direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×Dic7, C7⋊C20, C35⋊4C4, C14.C10, C70.2C2, C10.2D7, C2.(C5×D7), SmallGroup(140,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C5×Dic7 |
Generators and relations for C5×Dic7
G = < a,b,c | a5=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic7
►Regular action on 140 points
Generators in S140
(1 60 51 29 28)(2 61 52 30 15)(3 62 53 31 16)(4 63 54 32 17)(5 64 55 33 18)(6 65 56 34 19)(7 66 43 35 20)(8 67 44 36 21)(9 68 45 37 22)(10 69 46 38 23)(11 70 47 39 24)(12 57 48 40 25)(13 58 49 41 26)(14 59 50 42 27)(71 127 120 106 92)(72 128 121 107 93)(73 129 122 108 94)(74 130 123 109 95)(75 131 124 110 96)(76 132 125 111 97)(77 133 126 112 98)(78 134 113 99 85)(79 135 114 100 86)(80 136 115 101 87)(81 137 116 102 88)(82 138 117 103 89)(83 139 118 104 90)(84 140 119 105 91)
(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 78 8 71)(2 77 9 84)(3 76 10 83)(4 75 11 82)(5 74 12 81)(6 73 13 80)(7 72 14 79)(15 98 22 91)(16 97 23 90)(17 96 24 89)(18 95 25 88)(19 94 26 87)(20 93 27 86)(21 92 28 85)(29 99 36 106)(30 112 37 105)(31 111 38 104)(32 110 39 103)(33 109 40 102)(34 108 41 101)(35 107 42 100)(43 121 50 114)(44 120 51 113)(45 119 52 126)(46 118 53 125)(47 117 54 124)(48 116 55 123)(49 115 56 122)(57 137 64 130)(58 136 65 129)(59 135 66 128)(60 134 67 127)(61 133 68 140)(62 132 69 139)(63 131 70 138)
G:=sub<Sym(140)| (1,60,51,29,28)(2,61,52,30,15)(3,62,53,31,16)(4,63,54,32,17)(5,64,55,33,18)(6,65,56,34,19)(7,66,43,35,20)(8,67,44,36,21)(9,68,45,37,22)(10,69,46,38,23)(11,70,47,39,24)(12,57,48,40,25)(13,58,49,41,26)(14,59,50,42,27)(71,127,120,106,92)(72,128,121,107,93)(73,129,122,108,94)(74,130,123,109,95)(75,131,124,110,96)(76,132,125,111,97)(77,133,126,112,98)(78,134,113,99,85)(79,135,114,100,86)(80,136,115,101,87)(81,137,116,102,88)(82,138,117,103,89)(83,139,118,104,90)(84,140,119,105,91), (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), (1,78,8,71)(2,77,9,84)(3,76,10,83)(4,75,11,82)(5,74,12,81)(6,73,13,80)(7,72,14,79)(15,98,22,91)(16,97,23,90)(17,96,24,89)(18,95,25,88)(19,94,26,87)(20,93,27,86)(21,92,28,85)(29,99,36,106)(30,112,37,105)(31,111,38,104)(32,110,39,103)(33,109,40,102)(34,108,41,101)(35,107,42,100)(43,121,50,114)(44,120,51,113)(45,119,52,126)(46,118,53,125)(47,117,54,124)(48,116,55,123)(49,115,56,122)(57,137,64,130)(58,136,65,129)(59,135,66,128)(60,134,67,127)(61,133,68,140)(62,132,69,139)(63,131,70,138)>;
