direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C7⋊C16, C14⋊C16, C56.3C4, C28.3C8, C8.21D14, C8.4Dic7, C56.21C22, C7⋊2(C2×C16), C4.3(C7⋊C8), (C2×C8).9D7, (C2×C56).9C2, (C2×C14).2C8, C14.8(C2×C8), C28.38(C2×C4), (C2×C28).11C4, C22.2(C7⋊C8), (C2×C4).8Dic7, C4.10(C2×Dic7), C2.2(C2×C7⋊C8), SmallGroup(224,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×C7⋊C16 |
Generators and relations for C2×C7⋊C16
G = < a,b,c | a2=b7=c16=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C7⋊C16
►Regular action on 224 points
Generators in S224
(1 78)(2 79)(3 80)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 171)(18 172)(19 173)(20 174)(21 175)(22 176)(23 161)(24 162)(25 163)(26 164)(27 165)(28 166)(29 167)(30 168)(31 169)(32 170)(33 200)(34 201)(35 202)(36 203)(37 204)(38 205)(39 206)(40 207)(41 208)(42 193)(43 194)(44 195)(45 196)(46 197)(47 198)(48 199)(49 109)(50 110)(51 111)(52 112)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(61 105)(62 106)(63 107)(64 108)(81 190)(82 191)(83 192)(84 177)(85 178)(86 179)(87 180)(88 181)(89 182)(90 183)(91 184)(92 185)(93 186)(94 187)(95 188)(96 189)(113 158)(114 159)(115 160)(116 145)(117 146)(118 147)(119 148)(120 149)(121 150)(122 151)(123 152)(124 153)(125 154)(126 155)(127 156)(128 157)(129 213)(130 214)(131 215)(132 216)(133 217)(134 218)(135 219)(136 220)(137 221)(138 222)(139 223)(140 224)(141 209)(142 210)(143 211)(144 212)
(1 40 133 175 88 50 152)(2 153 51 89 176 134 41)(3 42 135 161 90 52 154)(4 155 53 91 162 136 43)(5 44 137 163 92 54 156)(6 157 55 93 164 138 45)(7 46 139 165 94 56 158)(8 159 57 95 166 140 47)(9 48 141 167 96 58 160)(10 145 59 81 168 142 33)(11 34 143 169 82 60 146)(12 147 61 83 170 144 35)(13 36 129 171 84 62 148)(14 149 63 85 172 130 37)(15 38 131 173 86 64 150)(16 151 49 87 174 132 39)(17 177 106 119 74 203 213)(18 214 204 75 120 107 178)(19 179 108 121 76 205 215)(20 216 206 77 122 109 180)(21 181 110 123 78 207 217)(22 218 208 79 124 111 182)(23 183 112 125 80 193 219)(24 220 194 65 126 97 184)(25 185 98 127 66 195 221)(26 222 196 67 128 99 186)(27 187 100 113 68 197 223)(28 224 198 69 114 101 188)(29 189 102 115 70 199 209)(30 210 200 71 116 103 190)(31 191 104 117 72 201 211)(32 212 202 73 118 105 192)
(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)| (1,78)(2,79)(3,80)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,171)(18,172)(19,173)(20,174)(21,175)(22,176)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,167)(30,168)(31,169)(32,170)(33,200)(34,201)(35,202)(36,203)(37,204)(38,205)(39,206)(40,207)(41,208)(42,193)(43,194)(44,195)(45,196)(46,197)(47,198)(48,199)(49,109)(50,110)(51,111)(52,112)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(61,105)(62,106)(63,107)(64,108)(81,190)(82,191)(83,192)(84,177)(85,178)(86,179)(87,180)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(113,158)(114,159)(115,160)(116,145)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,213)(130,214)(131,215)(132,216)(133,217)(134,218)(135,219)(136,220)(137,221)(138,222)(139,223)(140,224)