direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×C7⋊C8, C28⋊2C8, C42.6D7, C14.1C42, C7⋊1(C4×C8), C14.6(C2×C8), (C4×C28).6C2, C4.18(C4×D7), (C2×C28).10C4, C28.23(C2×C4), (C2×C4).87D14, (C2×C4).7Dic7, C2.1(C4×Dic7), C22.6(C2×Dic7), (C2×C28).101C22, C2.1(C2×C7⋊C8), (C2×C7⋊C8).11C2, (C2×C14).24(C2×C4), SmallGroup(224,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C4×C7⋊C8 |
Generators and relations for C4×C7⋊C8
G = < a,b,c | a4=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C4×C7⋊C8
►Regular action on 224 points
Generators in S224
(1 179 22 107)(2 180 23 108)(3 181 24 109)(4 182 17 110)(5 183 18 111)(6 184 19 112)(7 177 20 105)(8 178 21 106)(9 221 103 59)(10 222 104 60)(11 223 97 61)(12 224 98 62)(13 217 99 63)(14 218 100 64)(15 219 101 57)(16 220 102 58)(25 214 66 52)(26 215 67 53)(27 216 68 54)(28 209 69 55)(29 210 70 56)(30 211 71 49)(31 212 72 50)(32 213 65 51)(33 83 172 41)(34 84 173 42)(35 85 174 43)(36 86 175 44)(37 87 176 45)(38 88 169 46)(39 81 170 47)(40 82 171 48)(73 89 155 114)(74 90 156 115)(75 91 157 116)(76 92 158 117)(77 93 159 118)(78 94 160 119)(79 95 153 120)(80 96 154 113)(121 148 198 166)(122 149 199 167)(123 150 200 168)(124 151 193 161)(125 152 194 162)(126 145 195 163)(127 146 196 164)(128 147 197 165)(129 143 185 204)(130 144 186 205)(131 137 187 206)(132 138 188 207)(133 139 189 208)(134 140 190 201)(135 141 191 202)(136 142 192 203)
(1 15 55 80 188 41 122)(2 123 42 189 73 56 16)(3 9 49 74 190 43 124)(4 125 44 191 75 50 10)(5 11 51 76 192 45 126)(6 127 46 185 77 52 12)(7 13 53 78 186 47 128)(8 121 48 187 79 54 14)(17 194 86 135 157 212 104)(18 97 213 158 136 87 195)(19 196 88 129 159 214 98)(20 99 215 160 130 81 197)(21 198 82 131 153 216 100)(22 101 209 154 132 83 199)(23 200 84 133 155 210 102)(24 103 211 156 134 85 193)(25 224 184 146 38 204 93)(26 94 205 39 147 177 217)(27 218 178 148 40 206 95)(28 96 207 33 149 179 219)(29 220 180 150 34 208 89)(30 90 201 35 151 181 221)(31 222 182 152 36 202 91)(32 92 203 37 145 183 223)(57 69 113 138 172 167 107)(58 108 168 173 139 114 70)(59 71 115 140 174 161 109)(60 110 162 175 141 116 72)(61 65 117 142 176 163 111)(62 112 164 169 143 118 66)(63 67 119 144 170 165 105)(64 106 166 171 137 120 68)
(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)(193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216)(217 218 219 220 221 222 223 224)
G:=sub<Sym(224)| (1,179,22,107)(2,180,23,108)(3,181,24,109)(4,182,17,110)(5,183,18,111)(6,184,19,112)(7,177,20,105)(8,178,21,106)(9,221,103,59)(10,222,104,60)(11,223,97,61)(12,224,98,62)(13,217,99,63)(14,218,100,64)(15,219,101,57)(16,220,102,58)(25,214,66,52)(26,215,67,53)(27,216,68,54)(28,209,69,55)(29,210,70,56)(30,211,71,49)(31,212,72,50)(32,213,65,51)(33,83,172,41)(34,84,173,42)(35,85,174,43)(36,86,175,44)(37,87,176,45)(38,88,169,46)(39,81,170,47)(40,82,171,48)(73,89,155,114)(74,90,156,115)(75,91,157,116)(76,92,158,117)(77,93,159,118)(78,94,160,119)(79,95,153,120)(80,96,154,113)(121,148,198,166)(122,149,199,167)(123,150,200,168)(124,151,193,161)(125,152,194,162)(126,145,195,163)(127,146,196,164)(128,147,197,165)(129,143,185,204)(130,144,186,205)(131,137,187,206)(132