G:=Group( (1,60,51,29,28)(2,61,52,30,15)(3,62,53,31,16)(4,63,54,32,17)(5,64,55,33,18)(6,65,56,34,19)(7,66,43,35,20)(8,67,44,36,21)(9,68,45,37,22)(10,69,46,38,23)(11,70,47,39,24)(12,57,48,40,25)(13,58,49,41,26)(14,59,50,42,27)(71,127,120,106,92)(72,128,121,107,93)(73,129,122,108,94)(74,130,123,109,95)(75,131,124,110,96)(76,132,125,111,97)(77,133,126,112,98)(78,134,113,99,85)(79,135,114,100,86)(80,136,115,101,87)(81,137,116,102,88)(82,138,117,103,89)(83,139,118,104,90)(84,140,119,105,91), (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), (1,78,8,71)(2,77,9,84)(3,76,10,83)(4,75,11,82)(5,74,12,81)(6,73,13,80)(7,72,14,79)(15,98,22,91)(16,97,23,90)(17,96,24,89)(18,95,25,88)(19,94,26,87)(20,93,27,86)(21,92,28,85)(29,99,36,106)(30,112,37,105)(31,111,38,104)(32,110,39,103)(33,109,40,102)(34,108,41,101)(35,107,42,100)(43,121,50,114)(44,120,51,113)(45,119,52,126)(46,118,53,125)(47,117,54,124)(48,116,55,123)(49,115,56,122)(57,137,64,130)(58,136,65,129)(59,135,66,128)(60,134,67,127)(61,133,68,140)(62,132,69,139)(63,131,70,138) );
G=PermutationGroup([[(1,60,51,29,28),(2,61,52,30,15),(3,62,53,31,16),(4,63,54,32,17),(5,64,55,33,18),(6,65,56,34,19),(7,66,43,35,20),(8,67,44,36,21),(9,68,45,37,22),(10,69,46,38,23),(11,70,47,39,24),(12,57,48,40,25),(13,58,49,41,26),(14,59,50,42,27),(71,127,120,106,92),(72,128,121,107,93),(73,129,122,108,94),(74,130,123,109,95),(75,131,124,110,96),(76,132,125,111,97),(77,133,126,112,98),(78,134,113,99,85),(79,135,114,100,86),(80,136,115,101,87),(81,137,116,102,88),(82,138,117,103,89),(83,139,118,104,90),(84,140,119,105,91)], [(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)], [(1,78,8,71),(2,77,9,84),(3,76,10,83),(4,75,11,82),(5,74,12,81),(6,73,13,80),(7,72,14,79),(15,98,22,91),(16,97,23,90),(17,96,24,89),(18,95,25,88),(19,94,26,87),(20,93,27,86),(21,92,28,85),(29,99,36,106),(30,112,37,105),(31,111,38,104),(32,110,39,103),(33,109,40,102),(34,108,41,101),(35,107,42,100),(43,121,50,114),(44,120,51,113),(45,119,52,126),(46,118,53,125),(47,117,54,124),(48,116,55,123),(49,115,56,122),(57,137,64,130),(58,136,65,129),(59,135,66,128),(60,134,67,127),(61,133,68,140),(62,132,69,139),(63,131,70,138)]])
C5×Dic7 is a maximal subgroup of
D70.C2 C7⋊D20 C35⋊Q8 D7×C20
50 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 14A | 14B | 14C | 20A | ··· | 20H | 35A | ··· | 35L | 70A | ··· | 70L |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 14 | 14 | 14 | 20 | ··· | 20 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | D7 | Dic7 | C5×D7 | C5×Dic7 |
| kernel | C5×Dic7 | C70 | C35 | Dic7 | C14 | C7 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 3 | 3 | 12 | 12 |
Matrix representation of C5×Dic7 ►in GL2(𝔽41) generated by
| 10 | 0 |
| 0 | 10 |
| 1 | 17 |
| 15 | 10 |
| 9 | 30 |
| 0 | 32 |
G:=sub<GL(2,GF(41))| [10,0,0,10],[1,15,17,10],[9,0,30,32] >;
C5×Dic7 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_7 % in TeX
G:=Group("C5xDic7"); // GroupNames label
G:=SmallGroup(140,2);
// by ID
G=gap.SmallGroup(140,2);
# by ID
G:=PCGroup([4,-2,-5,-2,-7,40,1923]);
// Polycyclic
G:=Group<a,b,c|a^5=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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