(141,209)(142,210)(143,211)(144,212), (1,40,133,175,88,50,152)(2,153,51,89,176,134,41)(3,42,135,161,90,52,154)(4,155,53,91,162,136,43)(5,44,137,163,92,54,156)(6,157,55,93,164,138,45)(7,46,139,165,94,56,158)(8,159,57,95,166,140,47)(9,48,141,167,96,58,160)(10,145,59,81,168,142,33)(11,34,143,169,82,60,146)(12,147,61,83,170,144,35)(13,36,129,171,84,62,148)(14,149,63,85,172,130,37)(15,38,131,173,86,64,150)(16,151,49,87,174,132,39)(17,177,106,119,74,203,213)(18,214,204,75,120,107,178)(19,179,108,121,76,205,215)(20,216,206,77,122,109,180)(21,181,110,123,78,207,217)(22,218,208,79,124,111,182)(23,183,112,125,80,193,219)(24,220,194,65,126,97,184)(25,185,98,127,66,195,221)(26,222,196,67,128,99,186)(27,187,100,113,68,197,223)(28,224,198,69,114,101,188)(29,189,102,115,70,199,209)(30,210,200,71,116,103,190)(31,191,104,117,72,201,211)(32,212,202,73,118,105,192), (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( (1,78)(2,79)(3,80)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,71)(11,72)(12,73)(13,74)(14,75)(15,76)(16,77)(17,171)(18,172)(19,173)(20,174)(21,175)(22,176)(23,161)(24,162)(25,163)(26,164)(27,165)(28,166)(29,167)(30,168)(31,169)(32,170)(33,200)(34,201)(35,202)(36,203)(37,204)(38,205)(39,206)(40,207)(41,208)(42,193)(43,194)(44,195)(45,196)(46,197)(47,198)(48,199)(49,109)(50,110)(51,111)(52,112)(53,97)(54,98)(55,99)(56,100)(57,101)(58,102)(59,103)(60,104)(61,105)(62,106)(63,107)(64,108)(81,190)(82,191)(83,192)(84,177)(85,178)(86,179)(87,180)(88,181)(89,182)(90,183)(91,184)(92,185)(93,186)(94,187)(95,188)(96,189)(113,158)(114,159)(115,160)(116,145)(117,146)(118,147)(119,148)(120,149)(121,150)(122,151)(123,152)(124,153)(125,154)(126,155)(127,156)(128,157)(129,213)(130,214)(131,215)(132,216)(133,217)(134,218)(135,219)(136,220)(137,221)(138,222)(139,223)(140,224)(141,209)(142,210)(143,211)(144,212), (1,40,133,175,88,50,152)(2,153,51,89,176,134,41)(3,42,135,161,90,52,154)(4,155,53,91,162,136,43)(5,44,137,163,92,54,156)(6,157,55,93,164,138,45)(7,46,139,165,94,56,158)(8,159,57,95,166,140,47)(9,48,141,167,96,58,160)(10,145,59,81,168,142,33)(11,34,143,169,82,60,146)(12,147,61,83,170,144,35)(13,36,129,171,84,62,148)(14,149,63,85,172,130,37)(15,38,131,173,86,64,150)(16,151,49,87,174,132,39)(17,177,106,119,74,203,213)(18,214,204,75,120,107,178)(19,179,108,121,76,205,215)(20,216,206,77,122,109,180)(21,181,110,123,78,207,217)(22,218,208,79,124,111,182)(23,183,112,125,80,193,219)(24,220,194,65,126,97,184)(25,185,98,127,66,195,221)(26,222,196,67,128,99,186)(27,187,100,113,68,197,223)(28,224,198,69,114,101,188)(29,189,102,115,70,199,209)(30,210,200,71,116,103,190)(31,191,104,117,72,201,211)(32,212,202,73,118,105,192), (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([[(1,78),(2,79),(3,80),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,71),(11,72),(12,73),(13,74),(14,75),(15,76),(16,77),(17,171),(18,172),(19,173),(20,174),(21,175),(22,176),(23,161),(24,162),(25,163),(26,164),(27,165),(28,166),(29,167),(30,168),(31,169),(32,170),(33,200),(34,201),(35,202),(36,203),(37,204),