direct product, abelian, monomial, 7-elementary
Aliases: C7×C28, SmallGroup(196,7)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C7×C28 |
| C1 — C7×C28 |
| C1 — C7×C28 |
Generators and relations for C7×C28
G = < a,b | a7=b28=1, ab=ba >
Smallest permutation representation of C7×C28
►Regular action on 196 points
Generators in S196
(1 150 77 170 43 86 132)(2 151 78 171 44 87 133)(3 152 79 172 45 88 134)(4 153 80 173 46 89 135)(5 154 81 174 47 90 136)(6 155 82 175 48 91 137)(7 156 83 176 49 92 138)(8 157 84 177 50 93 139)(9 158 57 178 51 94 140)(10 159 58 179 52 95 113)(11 160 59 180 53 96 114)(12 161 60 181 54 97 115)(13 162 61 182 55 98 116)(14 163 62 183 56 99 117)(15 164 63 184 29 100 118)(16 165 64 185 30 101 119)(17 166 65 186 31 102 120)(18 167 66 187 32 103 121)(19 168 67 188 33 104 122)(20 141 68 189 34 105 123)(21 142 69 190 35 106 124)(22 143 70 191 36 107 125)(23 144 71 192 37 108 126)(24 145 72 193 38 109 127)(25 146 73 194 39 110 128)(26 147 74 195 40 111 129)(27 148 75 196 41 112 130)(28 149 76 169 42 85 131)
(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)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196)
G:=sub<Sym(196)| (1,150,77,170,43,86,132)(2,151,78,171,44,87,133)(3,152,79,172,45,88,134)(4,153,80,173,46,89,135)(5,154,81,174,47,90,136)(6,155,82,175,48,91,137)(7,156,83,176,49,92,138)(8,157,84,177,50,93,139)(9,158,57,178,51,94,140)(10,159,58,179,52,95,113)(11,160,59,180,53,96,114)(12,161,60,181,54,97,115)(13,162,61,182,55,98,116)(14,163,62,183,56,99,117)(15,164,63,184,29,100,118)(16,165,64,185,30,101,119)(17,166,65,186,31,102,120)(18,167,66,187,32,103,121)(19,168,67,188,33,104,122)(20,141,68,189,34,105,123)(21,142,69,190,35,106,124)(22,143,70,191,36,107,125)(23,144,71,192,37,108,126)(24,145,72,193,38,109,127)(25,146,73,194,39,110,128)(26,147,74,195,40,111,129)(27,148,75,196,41,112,130)(28,149,76,169,42,85,131), (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)>;
G:=Group( (1,150,77,170,43,86,132)(2,151,78,171,44,87,133)(3,152,79,172,45,88,134)(4,153,80,173,46,89,135)(5,154,81,174,47,90,136)(6,155,82,175,48,91,137)(7,156,83,176,49,92,138)(8,157,84,177,50,93,139)(9,158,57,178,51,94,140)(10,159,58,179,52,95,113)(11,160,59,180,53,96,114)(12,161,60,181,54,97,115)(13,162,61,182,55,98,116)(14,163,62,183,56,99,117)(15,164,63,184,29,100,118)(16,165,64,185,30,101,119)(17,166,65,186,31,102,120)(18,167,66,187,32,103,121)(19,168,67,188,33,104,122)(20,141,68,189,34,105,123)(21,142,69,190,35,106,124)(22,143,70,191,36,107,125)(23,144,71,192,37,108,126)(24,145,72,193,38,109,127)(25,146,73,194,39,110,128)(26,147,74,195,40,111,129)(27,148,75,196,41,112,130)(28,149,76,169,42,85,131), (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)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196) );
G=PermutationGroup([[(1,150,77,170,43,86,132),(2,151,78,171,44,87,133),(3,152,79,172,45,88,134),(4,153,80,173,46,89,135),(5,154,81,174,47,90,136),(6,155,82,175,48,91,137),(7,156,83,176,49,92,138),(8,157,84,177,50,93,139),(9,158,57,178,51,94,140),(10,159,58,179,52,95,113),(11,160,59,180,53,96,114),(12,161,60,181,54,97,115),(13,162,61,182,55,98,116),(14,163,62,183,56,99,117),(15,164,63,184,29,100,118),(16,165,64,185,30,101,119),(17,166,65,186,31,102,120),(18,167,66,187,32,103,121),(19,168,67,188,33,104,122),(20,141,68,189,34,105,123),(21,142,69,190,35,106,124),(22,143,70,191,36,107,125),(23,144,71,192,37,108,126),(24,145,72,193,38,109,127),(25,146,73,194,39,110,128),(26,147,74,195,40,111,129),(27,148,75,196,41,112,130),(28,149,76,169,42,85,131)], [(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),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196)]])
C7×C28 is a maximal subgroup of
C72⋊4C8 C72⋊4Q8 C28⋊D7
196 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | ··· | 7AV | 14A | ··· | 14AV | 28A | ··· | 28CR |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C7 | C14 | C28 |
| kernel | C7×C28 | C7×C14 | C72 | C28 | C14 | C7 |
| # reps | 1 | 1 | 2 | 48 | 48 | 96 |
Matrix representation of C7×C28 ►in GL2(𝔽29) generated by
| 25 | 0 |
| 0 | 24 |
| 12 | 0 |
| 0 | 21 |
G:=sub<GL(2,GF(29))| [25,0,0,24],[12,0,0,21] >;
C7×C28 in GAP, Magma, Sage, TeX
C_7\times C_{28} % in TeX
G:=Group("C7xC28"); // GroupNames label
G:=SmallGroup(196,7);
// by ID
G=gap.SmallGroup(196,7);
# by ID
G:=PCGroup([4,-2,-7,-7,-2,392]);
// Polycyclic
G:=Group<a,b|a^7=b^28=1,a*b=b*a>;
// generators/relations
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