direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C14×C8.C4, (C2×C56).29C4, (C2×C8).11C28, C8.17(C2×C28), C56.85(C2×C4), C4.77(D4×C14), C28.71(C4⋊C4), (C2×C28).79Q8, (C2×C28).538D4, C28.482(C2×D4), C23.9(C7×Q8), C22.1(Q8×C14), C4.28(C22×C28), (C22×C56).30C2, (C22×C8).12C14, (C22×C14).21Q8, (C2×C56).432C22, (C2×C28).902C23, C28.186(C22×C4), M4(2).9(C2×C14), (C14×M4(2)).33C2, (C2×M4(2)).15C14, (C22×C28).590C22, (C7×M4(2)).43C22, C4.22(C7×C4⋊C4), C2.15(C14×C4⋊C4), C14.71(C2×C4⋊C4), (C2×C4).74(C7×D4), (C2×C4).21(C7×Q8), (C2×C8).90(C2×C14), (C2×C4).77(C2×C28), C22.11(C7×C4⋊C4), (C2×C14).14(C2×Q8), (C2×C14).65(C4⋊C4), (C2×C28).338(C2×C4), (C2×C4).77(C22×C14), (C22×C4).119(C2×C14), SmallGroup(448,837)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14×C8.C4
G = < a,b,c | a14=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 130 in 106 conjugacy classes, 82 normal (42 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C8, C2×C4, C23, C14, C14, C14, C2×C8, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C28, C2×C14, C2×C14, C8.C4, C22×C8, C2×M4(2), C56, C56, C2×C28, C22×C14, C2×C8.C4, C2×C56, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C22×C28, C7×C8.C4, C22×C56, C14×M4(2), C14×C8.C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C8.C4, C2×C4⋊C4, C2×C28, C7×D4, C7×Q8, C22×C14, C2×C8.C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C7×C8.C4, C14×C4⋊C4, C14×C8.C4
►On 224 points
Generators in S224
(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)
(1 214 130 16 67 190 110 124)(2 215 131 17 68 191 111 125)(3 216 132 18 69 192 112 126)(4 217 133 19 70 193 99 113)(5 218 134 20 57 194 100 114)(6 219 135 21 58 195 101 115)(7 220 136 22 59 196 102 116)(8 221 137 23 60 183 103 117)(9 222 138 24 61 184 104 118)(10 223 139 25 62 185 105 119)(11 224 140 26 63 186 106 120)(12 211 127 27 64 187 107 121)(13 212 128 28 65 188 108 122)(14 213 129 15 66 189 109 123)(29 93 78 44 178 155 146 199)(30 94 79 45 179 156 147 200)(31 95 80 46 180 157 148 201)(32 96 81 47 181 158 149 202)(33 97 82 48 182 159 150 203)(34 98 83 49 169 160 151 204)(35 85 84 50 170 161 152 205)(36 86 71 51 171 162 153 206)(37 87 72 52 172 163 154 207)(38 88 73 53 173 164 141 208)(39 89 74 54 174 165 142 209)(40 90 75 55 175 166 143 210)(41 91 76 56 176 167 144 197)(42 92 77 43 177 168 145 198)
(1 181 130 81 67 32 110 149)(2 182 131 82 68 33 111 150)(3 169 132 83 69 34 112 151)(4 170 133 84 70 35 99 152)(5 171 134 71 57 36 100 153)(6 172 135 72 58 37 101 154)(7 173 136 73 59 38 102 141)(8 174 137 74 60 39 103 142)(9 175 138 75 61 40 104 143)(10 176 139 76 62 41 105 144)(11 177 140 77 63 42 106 145)(12 178 127 78 64 29 107 146)(13 179 128 79 65 30 108 147)(14 180 129 80 66 31 109 148)(15 95 189 201 123 157 213 46)(16 96 190 202 124 158 214 47)(17 97 191 203 125 159 215 48)(18 98 192 204 126 160 216 49)(19 85 193 205 113 161 217 50)(20 86 194 206 114 162 218 51)(21 87 195 207 115 163 219 52)(22 88 196 208 116 164 220 53)(23 89 183 209 117 165 221 54)(24 90 184 210 118 166 222 55)(25 91 185 197 119 167 223 56)(26 92 186 198 120 168 224 43)(27 93 187 199 121 155 211 44)(28 94 188 200 122 156 212 45)