(38,205),(39,206),(40,207),(41,208),(42,193),(43,194),(44,195),(45,196),(46,197),(47,198),(48,199),(49,109),(50,110),(51,111),(52,112),(53,97),(54,98),(55,99),(56,100),(57,101),(58,102),(59,103),(60,104),(61,105),(62,106),(63,107),(64,108),(81,190),(82,191),(83,192),(84,177),(85,178),(86,179),(87,180),(88,181),(89,182),(90,183),(91,184),(92,185),(93,186),(94,187),(95,188),(96,189),(113,158),(114,159),(115,160),(116,145),(117,146),(118,147),(119,148),(120,149),(121,150),(122,151),(123,152),(124,153),(125,154),(126,155),(127,156),(128,157),(129,213),(130,214),(131,215),(132,216),(133,217),(134,218),(135,219),(136,220),(137,221),(138,222),(139,223),(140,224),(141,209),(142,210),(143,211),(144,212)], [(1,40,133,175,88,50,152),(2,153,51,89,176,134,41),(3,42,135,161,90,52,154),(4,155,53,91,162,136,43),(5,44,137,163,92,54,156),(6,157,55,93,164,138,45),(7,46,139,165,94,56,158),(8,159,57,95,166,140,47),(9,48,141,167,96,58,160),(10,145,59,81,168,142,33),(11,34,143,169,82,60,146),(12,147,61,83,170,144,35),(13,36,129,171,84,62,148),(14,149,63,85,172,130,37),(15,38,131,173,86,64,150),(16,151,49,87,174,132,39),(17,177,106,119,74,203,213),(18,214,204,75,120,107,178),(19,179,108,121,76,205,215),(20,216,206,77,122,109,180),(21,181,110,123,78,207,217),(22,218,208,79,124,111,182),(23,183,112,125,80,193,219),(24,220,194,65,126,97,184),(25,185,98,127,66,195,221),(26,222,196,67,128,99,186),(27,187,100,113,68,197,223),(28,224,198,69,114,101,188),(29,189,102,115,70,199,209),(30,210,200,71,116,103,190),(31,191,104,117,72,201,211),(32,212,202,73,118,105,192)], [(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)]])
C2×C7⋊C16 is a maximal subgroup of
C56.C8 C28⋊C16 C8.4Dic14 C8.5Dic14 C14.D16 C56.6D4 C8.7Dic14 Dic28.C4 C16×Dic7 Dic7⋊C16 C112⋊9C4 D14⋊C16 Dic14.C8 C56.91D4 C56.92D4 C14.SD32 C14.Q32 C28.58D8 D7×C2×C16 C16.12D14 C56.70C23 C56.30C23
C2×C7⋊C16 is a maximal quotient of
C28⋊C16 C7⋊M6(2) C56.91D4
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 16A | ··· | 16P | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 2 | ··· | 2 | 2 | ··· | 2 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D7 | Dic7 | D14 | Dic7 | C7⋊C8 | C7⋊C8 | C7⋊C16 |
| kernel | C2×C7⋊C16 | C7⋊C16 | C2×C56 | C56 | C2×C28 | C28 | C2×C14 | C14 | C2×C8 | C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 3 | 3 | 3 | 3 | 6 | 6 | 24 |
Matrix representation of C2×C7⋊C16 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 1 | 0 | 0 |
| 0 | 79 | 112 |
| 0 | 1 | 0 |
| 48 | 0 | 0 |
| 0 | 101 | 84 |
| 0 | 40 | 12 |
G:=sub<GL(3,GF(113))| [1,0,0,0,112,0,0,0,112],[1,0,0,0,79,1,0,112,0],[48,0,0,0,101,40,0,84,12] >;
C2×C7⋊C16 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes C_{16} % in TeX
G:=Group("C2xC7:C16"); // GroupNames label
G:=SmallGroup(224,17);
// by ID
G=gap.SmallGroup(224,17);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,50,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^2=b^7=c^16=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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