metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.2D4, C4.10D28, C14.4Q16, Dic14⋊3C4, C14.5SD16, C4⋊C4.3D7, C4.2(C4×D7), C28.4(C2×C4), C7⋊1(Q8⋊C4), (C2×C14).31D4, (C2×C4).36D14, C2.6(D14⋊C4), C2.2(D4.D7), C2.2(C7⋊Q16), C14.4(C22⋊C4), (C2×C28).11C22, (C2×Dic14).5C2, C22.15(C7⋊D4), (C2×C7⋊C8).3C2, (C7×C4⋊C4).3C2, SmallGroup(224,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.Q16
G = < a,b,c | a14=b8=1, c2=a7b4, bab-1=a-1, ac=ca, cbc-1=a7b-1 >
Smallest permutation representation of C14.Q16
►Regular action 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 68 99 147 174 91 131 124)(2 67 100 146 175 90 132 123)(3 66 101 145 176 89 133 122)(4 65 102 144 177 88 134 121)(5 64 103 143 178 87 135 120)(6 63 104 142 179 86 136 119)(7 62 105 141 180 85 137 118)(8 61 106 154 181 98 138 117)(9 60 107 153 182 97 139 116)(10 59 108 152 169 96 140 115)(11 58 109 151 170 95 127 114)(12 57 110 150 171 94 128 113)(13 70 111 149 172 93 129 126)(14 69 112 148 173 92 130 125)(15 189 33 223 77 157 209 45)(16 188 34 222 78 156 210 44)(17 187 35 221 79 155 197 43)(18 186 36 220 80 168 198 56)(19 185 37 219 81 167 199 55)(20 184 38 218 82 166 200 54)(21 183 39 217 83 165 201 53)(22 196 40 216 84 164 202 52)(23 195 41 215 71 163 203 51)(24 194 42 214 72 162 204 50)(25 193 29 213 73 161 205 49)(26 192 30 212 74 160 206 48)(27 191 31 211 75 159 207 47)(28 190 32 224 76 158 208 46)
(1 197 181 42)(2 198 182 29)(3 199 169 30)(4 200 170 31)(5 201 171 32)(6 202 172 33)(7 203 173 34)(8 204 174 35)(9 205 175 36)(10 206 176 37)(11 207 177 38)(12 208 178 39)(13 209 179 40)(14 210 180 41)(15 104 84 129)(16 105 71 130)(17 106 72 131)(18 107 73 132)(19 108 74 133)(20 109 75 134)(21 110 76 135)(22 111 77 136)(23 112 78 137)(24 99 79 138)(25 100 80 139)(26 101 81 140)(27 102 82 127)(28 103 83 128)(43 147 214 117)(44 148 215 118)(45 149 216 119)(46 150 217 120)(47 151 218 121)(48 152 219 122)(49 153 220 123)(50 154 221 124)(51 141 222 125)(52 142 223 126)(53 143 224 113)(54 144 211 114)(55 145 212 115)(56 146 213 116)(57 165 87 190)(58 166 88 191)(59 167 89 192)(60 168 90 193)(61 155 91 194)(62 156 92 195)(63 157 93 196)(64 158 94 183)(65 159 95 184)(66 160 96 185)(67 161 97 186)(68 162 98 187)(69 163 85 188)(70 164 86 189)
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), (1,68,99,147,174,91,131,124)(2,67,100,146,175,90,132,123)(3,66,101,145,176,89,133,122)(4,65,102,144,177,88,134,121)(5,64,103,143,178,87,135,120)(6,63,104,142,179,86,136,119)(7,62,105,141,180,85,137,118)(8,61,106,154,181,98,138,117)(9,60,107,153,182,97,139,116)(10,59,108,152,169,96,140,115)(11,58,109,151,170,95,127,114)(12,57,110,150,171,94,128,113)(13,70,111,149,172,93,129,126)(14,69,112,148,173,92,130,125)(15,189,33,223,77,157,209,45)(16,188,34,222,78,156,210,44)(17,187,35,221,79,155,197,43)(18,186,36,220,80,168,198,56)(19,185,37,219,81,167,199,55)(20,184,38,218,82,166,200,54)(21,183,39,217,83,165,201,53)(22,196,40,216,84,164,202,52)(23,195,41,215,71,163,203,51)(24,194,42,214,72,162,204,50)(25,193,29,213,73,161,205,49)(26,192,30,212,74,160,206,48)(27,191,31,211,75,159,207,47)(28,190,32,224,76,158,208,46), (1,197,181,42)(2,198,182,29)(3,199,169,30)(4,200,170,31)(5,201,171,32)(6,202,172,33)(7,203,173,34)(8,204,174,35)(9,205,175,36)(10,206,176,37)(11,207,177,38)(12,208,178,39)(13,209,179,40)(14,210,180,41)(15,104,84,129)(16,105,71,130)(17,106,72,131)(18,107,73,132)(19,108,74,133)(20,109,75,134)(21,110,76,135)(22,111,77,136)(23,112,78,137)(24,99,79,138)(25,100,80,139)(26,101,81,140)(27,102,82,127)(28,103,83,128)(43,147,214,117)(44,148,215,118)(45,149,216,119)(46,150,217,120)(47,151,218,121)(48,152,219,122)(49,153,220,123)(50,154,221,124)(51,141,222,125)(52,142,223,126)(53,143,224,113)(54,144,211,114)(55,145,212,115)(56,146,213,116)(57,165,87,190)(58,166,88,191)(59,167,89,192)(60,168,90,193)(61,155,91,194)(62,156,92,195)(63,157,93,196)(64,158,94,183)(65,159,95,184)(66,160,96,185)(67,161,97,186)(68,162,98,187)(69,163,85,188)(70,164,86,189)>;
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), (1,68,99,147,174,91,131,124)(2,67,100,146,175,90,132,123)(3,66,101,145,176,89,133,122)(4,65,102,144,177,88,134,121)(5,64,103,143,178,87,135,120)(6,63,104,142,179,86,136,119)(7,62,105,141,180,85,137,118)(8,61,106,154,181,98,138,117)(9,60,107,153,182,97,139,116)(10,59,108,152,169,96,140,115)(11,58,109,151,170,95,127,114)(12,57,110,150,171,94,128,113)(13,70,111,149,172,93,129,126)(14,69,112,148,173,92,130,125)(15,189,33,223,77,157,209,45)(16,188,34,222,78,156,210,44)(17,187,35,221,79,155,197,43)(18,186,36,220,80,168,198,56)(19,185,37,219,81,167,199,55)(20,184,38,218,82,166,200,54)(21,183,39,217,83,165,201,53)(22,196,40,216,84,164,202,52)(23,195,41,215,71,163,203,51)(24,194,42,214,72,162,204,50)(25,193,29,213,73,161,205,49)(26,192,30,212,74,160,206,48)(27,191,31,211,75,159,207,47)(28,190,32,224,76,158,208,46), (1,197,181,42)(2,198,182,29)(3,199,169,30)(4,200,170,31)(5,201,171,32)(6,202,172,33)(7,203,173,34)(8,204,174,35)(9,205,175,36)(10,206,176,37)(11,207,177,38)(12,208,178,39)(13,209,179,40)(14,210,180,41)(15,104,84,129)(16,105,71,130)(17,106,72,131)(18,107,73,132)(19,108,74,133)(20,109,75,134)(21,110,76,135)(22,111,77,136)(23,112,78,137)(24,99,79,138)(25,100,80,139)(26,101,81,140)(27,102,82,127)(28,103,83,128)(43,147,214,117)(44,148,215,118)(45,149,216,119)(46,150,217,120)(47,151,218,121)(48,152,219,122)(49,153,220,123)(50,154,221,124)(51,141,222,125)(52,142,223,126)(53,143,224,113)(54,144,211,114)(55,145,212,115)(56,146,213,116)(57,165,87,190)(58,166,88,191)(59,167,89,192)(60,168,90,193)(61,155,91,194)(62,156,92,195)(63,157,93,196)(64,158,94,183)(65,159,95,184)(66,160,96,185)(67,161,97,186)(68,162,98,187)(69,163,85,188)(70,164,86,189) );
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)], [(1,68,99,147,174,91,131,124),(2,67,100,146,175,90,132,123),(3,66,101,145,176,89,133,122),(4,65,102,144,177,88,134,121),(5,64,103,143,178,87,135,120),(6,63,104,142,179,86,136,119),(7,62,105,141,180,85,137,118),(8,61,106,154,181,98,138,117),(9,60,107,153,182,97,139,116),(10,59,108,152,169,96,140,115),(11,58,109,151,170,95,127,114),(12,57,110,150,171,94,128,113),(13,70,111,149,172,93,129,126),(14,69,112,148,173,92,130,125),(15,189,33,223,77,157,209,45),(16,188,34,222,78