,138,188,207)(133,139,189,208)(134,140,190,201)(135,141,191,202)(136,142,192,203), (1,15,55,80,188,41,122)(2,123,42,189,73,56,16)(3,9,49,74,190,43,124)(4,125,44,191,75,50,10)(5,11,51,76,192,45,126)(6,127,46,185,77,52,12)(7,13,53,78,186,47,128)(8,121,48,187,79,54,14)(17,194,86,135,157,212,104)(18,97,213,158,136,87,195)(19,196,88,129,159,214,98)(20,99,215,160,130,81,197)(21,198,82,131,153,216,100)(22,101,209,154,132,83,199)(23,200,84,133,155,210,102)(24,103,211,156,134,85,193)(25,224,184,146,38,204,93)(26,94,205,39,147,177,217)(27,218,178,148,40,206,95)(28,96,207,33,149,179,219)(29,220,180,150,34,208,89)(30,90,201,35,151,181,221)(31,222,182,152,36,202,91)(32,92,203,37,145,183,223)(57,69,113,138,172,167,107)(58,108,168,173,139,114,70)(59,71,115,140,174,161,109)(60,110,162,175,141,116,72)(61,65,117,142,176,163,111)(62,112,164,169,143,118,66)(63,67,119,144,170,165,105)(64,106,166,171,137,120,68), (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)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224)>;
G:=Group( (1,179,22,107)(2,180,23,108)(3,181,24,109)(4,182,17,110)(5,183,18,111)(6,184,19,112)(7,177,20,105)(8,178,21,106)(9,221,103,59)(10,222,104,60)(11,223,97,61)(12,224,98,62)(13,217,99,63)(14,218,100,64)(15,219,101,57)(16,220,102,58)(25,214,66,52)(26,215,67,53)(27,216,68,54)(28,209,69,55)(29,210,70,56)(30,211,71,49)(31,212,72,50)(32,213,65,51)(33,83,172,41)(34,84,173,42)(35,85,174,43)(36,86,175,44)(37,87,176,45)(38,88,169,46)(39,81,170,47)(40,82,171,48)(73,89,155,114)(74,90,156,115)(75,91,157,116)(76,92,158,117)(77,93,159,118)(78,94,160,119)(79,95,153,120)(80,96,154,113)(121,148,198,166)(122,149,199,167)(123,150,200,168)(124,151,193,161)(125,152,194,162)(126,145,195,163)(127,146,196,164)(128,147,197,165)(129,143,185,204)(130,144,186,205)(131,137,187,206)(132,138,188,207)(133,139,189,208)(134,140,190,201)(135,141,191,202)(136,142,192,203), (1,15,55,80,188,41,122)(2,123,42,189,73,56,16)(3,9,49,74,190,43,124)(4,125,44,191,75,50,10)(5,11,51,76,192,45,126)(6,127,46,185,77,52,12)(7,13,53,78,186,47,128)(8,121,48,187,79,54,14)(17,194,86,135,157,212,104)(18,97,213,158,136,87,195)(19,196,88,129,159,214,98)(20,99,215,160,130,81,197)(21,198,82,131,153,216,100)(22,101,209,154,132,83,199)(23,200,84,133,155,210,102)(24,103,211,156,134,85,193)(25,224,184,146,38,204,93)(26,94,205,39,147,177,217)(27,218,178,148,40,206,95)(28,96,207,33,149,179,219)(29,220,180,150,34,208,89)(30,90,201,35,151,181,221)(31,222,182,152,36,202,91)(32,92,203,37,145,183,223)(57,69,113,138,172,167,107)(58,108,168,173,139,114,70)(59,71,115,140,174,161,109)(60,110,162,175,141,116,72)(61,65,117,142,176,163,111)(62,112,164,169,143,118,66)(63,67,119,144,170,165,105)(64,106,166,171,137,120,68), (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)(193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216)(217,218,219,220,221,222,223,224) );
G=PermutationGroup([[(1,179,22,107),(2,180,23,108),(3,181,24,109),(4,182,17,110),(5,183,18,111),(6,184,19,112),(7,177,20,105),(8,178,21,106),(9,221,103,59),(10,222,104,60),(11,223,97,61),(12,224,98,62),(13,217,99,63),(14,218,100,64),(15,219,101,57),(16,220,102,58),(25,214,66,52),(26,215,67,53),(27,216,68,54),(28,209,69,55),(29,210,70,56),(30,211,71,49),(31,212,72,50),(32,213,65,51),(33,83,172,41),(34,84,173,42),(35,85,174,43),(36,86,175,44),(37,87,176,45),(38,88,169,46),(39,81,170,47),(40,82,171,48),(73,89,155,114),(74,90,156,115),(75,91,157,116),(76,92,158,117),(77,93,159,118),(78,94,160,119),(79,95,153,120),(80,96,154,113),(121,148,198,166),(122,149,199,167),(123,150,200,168),(124,151,193,161),(125,152,194,162),(126,145,195,163),(127,146,196,164),(128,147,197,165),(129,143,185,204),(130,144,186,205),(131,137,187,206),(132,138,188,207),(133,139,189,208),(134,140,190,201),(135,141,191,202),(136,142,192,203)], [(1,15,55,80,188,41,122),(2,123,42,189,73,56,16),(3,9,49,74,190,43,124),(4,125,44,191,75,50,10),(5,11,51,76,192,45,126),(6,127,46,185,77,52,12),(7,13,53,78,186,47,128),(8,121,48,187,79,54,14),(17,194,86,135,157,212,104),(18,97,213,158,136,87,195),(19,196,88,129,159,214,98),(20,99,215,160,130,81,197),(21,198,82,131,153,216,100),(22,101,209,154,132,83,199),(23,200,84,133,155,210,102),(24,103,211,156,134,85,193),(25,224,184,146,38,204,93),(26,94,205,39,147,177,217),(27,218,178,148,40,206,95),(28,96,207,33,149,179,219),(29,220,180,150,34,208,89),(30,90,201,35,151,181,221),(31,222,182,152,36,202,91),(32,92,203,37,145,183,223),(57,69,113,138,172,167,107),(58,108,168,173,139,114,70),(59,71,115,140,174,161,109),(60,110,162,175,141,116,72),(61,65,117,142,176,163,111),(62,112,164,169,143,118,66),(63,67,119,144,170,165,105),(64,106,166,171,137,120,68)], [(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),(193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216),(217,218,219,220,221,222,223,224)]])
C4×C7⋊C8 is a maximal subgroup of
C42.279D14 C56⋊C8 C28.53D8 C28.39SD16 D28⋊2C8 Dic14⋊2C8 C28.57D8 C28.26Q16 D7×C4×C8 C42.282D14 D14.4C42 C42.185D14 C42.196D14 Dic14⋊C8 C28.M4(2) D28⋊C8 C28⋊2M4(2) C42.6Dic7 C28.5C42 C42.187D14 C28⋊3M4(2) C42.210D14 C42.213D14 C42.214D14 C42.215D14 C42.216D14 C28.16D8 C28⋊D8 C28⋊4SD16 C28.17D8 C28.SD16 C28.Q16 C28⋊6SD16 C28.D8 C28⋊3Q16 C28.11Q16
C4×C7⋊C8 is a maximal quotient of
C56⋊C8 C56.C8 (C2×C28)⋊3C8
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8P | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D7 | Dic7 | D14 | C7⋊C8 | C4×D7 |
| kernel | C4×C7⋊C8 | C2×C7⋊C8 | C4×C28 | C7⋊C8 | C2×C28 | C28 | C42 | C2×C4 | C2×C4 | C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 3 | 6 | 3 | 24 | 12 |
Matrix representation of C4×C7⋊C8 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 25 | 112 |
| 0 | 0 | 26 | 112 |
| 69 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 10 | 37 |
| 0 | 0 | 92 | 103 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,15,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,25,26,0,0,112,112],[69,0,0,0,0,112,0,0,0,0,10,92,0,0,37,103] >;
C4×C7⋊C8 in GAP, Magma, Sage, TeX
C_4\times C_7\rtimes C_8
% in TeX
G:=Group("C4xC7:C8"); // GroupNames label
G:=SmallGroup(224,8);
// by ID
G=gap.SmallGroup(224,8);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,55,86,6917]);
// Polycyclic
G:=Group<a,b,c|a^4=b^7=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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