G:=sub<Sym(224)| (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), (1,214,130,16,67,190,110,124)(2,215,131,17,68,191,111,125)(3,216,132,18,69,192,112,126)(4,217,133,19,70,193,99,113)(5,218,134,20,57,194,100,114)(6,219,135,21,58,195,101,115)(7,220,136,22,59,196,102,116)(8,221,137,23,60,183,103,117)(9,222,138,24,61,184,104,118)(10,223,139,25,62,185,105,119)(11,224,140,26,63,186,106,120)(12,211,127,27,64,187,107,121)(13,212,128,28,65,188,108,122)(14,213,129,15,66,189,109,123)(29,93,78,44,178,155,146,199)(30,94,79,45,179,156,147,200)(31,95,80,46,180,157,148,201)(32,96,81,47,181,158,149,202)(33,97,82,48,182,159,150,203)(34,98,83,49,169,160,151,204)(35,85,84,50,170,161,152,205)(36,86,71,51,171,162,153,206)(37,87,72,52,172,163,154,207)(38,88,73,53,173,164,141,208)(39,89,74,54,174,165,142,209)(40,90,75,55,175,166,143,210)(41,91,76,56,176,167,144,197)(42,92,77,43,177,168,145,198), (1,181,130,81,67,32,110,149)(2,182,131,82,68,33,111,150)(3,169,132,83,69,34,112,151)(4,170,133,84,70,35,99,152)(5,171,134,71,57,36,100,153)(6,172,135,72,58,37,101,154)(7,173,136,73,59,38,102,141)(8,174,137,74,60,39,103,142)(9,175,138,75,61,40,104,143)(10,176,139,76,62,41,105,144)(11,177,140,77,63,42,106,145)(12,178,127,78,64,29,107,146)(13,179,128,79,65,30,108,147)(14,180,129,80,66,31,109,148)(15,95,189,201,123,157,213,46)(16,96,190,202,124,158,214,47)(17,97,191,203,125,159,215,48)(18,98,192,204,126,160,216,49)(19,85,193,205,113,161,217,50)(20,86,194,206,114,162,218,51)(21,87,195,207,115,163,219,52)(22,88,196,208,116,164,220,53)(23,89,183,209,117,165,221,54)(24,90,184,210,118,166,222,55)(25,91,185,197,119,167,223,56)(26,92,186,198,120,168,224,43)(27,93,187,199,121,155,211,44)(28,94,188,200,122,156,212,45)>;
G:=Group( (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), (1,214,130,16,67,190,110,124)(2,215,131,17,68,191,111,125)(3,216,132,18,69,192,112,126)(4,217,133,19,70,193,99,113)(5,218,134,20,57,194,100,114)(6,219,135,21,58,195,101,115)(7,220,136,22,59,196,102,116)(8,221,137,23,60,183,103,117)(9,222,138,24,61,184,104,118)(10,223,139,25,62,185,105,119)(11,224,140,26,63,186,106,120)(12,211,127,27,64,187,107,121)(13,212,128,28,65,188,108,122)(14,213,129,15,66,189,109,123)(29,93,78,44,178,155,146,199)(30,94,79,45,179,156,147,200)(31,95,80,46,180,157,148,201)(32,96,81,47,181,158,149,202)(33,97,82,48,182,159,150,203)(34,98,83,49,169,160,151,204)(35,85,84,50,170,161,152,205)(36,86,71,51,171,162,153,206)(37,87,72,52,172,163,154,207)(38,88,73,53,173,164,141,208)(39,89,74,54,174,165,142,209)(40,90,75,55,175,166,143,210)(41,91,76,56,176,167,144,197)(42,92,77,43,177,168,145,198), (1,181,130,81,67,32,110,149)(2,182,131,82,68,33,111,150)(3,169,132,83,69,34,112,151)(4,170,133,84,70,35,99,152)(5,171,134,71,57,36,100,153)(6,172,135,72,58,37,101,154)(7,173,136,73,59,38,102,141)(8,174,137,74,60,39,103,142)(9,175,138,75,61,40,104,143)(10,176,139,76,62,41,105,144)(11,177,140,77,63,42,106,145)(12,178,127,78,64,29,107,146)(13,179,128,79,65,30,108,147)(14,180,129,80,66,31,109,148)(15,95,189,201,123,157,213,46)(16,96,190,202,124,158,214,47)(17,97,191,203,125,159,215,48)(18,98,192,204,126,160,216,49)(19,85,193,205,113,161,217,50)(20,86,194,206,114,162,218,51)(21,87,195,207,115,163,219,52)(22,88,196,208,116,164,220,53)(23,89,183,209,117,165,221,54)(24,90,184,210,118,166,222,55)(25,91,185,197,119,167,223,56)(26,92,186,198,120,168,224,43)(27,93,187,199,121,155,211,44)(28,94,188,200,122,156,212,45) );