,156,210,44),(17,187,35,221,79,155,197,43),(18,186,36,220,80,168,198,56),(19,185,37,219,81,167,199,55),(20,184,38,218,82,166,200,54),(21,183,39,217,83,165,201,53),(22,196,40,216,84,164,202,52),(23,195,41,215,71,163,203,51),(24,194,42,214,72,162,204,50),(25,193,29,213,73,161,205,49),(26,192,30,212,74,160,206,48),(27,191,31,211,75,159,207,47),(28,190,32,224,76,158,208,46)], [(1,197,181,42),(2,198,182,29),(3,199,169,30),(4,200,170,31),(5,201,171,32),(6,202,172,33),(7,203,173,34),(8,204,174,35),(9,205,175,36),(10,206,176,37),(11,207,177,38),(12,208,178,39),(13,209,179,40),(14,210,180,41),(15,104,84,129),(16,105,71,130),(17,106,72,131),(18,107,73,132),(19,108,74,133),(20,109,75,134),(21,110,76,135),(22,111,77,136),(23,112,78,137),(24,99,79,138),(25,100,80,139),(26,101,81,140),(27,102,82,127),(28,103,83,128),(43,147,214,117),(44,148,215,118),(45,149,216,119),(46,150,217,120),(47,151,218,121),(48,152,219,122),(49,153,220,123),(50,154,221,124),(51,141,222,125),(52,142,223,126),(53,143,224,113),(54,144,211,114),(55,145,212,115),(56,146,213,116),(57,165,87,190),(58,166,88,191),(59,167,89,192),(60,168,90,193),(61,155,91,194),(62,156,92,195),(63,157,93,196),(64,158,94,183),(65,159,95,184),(66,160,96,185),(67,161,97,186),(68,162,98,187),(69,163,85,188),(70,164,86,189)]])
C14.Q16 is a maximal subgroup of
C28⋊Q8⋊C2 Dic14.D4 (C8×Dic7)⋊C2 D4⋊(C4×D7) D4⋊2D7⋊C4 D4⋊3D28 D4.D28 D28.D4 Dic7.1Q16 Dic7⋊Q16 C56⋊C4.C2 D7×Q8⋊C4 (Q8×D7)⋊C4 Q8⋊2D28 D14⋊4Q16 Dic7⋊SD16 Dic7⋊8SD16 Dic28⋊9C4 Dic14⋊Q8 Dic14.Q8 D14.2SD16 C8⋊8D28 C28.(C4○D4) C8.2D28 Dic28⋊6C4 Dic14⋊2Q8 Dic14.2Q8 D14.2Q16 C8⋊3D28 D14⋊2Q16 C2.D8⋊7D7 C56⋊C2⋊C4 C4○D28⋊C4 C4⋊C4.230D14 C4⋊C4.231D14 C4⋊C4.233D14 C4.(C2×D28) (C2×C4).47D28 D4.1D28 C4×D4.D7 C42.51D14 D4.2D28 Q8.1D28 C4×C7⋊Q16 C42.59D14 C28⋊7Q16 D28⋊17D4 Dic14⋊17D4 C7⋊C8⋊23D4 C7⋊C8⋊5D4 D28.37D4 Dic14.37D4 C7⋊C8.29D4 C7⋊C8.6D4 Dic14.4Q8 C42.216D14 C42.71D14 C42.82D14 Dic14⋊5Q8 C28.11Q16 Dic14⋊6Q8
C14.Q16 is a maximal quotient of
C4⋊Dic7⋊C4 C4.Dic28 Dic14⋊2C8 C28.2D8 C28.C42
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | SD16 | Q16 | D14 | C4×D7 | D28 | C7⋊D4 | D4.D7 | C7⋊Q16 |
| kernel | C14.Q16 | C2×C7⋊C8 | C7×C4⋊C4 | C2×Dic14 | Dic14 | C28 | C2×C14 | C4⋊C4 | C14 | C14 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 3 | 3 |
Matrix representation of C14.Q16 ►in GL4(𝔽113) generated by
| 25 | 25 | 0 | 0 |
| 88 | 79 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 105 | 0 | 0 |
| 91 | 86 | 0 | 0 |
| 0 | 0 | 0 | 105 |
| 0 | 0 | 99 | 51 |
| 104 | 46 | 0 | 0 |
| 67 | 9 | 0 | 0 |
| 0 | 0 | 30 | 10 |
| 0 | 0 | 102 | 83 |
G:=sub<GL(4,GF(113))| [25,88,0,0,25,79,0,0,0,0,1,0,0,0,0,1],[27,91,0,0,105,86,0,0,0,0,0,99,0,0,105,51],[104,67,0,0,46,9,0,0,0,0,30,102,0,0,10,83] >;
C14.Q16 in GAP, Magma, Sage, TeX
C_{14}.Q_{16} % in TeX
G:=Group("C14.Q16"); // GroupNames label
G:=SmallGroup(224,16);
// by ID
G=gap.SmallGroup(224,16);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,121,31,579,297,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^14=b^8=1,c^2=a^7*b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^7*b^-1>;
// generators/relations
Export