metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.7Q8, C28.55D8, C8.7Dic14, C7⋊C16.1C4, C8.26(C4×D7), C28.4(C4⋊C4), C56.11(C2×C4), (C2×C28).93D4, (C2×C14).2Q16, C7⋊1(C8.4Q8), C4.28(D4⋊D7), (C2×C8).248D14, C14.4(C2.D8), C8.C4.1D7, C56.C4.3C2, C4.4(Dic7⋊C4), (C2×C56).36C22, C2.5(C28.Q8), C22.1(C7⋊Q16), (C2×C7⋊C16).1C2, (C7×C8.C4).1C2, (C2×C4).117(C7⋊D4), SmallGroup(448,50)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.7Dic14
G = < a,b,c | a8=1, b28=a4, c2=ab14, bab-1=a3, ac=ca, cbc-1=a5b27 >
Smallest permutation representation of C8.7Dic14
►On 224 points
Generators in S224
(1 214 15 172 29 186 43 200)(2 173 44 215 30 201 16 187)(3 216 17 174 31 188 45 202)(4 175 46 217 32 203 18 189)(5 218 19 176 33 190 47 204)(6 177 48 219 34 205 20 191)(7 220 21 178 35 192 49 206)(8 179 50 221 36 207 22 193)(9 222 23 180 37 194 51 208)(10 181 52 223 38 209 24 195)(11 224 25 182 39 196 53 210)(12 183 54 169 40 211 26 197)(13 170 27 184 41 198 55 212)(14 185 56 171 42 213 28 199)(57 146 71 160 85 118 99 132)(58 161 100 147 86 133 72 119)(59 148 73 162 87 120 101 134)(60 163 102 149 88 135 74 121)(61 150 75 164 89 122 103 136)(62 165 104 151 90 137 76 123)(63 152 77 166 91 124 105 138)(64 167 106 153 92 139 78 125)(65 154 79 168 93 126 107 140)(66 113 108 155 94 141 80 127)(67 156 81 114 95 128 109 142)(68 115 110 157 96 143 82 129)(69 158 83 116 97 130 111 144)(70 117 112 159 98 145 84 131)
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 85 172 132 43 71 214 118 29 57 200 160 15 99 186 146)(2 131 187 84 16 145 201 98 30 159 215 112 44 117 173 70)(3 111 174 158 45 97 216 144 31 83 202 130 17 69 188 116)(4 157 189 110 18 115 203 68 32 129 217 82 46 143 175 96)(5 81 176 128 47 67 218 114 33 109 204 156 19 95 190 142)(6 127 191 80 20 141 205 94 34 155 219 108 48 113 177 66)(7 107 178 154 49 93 220 140 35 79 206 126 21 65 192 168)(8 153 193 106 22 167 207 64 36 125 221 78 50 139 179 92)(9 77 180 124 51 63 222 166 37 105 208 152 23 91 194 138)(10 123 195 76 24 137 209 90 38 151 223 104 52 165 181 62)(11 103 182 150 53 89 224 136 39 75 210 122 25 61 196 164)(12 149 197 102 26 163 211 60 40 121 169 74 54 135 183 88)(13 73 184 120 55 59 170 162 41 101 212 148 27 87 198 134)(14 119 199 72 28 133 213 86 42 147 171 100 56 161 185 58)
G:=sub<Sym(224)| (1,214,15,172,29,186,43,200)(2,173,44,215,30,201,16,187)(3,216,17,174,31,188,45,202)(4,175,46,217,32,203,18,189)(5,218,19,176,33,190,47,204)(6,177,48,219,34,205,20,191)(7,220,21,178,35,192,49,206)(8,179,50,221,36,207,22,193)(9,222,23,180,37,194,51,208)(10,181,52,223,38,209,24,195)(11,224,25,182,39,196,53,210)(12,183,54,169,40,211,26,197)(13,170,27,184,41,198,55,212)(14,185,56,171,42,213,28,199)(57,146,71,160,85,118,99,132)(58,161,100,147,86,133,72,119)(59,148,73,162,87,120,101,134)(60,163,102,149,88,135,74,121)(61,150,75,164,89,122,103,136)(62,165,104,151,90,137,76,123)(63,152,77,166,91,124,105,138)(64,167,106,153,92,139,78,125)(65,154,79,168,93,126,107,140)(66,113,108,155,94,141,80,127)(67,156,81,114,95,128,109,142)(68,115,110,157,96,143,82,129)(69,158,83,116,97,130,111,144)(70,117,112,159,98,145,84,131), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,85,172,132,43,71,214,118,29,57,200,160,15,99,186,146)(2,131,187,84,16,145,201,98,30,159,215,112,44,117,173,70)(3,111,174,158,45,97,216,144,31,83,202,130,17,69,188,116)(4,157,189,110,18,115,203,68,32,129,217,82,46,143,175,96)(5,81,176,128,47,67,218,114,33,109,204,156,19,95,190,142)(6,127,191,80,20,141,205,94,34,155,219,108,48,113,177,66)(7,107,178,154,49,93,220,140,35,79,206,126,21,65,192,168)(8,153,193,106,22,167,207,64,36,125,221,78,50,139,179,92)(9,77,180,124,51,63,222,166,37,105,208,152,23,91,194,138)(10,123,195,76,24,137,209,90,38,151,223,104,52,165,181,62)(11,103,182,150,53,89,224,136,39,75,210,122,25,61,196,164)(12,149,197,102,26,163,211,60,40,121,169,74,54,135,183,88)(13,73,184,120,55,59,170,162,41,101,212,148,27,87,198,134)(14,119,199,72,28,133,213,86,42,147,171,100,56,161,185,58)>;