G=PermutationGroup([[(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)], [(1,214,130,16,67,190,110,124),(2,215,131,17,68,191,111,125),(3,216,132,18,69,192,112,126),(4,217,133,19,70,193,99,113),(5,218,134,20,57,194,100,114),(6,219,135,21,58,195,101,115),(7,220,136,22,59,196,102,116),(8,221,137,23,60,183,103,117),(9,222,138,24,61,184,104,118),(10,223,139,25,62,185,105,119),(11,224,140,26,63,186,106,120),(12,211,127,27,64,187,107,121),(13,212,128,28,65,188,108,122),(14,213,129,15,66,189,109,123),(29,93,78,44,178,155,146,199),(30,94,79,45,179,156,147,200),(31,95,80,46,180,157,148,201),(32,96,81,47,181,158,149,202),(33,97,82,48,182,159,150,203),(34,98,83,49,169,160,151,204),(35,85,84,50,170,161,152,205),(36,86,71,51,171,162,153,206),(37,87,72,52,172,163,154,207),(38,88,73,53,173,164,141,208),(39,89,74,54,174,165,142,209),(40,90,75,55,175,166,143,210),(41,91,76,56,176,167,144,197),(42,92,77,43,177,168,145,198)], [(1,181,130,81,67,32,110,149),(2,182,131,82,68,33,111,150),(3,169,132,83,69,34,112,151),(4,170,133,84,70,35,99,152),(5,171,134,71,57,36,100,153),(6,172,135,72,58,37,101,154),(7,173,136,73,59,38,102,141),(8,174,137,74,60,39,103,142),(9,175,138,75,61,40,104,143),(10,176,139,76,62,41,105,144),(11,177,140,77,63,42,106,145),(12,178,127,78,64,29,107,146),(13,179,128,79,65,30,108,147),(14,180,129,80,66,31,109,148),(15,95,189,201,123,157,213,46),(16,96,190,202,124,158,214,47),(17,97,191,203,125,159,215,48),(18,98,192,204,126,160,216,49),(19,85,193,205,113,161,217,50),(20,86,194,206,114,162,218,51),(21,87,195,207,115,163,219,52),(22,88,196,208,116,164,220,53),(23,89,183,209,117,165,221,54),(24,90,184,210,118,166,222,55),(25,91,185,197,119,167,223,56),(26,92,186,198,120,168,224,43),(27,93,187,199,121,155,211,44),(28,94,188,200,122,156,212,45)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AJ | 56A | ··· | 56AV | 56AW | ··· | 56CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | D4 | Q8 | Q8 | C8.C4 | C7×D4 | C7×Q8 | C7×Q8 | C7×C8.C4 |
| kernel | C14×C8.C4 | C7×C8.C4 | C22×C56 | C14×M4(2) | C2×C56 | C2×C8.C4 | C8.C4 | C22×C8 | C2×M4(2) | C2×C8 | C2×C28 | C2×C28 | C22×C14 | C14 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 2 | 8 | 6 | 24 | 6 | 12 | 48 | 2 | 1 | 1 | 8 | 12 | 6 | 6 | 48 |
Matrix representation of C14×C8.C4 ►in GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 30 | 0 |
| 0 | 0 | 30 |
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 38 | 44 |
| 15 | 0 | 0 |
| 0 | 29 | 2 |
| 0 | 24 | 84 |
G:=sub<GL(3,GF(113))| [112,0,0,0,30,0,0,0,30],[1,0,0,0,18,38,0,0,44],[15,0,0,0,29,24,0,2,84] >;
C14×C8.C4 in GAP, Magma, Sage, TeX
C_{14}\times C_8.C_4 % in TeX
G:=Group("C14xC8.C4"); // GroupNames label
G:=SmallGroup(448,837);
// by ID
G=gap.SmallGroup(448,837);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,9804,172,124]);
// Polycyclic
G:=Group<a,b,c|a^14=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations