metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C35⋊3D4, C7⋊2D20, Dic7⋊D5, D70⋊4C2, D10⋊2D7, C14.6D10, C10.6D14, C70.6C22, C5⋊1(C7⋊D4), C2.6(D5×D7), (D5×C14)⋊2C2, (C5×Dic7)⋊3C2, SmallGroup(280,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊D20
G = < a,b,c | a7=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C7⋊D20
►On 140 points
Generators in S140
(1 90 32 63 129 103 53)(2 54 104 130 64 33 91)(3 92 34 65 131 105 55)(4 56 106 132 66 35 93)(5 94 36 67 133 107 57)(6 58 108 134 68 37 95)(7 96 38 69 135 109 59)(8 60 110 136 70 39 97)(9 98 40 71 137 111 41)(10 42 112 138 72 21 99)(11 100 22 73 139 113 43)(12 44 114 140 74 23 81)(13 82 24 75 121 115 45)(14 46 116 122 76 25 83)(15 84 26 77 123 117 47)(16 48 118 124 78 27 85)(17 86 28 79 125 119 49)(18 50 120 126 80 29 87)(19 88 30 61 127 101 51)(20 52 102 128 62 31 89)
(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)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)
(1 5)(2 4)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(21 118)(22 117)(23 116)(24 115)(25 114)(26 113)(27 112)(28 111)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 120)(40 119)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 100)(48 99)(49 98)(50 97)(51 96)(52 95)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 135)(62 134)(63 133)(64 132)(65 131)(66 130)(67 129)(68 128)(69 127)(70 126)(71 125)(72 124)(73 123)(74 122)(75 121)(76 140)(77 139)(78 138)(79 137)(80 136)
G:=sub<Sym(140)| (1,90,32,63,129,103,53)(2,54,104,130,64,33,91)(3,92,34,65,131,105,55)(4,56,106,132,66,35,93)(5,94,36,67,133,107,57)(6,58,108,134,68,37,95)(7,96,38,69,135,109,59)(8,60,110,136,70,39,97)(9,98,40,71,137,111,41)(10,42,112,138,72,21,99)(11,100,22,73,139,113,43)(12,44,114,140,74,23,81)(13,82,24,75,121,115,45)(14,46,116,122,76,25,83)(15,84,26,77,123,117,47)(16,48,118,124,78,27,85)(17,86,28,79,125,119,49)(18,50,120,126,80,29,87)(19,88,30,61,127,101,51)(20,52,102,128,62,31,89), (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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,120)(40,119)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,135)(62,134)(63,133)(64,132)(65,131)(66,130)(67,129)(68,128)(69,127)(70,126)(71,125)(72,124)(73,123)(74,122)(75,121)(76,140)(77,139)(78,138)(79,137)(80,136)>;
G:=Group( (1,90,32,63,129,103,53)(2,54,104,130,64,33,91)(3,92,34,65,131,105,55)(4,56,106,132,66,35,93)(5,94,36,67,133,107,57)(6,58,108,134,68,37,95)(7,96,38,69,135,109,59)(8,60,110,136,70,39,97)(9,98,40,71,137,111,41)(10,42,112,138,72,21,99)(11,100,22,73,139,113,43)(12,44,114,140,74,23,81)(13,82,24,75,121,115,45)(14,46,116,122,76,25,83)(15,84,26,77,123,117,47)(16,48,118,124,78,27,85)(17,86,28,79,125,119,49)(18,50,120,126,80,29,87)(19,88,30,61,127,101,51)(20,52,102,128,62,31,89), (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)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140), (1,5)(2,4)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(21,118)(22,117)(23,116)(24,115)(25,114)(26,113)(27,112)(28,111)(29,110)(30,109)(31,108)(32,107)(33,106)(34,105)(35,104)(36,103)(37,102)(38,101)(39,120)(40,119)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,135)(62,134)(63,133)(64,132)(65,131)(66,130)(67,129)(68,128)(69,127)(70,126)(71,125)(72,124)(73,123)(74,122)(75,121)(76,140)(77,139)(78,138)(79,137)(80,136) );
G=PermutationGroup([[(1,90,32,63,129,103,53),(2,54,104,130,64,33,91),(3,92,34,65,131,105,55),(4,56,106,132,66,35,93),(5,94,36,67,133,107,57),(6,58,108,134,68,37,95),(7,96,38,69,135,109,59),(8,60,110,136,70,39,97),(9,98,40,71,137,111,41),(10,42,112,138,72,21,99),(11,100,22,73,139,113,43),(12,44,114,140,74,23,81),(13,82,24,75,121,115,45),(14,46,116,122,76,25,83),(15,84,26,77,123,117,47),(16,48,118,124,78,27,85),(17,86,28,79,125,119,49),(18,50,120,126,80,29,87),(19,88,30,61,127,101,51),(20,52,102,128,62,31,89)], [(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),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)], [(1,5),(2,4),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(21,118),(22,117),(23,116),(24,115),(25,114),(26,113),(27,112),(28,111),(29,110),(30,109),(31,108),(32,107),(33,106),(34,105),(35,104),(36,103),(37,102),(38,101),(39,120),(40,119),(41,86),(42,85),(43,84),(44,83),(45,82),(46,81),(47,100),(48,99),(49,98),(50,97),(51,96),(52,95),(53,94),(54,93),(55,92),(56,91),(57,90),(58,89),(59,88),(60,87),(61,135),(62,134),(63,133),(64,132),(65,131),(66,130),(67,129),(68,128),(69,127),(70,126),(71,125),(72,124),(73,123),(74,122),(75,121),(76,140),(77,139),(78,138),(79,137),(80,136)]])
37 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 14D | ··· | 14I | 20A | 20B | 20C | 20D | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 14 | ··· | 14 | 20 | 20 | 20 | 20 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 10 | 70 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 14 | 14 | 14 | 14 | 4 | ··· | 4 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D5 | D7 | D10 | D14 | D20 | C7⋊D4 | D5×D7 | C7⋊D20 |
| kernel | C7⋊D20 | C5×Dic7 | D5×C14 | D70 | C35 | Dic7 | D10 | C14 | C10 | C7 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 6 | 6 | 6 |
Matrix representation of C7⋊D20 ►in GL6(𝔽281)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 279 | 234 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 242 | 236 | 0 | 0 | 0 | 0 |
| 222 | 76 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 6 | 0 | 0 |
| 0 | 0 | 34 | 239 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 280 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 280 | 0 | 0 | 0 | 0 | 0 |
| 65 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 239 | 275 | 0 | 0 |
| 0 | 0 | 247 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 280 |
G:=sub<GL(6,GF(281))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,279,0,0,0,0,1,234,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[242,222,0,0,0,0,236,76,0,0,0,0,0,0,42,34,0,0,0,0,6,239,0,0,0,0,0,0,0,1,0,0,0,0,280,0],[280,65,0,0,0,0,0,1,0,0,0,0,0,0,239,247,0,0,0,0,275,42,0,0,0,0,0,0,1,0,0,0,0,0,0,280] >;
C7⋊D20 in GAP, Magma, Sage, TeX
C_7\rtimes D_{20} % in TeX
G:=Group("C7:D20"); // GroupNames label
G:=SmallGroup(280,12);
// by ID
G=gap.SmallGroup(280,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,20,61,328,6004]);
// Polycyclic
G:=Group<a,b,c|a^7=b^20=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export