metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C35⋊2D4, C5⋊2D28, Dic5⋊D7, D14⋊2D5, D70⋊3C2, C10.5D14, C14.5D10, C70.5C22, C7⋊1(C5⋊D4), C2.5(D5×D7), (C10×D7)⋊2C2, (C7×Dic5)⋊3C2, SmallGroup(280,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊D28
G = < a,b,c | a5=b28=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C5⋊D28
►On 140 points
Generators in S140
(1 93 129 70 50)(2 51 71 130 94)(3 95 131 72 52)(4 53 73 132 96)(5 97 133 74 54)(6 55 75 134 98)(7 99 135 76 56)(8 29 77 136 100)(9 101 137 78 30)(10 31 79 138 102)(11 103 139 80 32)(12 33 81 140 104)(13 105 113 82 34)(14 35 83 114 106)(15 107 115 84 36)(16 37 57 116 108)(17 109 117 58 38)(18 39 59 118 110)(19 111 119 60 40)(20 41 61 120 112)(21 85 121 62 42)(22 43 63 122 86)(23 87 123 64 44)(24 45 65 124 88)(25 89 125 66 46)(26 47 67 126 90)(27 91 127 68 48)(28 49 69 128 92)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 92)(30 91)(31 90)(32 89)(33 88)(34 87)(35 86)(36 85)(37 112)(38 111)(39 110)(40 109)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 100)(50 99)(51 98)(52 97)(53 96)(54 95)(55 94)(56 93)(57 120)(58 119)(59 118)(60 117)(61 116)(62 115)(63 114)(64 113)(65 140)(66 139)(67 138)(68 137)(69 136)(70 135)(71 134)(72 133)(73 132)(74 131)(75 130)(76 129)(77 128)(78 127)(79 126)(80 125)(81 124)(82 123)(83 122)(84 121)
G:=sub<Sym(140)| (1,93,129,70,50)(2,51,71,130,94)(3,95,131,72,52)(4,53,73,132,96)(5,97,133,74,54)(6,55,75,134,98)(7,99,135,76,56)(8,29,77,136,100)(9,101,137,78,30)(10,31,79,138,102)(11,103,139,80,32)(12,33,81,140,104)(13,105,113,82,34)(14,35,83,114,106)(15,107,115,84,36)(16,37,57,116,108)(17,109,117,58,38)(18,39,59,118,110)(19,111,119,60,40)(20,41,61,120,112)(21,85,121,62,42)(22,43,63,122,86)(23,87,123,64,44)(24,45,65,124,88)(25,89,125,66,46)(26,47,67,126,90)(27,91,127,68,48)(28,49,69,128,92), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,133)(73,132)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125)(81,124)(82,123)(83,122)(84,121)>;
G:=Group( (1,93,129,70,50)(2,51,71,130,94)(3,95,131,72,52)(4,53,73,132,96)(5,97,133,74,54)(6,55,75,134,98)(7,99,135,76,56)(8,29,77,136,100)(9,101,137,78,30)(10,31,79,138,102)(11,103,139,80,32)(12,33,81,140,104)(13,105,113,82,34)(14,35,83,114,106)(15,107,115,84,36)(16,37,57,116,108)(17,109,117,58,38)(18,39,59,118,110)(19,111,119,60,40)(20,41,61,120,112)(21,85,121,62,42)(22,43,63,122,86)(23,87,123,64,44)(24,45,65,124,88)(25,89,125,66,46)(26,47,67,126,90)(27,91,127,68,48)(28,49,69,128,92), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,100)(50,99)(51,98)(52,97)(53,96)(54,95)(55,94)(56,93)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,140)(66,139)(67,138)(68,137)(69,136)(70,135)(71,134)(72,133)(73,132)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125)(81,124)(82,123)(83,122)(84,121) );
G=PermutationGroup([[(1,93,129,70,50),(2,51,71,130,94),(3,95,131,72,52),(4,53,73,132,96),(5,97,133,74,54),(6,55,75,134,98),(7,99,135,76,56),(8,29,77,136,100),(9,101,137,78,30),(10,31,79,138,102),(11,103,139,80,32),(12,33,81,140,104),(13,105,113,82,34),(14,35,83,114,106),(15,107,115,84,36),(16,37,57,116,108),(17,109,117,58,38),(18,39,59,118,110),(19,111,119,60,40),(20,41,61,120,112),(21,85,121,62,42),(22,43,63,122,86),(23,87,123,64,44),(24,45,65,124,88),(25,89,125,66,46),(26,47,67,126,90),(27,91,127,68,48),(28,49,69,128,92)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,92),(30,91),(31,90),(32,89),(33,88),(34,87),(35,86),(36,85),(37,112),(38,111),(39,110),(40,109),(41,108),(42,107),(43,106),(44,105),(45,104),(46,103),(47,102),(48,101),(49,100),(50,99),(51,98),(52,97),(53,96),(54,95),(55,94),(56,93),(57,120),(58,119),(59,118),(60,117),(61,116),(62,115),(63,114),(64,113),(65,140),(66,139),(67,138),(68,137),(69,136),(70,135),(71,134),(72,133),(73,132),(74,131),(75,130),(76,129),(77,128),(78,127),(79,126),(80,125),(81,124),(82,123),(83,122),(84,121)]])
37 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 10E | 10F | 14A | 14B | 14C | 28A | ··· | 28F | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 2 | 2 | 4 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 10 | 10 | 14 | 14 | 14 | 28 | ··· | 28 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 14 | 70 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 10 | ··· | 10 | 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 | C5⋊D4 | D28 | D5×D7 | C5⋊D28 |
| kernel | C5⋊D28 | C7×Dic5 | C10×D7 | D70 | C35 | D14 | Dic5 | C14 | C10 | C7 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 6 | 6 | 6 |
Matrix representation of C5⋊D28 ►in GL4(𝔽281) generated by
| 0 | 1 | 0 | 0 |
| 280 | 37 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 160 | 204 | 0 | 0 |
| 223 | 121 | 0 | 0 |
| 0 | 0 | 40 | 47 |
| 0 | 0 | 234 | 1 |
| 280 | 0 | 0 | 0 |
| 244 | 1 | 0 | 0 |
| 0 | 0 | 241 | 234 |
| 0 | 0 | 40 | 40 |
G:=sub<GL(4,GF(281))| [0,280,0,0,1,37,0,0,0,0,1,0,0,0,0,1],[160,223,0,0,204,121,0,0,0,0,40,234,0,0,47,1],[280,244,0,0,0,1,0,0,0,0,241,40,0,0,234,40] >;
C5⋊D28 in GAP, Magma, Sage, TeX
C_5\rtimes D_{28} % in TeX
G:=Group("C5:D28"); // GroupNames label
G:=SmallGroup(280,11);
// by ID
G=gap.SmallGroup(280,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,61,26,328,6004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^28=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export