 70 58)(72 77)(73 76)(74 75)(78 85 99 92)(79 91 100 98)(80 90 101 97)(81 89 102 96)(82 88 103 95)(83 87 104 94)(84 86 105 93)
G:=sub<Sym(105)| (1,71,36)(2,72,37)(3,73,38)(4,74,39)(5,75,40)(6,76,41)(7,77,42)(8,78,43)(9,79,44)(10,80,45)(11,81,46)(12,82,47)(13,83,48)(14,84,49)(15,85,50)(16,86,51)(17,87,52)(18,88,53)(19,89,54)(20,90,55)(21,91,56)(22,92,57)(23,93,58)(24,94,59)(25,95,60)(26,96,61)(27,97,62)(28,98,63)(29,99,64)(30,100,65)(31,101,66)(32,102,67)(33,103,68)(34,104,69)(35,105,70), (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), (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14)(36,64,57,50,43)(37,65,58,51,44)(38,66,59,52,45)(39,67,60,53,46)(40,68,61,54,47)(41,69,62,55,48)(42,70,63,56,49)(71,99,92,85,78)(72,100,93,86,79)(73,101,94,87,80)(74,102,95,88,81)(75,103,96,89,82)(76,104,97,90,83)(77,105,98,91,84), (2,7)(3,6)(4,5)(8,15,29,22)(9,21,30,28)(10,20,31,27)(11,19,32,26)(12,18,33,25)(13,17,34,24)(14,16,35,23)(37,42)(38,41)(39,40)(43,50,64,57)(44,56,65,63)(45,55,66,62)(46,54,67,61)(47,53,68,60)(48,52,69,59)(49,51,70,58)(72,77)(73,76)(74,75)(78,85,99,92)(79,91,100,98)(80,90,101,97)(81,89,102,96)(82,88,103,95)(83,87,104,94)(84,86,105,93)>;
G:=Group( (1,71,36)(2,72,37)(3,73,38)(4,74,39)(5,75,40)(6,76,41)(7,77,42)(8,78,43)(9,79,44)(10,80,45)(11,81,46)(12,82,47)(13,83,48)(14,84,49)(15,85,50)(16,86,51)(17,87,52)(18,88,53)(19,89,54)(20,90,55)(21,91,56)(22,92,57)(23,93,58)(24,94,59)(25,95,60)(26,96,61)(27,97,62)(28,98,63)(29,99,64)(30,100,65)(31,101,66)(32,102,67)(33,103,68)(34,104,69)(35,105,70), (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), (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14)(36,64,57,50,43)(37,65,58,51,44)(38,66,59,52,45)(39,67,60,53,46)(40,68,61,54,47)(41,69,62,55,48)(42,70,63,56,49)(71,99,92,85,78)(72,100,93,86,79)(73,101,94,87,80)(74,102,95,88,81)(75,103,96,89,82)(76,104,97,90,83)(77,105,98,91,84), (2,7)(3,6)(4,5)(8,15,29,22)(9,21,30,28)(10,20,31,27)(11,19,32,26)(12,18,33,25)(13,17,34,24)(14,16,35,23)(37,42)(38,41)(39,40)(43,50,64,57)(44,56,65,63)(45,55,66,62)(46,54,67,61)(47,53,68,60)(48,52,69,59)(49,51,70,58)(72,77)(73,76)(74,75)(78,85,99,92)(79,91,100,98)(80,90,101,97)(81,89,102,96)(82,88,103,95)(83,87,104,94)(84,86,105,93) );
G=PermutationGroup([[(1,71,36),(2,72,37),(3,73,38),(4,74,39),(5,75,40),(6,76,41),(7,77,42),(8,78,43),(9,79,44),(10,80,45),(11,81,46),(12,82,47),(13,83,48),(14,84,49),(15,85,50),(16,86,51),(17,87,52),(18,88,53),(19,89,54),(20,90,55),(21,91,56),(22,92,57),(23,93,58),(24,94,59),(25,95,60),(26,96,61),(27,97,62),(28,98,63),(29,99,64),(30,100,65),(31,101,66),(32,102,67),(33,103,68),(34,104,69),(35,105,70)], [(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)], [(1,29,22,15,8),(2,30,23,16,9),(3,31,24,17,10),(4,32,25,18,11),(5,33,26,19,12),(6,34