direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×D35, C35⋊5C6, C15⋊2D7, C21⋊2D5, C105⋊2C2, C5⋊(C3×D7), C7⋊3(C3×D5), SmallGroup(210,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C3×D35 |
Generators and relations for C3×D35
G = < a,b,c | a3=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C3×D35
►On 105 points
Generators in S105
(1 91 47)(2 92 48)(3 93 49)(4 94 50)(5 95 51)(6 96 52)(7 97 53)(8 98 54)(9 99 55)(10 100 56)(11 101 57)(12 102 58)(13 103 59)(14 104 60)(15 105 61)(16 71 62)(17 72 63)(18 73 64)(19 74 65)(20 75 66)(21 76 67)(22 77 68)(23 78 69)(24 79 70)(25 80 36)(26 81 37)(27 82 38)(28 83 39)(29 84 40)(30 85 41)(31 86 42)(32 87 43)(33 88 44)(34 89 45)(35 90 46)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(58 70)(59 69)(60 68)(61 67)(62 66)(63 65)(71 75)(72 74)(76 105)(77 104)(78 103)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)
G:=sub<Sym(105)| (1,91,47)(2,92,48)(3,93,49)(4,94,50)(5,95,51)(6,96,52)(7,97,53)(8,98,54)(9,99,55)(10,100,56)(11,101,57)(12,102,58)(13,103,59)(14,104,60)(15,105,61)(16,71,62)(17,72,63)(18,73,64)(19,74,65)(20,75,66)(21,76,67)(22,77,68)(23,78,69)(24,79,70)(25,80,36)(26,81,37)(27,82,38)(28,83,39)(29,84,40)(30,85,41)(31,86,42)(32,87,43)(33,88,44)(34,89,45)(35,90,46), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,75)(72,74)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)>;
G:=Group( (1,91,47)(2,92,48)(3,93,49)(4,94,50)(5,95,51)(6,96,52)(7,97,53)(8,98,54)(9,99,55)(10,100,56)(11,101,57)(12,102,58)(13,103,59)(14,104,60)(15,105,61)(16,71,62)(17,72,63)(18,73,64)(19,74,65)(20,75,66)(21,76,67)(22,77,68)(23,78,69)(24,79,70)(25,80,36)(26,81,37)(27,82,38)(28,83,39)(29,84,40)(30,85,41)(31,86,42)(32,87,43)(33,88,44)(34,89,45)(35,90,46), (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), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)(58,70)(59,69)(60,68)(61,67)(62,66)(63,65)(71,75)(72,74)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91) );
G=PermutationGroup([[(1,91,47),(2,92,48),(3,93,49),(4,94,50),(5,95,51),(6,96,52),(7,97,53),(8,98,54),(9,99,55),(10,100,56),(11,101,57),(12,102,58),(13,103,59),(14,104,60),(15,105,61),(16,71,62),(17,72,63),(18,73,64),(19,74,65),(20,75,66),(21,76,67),(22,77,68),(23,78,69),(24,79,70),(25,80,36),(26,81,37),(27,82,38),(28,83,39),(29,84,40),(30,85,41),(31,86,42),(32,87,43),(33,88,44),(34,89,45),(35,90,46)], [(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)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(36,57),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47),(58,70),(59,69),(60,68),(61,67),(62,66),(63,65),(71,75),(72,74),(76,105),(77,104),(78,103),(79,102),(80,101),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91)]])
C3×D35 is a maximal subgroup of
C3×D5×D7 D15⋊D7
57 conjugacy classes
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 7A | 7B | 7C | 15A | 15B | 15C | 15D | 21A | ··· | 21F | 35A | ··· | 35L | 105A | ··· | 105X |
| order | 1 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 7 | 7 | 7 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 35 | ··· | 35 | 105 | ··· | 105 |
| size | 1 | 35 | 1 | 1 | 2 | 2 | 35 | 35 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | D5 | D7 | C3×D5 | C3×D7 | D35 | C3×D35 |
| kernel | C3×D35 | C105 | D35 | C35 | C21 | C15 | C7 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 3 | 4 | 6 | 12 | 24 |
Matrix representation of C3×D35 ►in GL2(𝔽211) generated by
| 196 | 0 |
| 0 | 196 |
| 127 | 186 |
| 50 | 148 |
| 187 | 137 |
| 176 | 24 |
G:=sub<GL(2,GF(211))| [196,0,0,196],[127,50,186,148],[187,176,137,24] >;
C3×D35 in GAP, Magma, Sage, TeX
C_3\times D_{35} % in TeX
G:=Group("C3xD35"); // GroupNames label
G:=SmallGroup(210,7);
// by ID
G=gap.SmallGroup(210,7);
# by ID
G:=PCGroup([4,-2,-3,-5,-7,290,2883]);
// Polycyclic
G:=Group<a,b,c|a^3=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export