direct product, abelian, monomial, 7-elementary
Aliases: C7×C21, SmallGroup(147,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C7×C21 |
| C1 — C7×C21 |
| C1 — C7×C21 |
Generators and relations for C7×C21
G = < a,b | a7=b21=1, ab=ba >
Smallest permutation representation of C7×C21
►Regular action on 147 points
Generators in S147
(1 28 58 122 91 130 78)(2 29 59 123 92 131 79)(3 30 60 124 93 132 80)(4 31 61 125 94 133 81)(5 32 62 126 95 134 82)(6 33 63 106 96 135 83)(7 34 43 107 97 136 84)(8 35 44 108 98 137 64)(9 36 45 109 99 138 65)(10 37 46 110 100 139 66)(11 38 47 111 101 140 67)(12 39 48 112 102 141 68)(13 40 49 113 103 142 69)(14 41 50 114 104 143 70)(15 42 51 115 105 144 71)(16 22 52 116 85 145 72)(17 23 53 117 86 146 73)(18 24 54 118 87 147 74)(19 25 55 119 88 127 75)(20 26 56 120 89 128 76)(21 27 57 121 90 129 77)
(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)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147)
G:=sub<Sym(147)| (1,28,58,122,91,130,78)(2,29,59,123,92,131,79)(3,30,60,124,93,132,80)(4,31,61,125,94,133,81)(5,32,62,126,95,134,82)(6,33,63,106,96,135,83)(7,34,43,107,97,136,84)(8,35,44,108,98,137,64)(9,36,45,109,99,138,65)(10,37,46,110,100,139,66)(11,38,47,111,101,140,67)(12,39,48,112,102,141,68)(13,40,49,113,103,142,69)(14,41,50,114,104,143,70)(15,42,51,115,105,144,71)(16,22,52,116,85,145,72)(17,23,53,117,86,146,73)(18,24,54,118,87,147,74)(19,25,55,119,88,127,75)(20,26,56,120,89,128,76)(21,27,57,121,90,129,77), (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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147)>;
G:=Group( (1,28,58,122,91,130,78)(2,29,59,123,92,131,79)(3,30,60,124,93,132,80)(4,31,61,125,94,133,81)(5,32,62,126,95,134,82)(6,33,63,106,96,135,83)(7,34,43,107,97,136,84)(8,35,44,108,98,137,64)(9,36,45,109,99,138,65)(10,37,46,110,100,139,66)(11,38,47,111,101,140,67)(12,39,48,112,102,141,68)(13,40,49,113,103,142,69)(14,41,50,114,104,143,70)(15,42,51,115,105,144,71)(16,22,52,116,85,145,72)(17,23,53,117,86,146,73)(18,24,54,118,87,147,74)(19,25,55,119,88,127,75)(20,26,56,120,89,128,76)(21,27,57,121,90,129,77), (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)(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147) );
G=PermutationGroup([[(1,28,58,122,91,130,78),(2,29,59,123,92,131,79),(3,30,60,124,93,132,80),(4,31,61,125,94,133,81),(5,32,62,126,95,134,82),(6,33,63,106,96,135,83),(7,34,43,107,97,136,84),(8,35,44,108,98,137,64),(9,36,45,109,99,138,65),(10,37,46,110,100,139,66),(11,38,47,111,101,140,67),(12,39,48,112,102,141,68),(13,40,49,113,103,142,69),(14,41,50,114,104,143,70),(15,42,51,115,105,144,71),(16,22,52,116,85,145,72),(17,23,53,117,86,146,73),(18,24,54,118,87,147,74),(19,25,55,119,88,127,75),(20,26,56,120,89,128,76),(21,27,57,121,90,129,77)], [(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),(127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147)]])
C7×C21 is a maximal subgroup of
C7⋊D21 C72⋊C9 C72⋊3C9
147 conjugacy classes
| class | 1 | 3A | 3B | 7A | ··· | 7AV | 21A | ··· | 21CR |
| order | 1 | 3 | 3 | 7 | ··· | 7 | 21 | ··· | 21 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
147 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C7 | C21 |
| kernel | C7×C21 | C72 | C21 | C7 |
| # reps | 1 | 2 | 48 | 96 |
Matrix representation of C7×C21 ►in GL2(𝔽43) generated by
| 35 | 0 |
| 0 | 16 |
| 13 | 0 |
| 0 | 21 |
G:=sub<GL(2,GF(43))| [35,0,0,16],[13,0,0,21] >;
C7×C21 in GAP, Magma, Sage, TeX
C_7\times C_{21} % in TeX
G:=Group("C7xC21"); // GroupNames label
G:=SmallGroup(147,6);
// by ID
G=gap.SmallGroup(147,6);
# by ID
G:=PCGroup([3,-3,-7,-7]);
// Polycyclic
G:=Group<a,b|a^7=b^21=1,a*b=b*a>;
// generators/relations
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