,27,20,13),(7,35,28,21,14),(36,64,57,50,43),(37,65,58,51,44),(38,66,59,52,45),(39,67,60,53,46),(40,68,61,54,47),(41,69,62,55,48),(42,70,63,56,49),(71,99,92,85,78),(72,100,93,86,79),(73,101,94,87,80),(74,102,95,88,81),(75,103,96,89,82),(76,104,97,90,83),(77,105,98,91,84)], [(2,7),(3,6),(4,5),(8,15,29,22),(9,21,30,28),(10,20,31,27),(11,19,32,26),(12,18,33,25),(13,17,34,24),(14,16,35,23),(37,42),(38,41),(39,40),(43,50,64,57),(44,56,65,63),(45,55,66,62),(46,54,67,61),(47,53,68,60),(48,52,69,59),(49,51,70,58),(72,77),(73,76),(74,75),(78,85,99,92),(79,91,100,98),(80,90,101,97),(81,89,102,96),(82,88,103,95),(83,87,104,94),(84,86,105,93)]])
51 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 15A | 15B | 21A | ··· | 21F | 35A | ··· | 35F | 42A | ··· | 42F | 105A | ··· | 105L |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 15 | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 42 | ··· | 42 | 105 | ··· | 105 |
| size | 1 | 5 | 1 | 1 | 35 | 35 | 4 | 5 | 5 | 2 | 2 | 2 | 35 | 35 | 35 | 35 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D7 | Dic7 | C3×D7 | C3×Dic7 | F5 | C3×F5 | C7⋊F5 | C3×C7⋊F5 |
| kernel | C3×C7⋊F5 | D5×C21 | C7⋊F5 | C105 | C7×D5 | C35 | C3×D5 | C15 | D5 | C5 | C21 | C7 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 3 | 3 | 6 | 6 | 1 | 2 | 6 | 12 |
Matrix representation of C3×C7⋊F5 ►in GL6(𝔽421)
| 20 | 0 | 0 | 0 | 0 | 0 |
| 0 | 20 | 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 |
| 116 | 420 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 420 | 0 | 0 |
| 0 | 0 | 1 | 403 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 420 |
| 0 | 0 | 0 | 0 | 1 | 403 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 369 | 263 | 1 | 0 |
| 0 | 0 | 158 | 51 | 0 | 1 |
| 0 | 0 | 420 | 0 | 0 | 0 |
| 0 | 0 | 0 | 420 | 0 | 0 |
| 392 | 0 | 0 | 0 | 0 | 0 |
| 4 | 29 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 403 | 420 | 18 |
| 0 | 0 | 0 | 420 | 0 | 1 |
| 0 | 0 | 0 | 0 | 369 | 252 |
| 0 | 0 | 0 | 0 | 158 | 52 |
G:=sub<GL(6,GF(421))| [20,0,0,0,0,0,0,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[116,1,0,0,0,0,420,0,0,0,0,0,0,0,0,1,0,0,0,0,420,403,0,0,0,0,0,0,0,1,0,0,0,0,420,403],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,369,158,420,0,0,0,263,51,0,420,0,0,1,0,0,0,0,0,0,1,0,0],[392,4,0,0,0,0,0,29,0,0,0,0,0,0,1,0,0,0,0,0,403,420,0,0,0,0,420,0,369,158,0,0,18,1,252,52] >;
C3×C7⋊F5 in GAP, Magma, Sage, TeX
C_3\times C_7\rtimes F_5
% in TeX
G:=Group("C3xC7:F5"); // GroupNames label
G:=SmallGroup(420,21);
// by ID
G=gap.SmallGroup(420,21);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,-7,30,723,173,9004]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^7=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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