G:=Group( (1,214,15,172,29,186,43,200)(2,173,44,215,30,201,16,187)(3,216,17,174,31,188,45,202)(4,175,46,217,32,203,18,189)(5,218,19,176,33,190,47,204)(6,177,48,219,34,205,20,191)(7,220,21,178,35,192,49,206)(8,179,50,221,36,207,22,193)(9,222,23,180,37,194,51,208)(10,181,52,223,38,209,24,195)(11,224,25,182,39,196,53,210)(12,183,54,169,40,211,26,197)(13,170,27,184,41,198,55,212)(14,185,56,171,42,213,28,199)(57,146,71,160,85,118,99,132)(58,161,100,147,86,133,72,119)(59,148,73,162,87,120,101,134)(60,163,102,149,88,135,74,121)(61,150,75,164,89,122,103,136)(62,165,104,151,90,137,76,123)(63,152,77,166,91,124,105,138)(64,167,106,153,92,139,78,125)(65,154,79,168,93,126,107,140)(66,113,108,155,94,141,80,127)(67,156,81,114,95,128,109,142)(68,115,110,157,96,143,82,129)(69,158,83,116,97,130,111,144)(70,117,112,159,98,145,84,131), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,85,172,132,43,71,214,118,29,57,200,160,15,99,186,146)(2,131,187,84,16,145,201,98,30,159,215,112,44,117,173,70)(3,111,174,158,45,97,216,144,31,83,202,130,17,69,188,116)(4,157,189,110,18,115,203,68,32,129,217,82,46,143,175,96)(5,81,176,128,47,67,218,114,33,109,204,156,19,95,190,142)(6,127,191,80,20,141,205,94,34,155,219,108,48,113,177,66)(7,107,178,154,49,93,220,140,35,79,206,126,21,65,192,168)(8,153,193,106,22,167,207,64,36,125,221,78,50,139,179,92)(9,77,180,124,51,63,222,166,37,105,208,152,23,91,194,138)(10,123,195,76,24,137,209,90,38,151,223,104,52,165,181,62)(11,103,182,150,53,89,224,136,39,75,210,122,25,61,196,164)(12,149,197,102,26,163,211,60,40,121,169,74,54,135,183,88)(13,73,184,120,55,59,170,162,41,101,212,148,27,87,198,134)(14,119,199,72,28,133,213,86,42,147,171,100,56,161,185,58) );
G=PermutationGroup([[(1,214,15,172,29,186,43,200),(2,173,44,215,30,201,16,187),(3,216,17,174,31,188,45,202),(4,175,46,217,32,203,18,189),(5,218,19,176,33,190,47,204),(6,177,48,219,34,205,20,191),(7,220,21,178,35,192,49,206),(8,179,50,221,36,207,22,193),(9,222,23,180,37,194,51,208),(10,181,52,223,38,209,24,195),(11,224,25,182,39,196,53,210),(12,183,54,169,40,211,26,197),(13,170,27,184,41,198,55,212),(14,185,56,171,42,213,28,199),(57,146,71,160,85,118,99,132),(58,161,100,147,86,133,72,119),(59,148,73,162,87,120,101,134),(60,163,102,149,88,135,74,121),(61,150,75,164,89,122,103,136),(62,165,104,151,90,137,76,123),(63,152,77,166,91,124,105,138),(64,167,106,153,92,139,78,125),(65,154,79,168,93,126,107,140),(66,113,108,155,94,141,80,127),(67,156,81,114,95,128,109,142),(68,115,110,157,96,143,82,129),(69,158,83,116,97,130,111,144),(70,117,112,159,98,145,84,131)], [(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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,85,172,132,43,71,214,118,29,57,200,160,15,99,186,146),(2,131,187,84,16,145,201,98,30,159,215,112,44,117,173,70),(3,111,174,158,45,97,216,144,31,83,202,130,17,69,188,116),(4,157,189,110,18,115,203,68,32,129,217,82,46,143,175,96),(5,81,176,128,47,67,218,114,33,109,204,156,19,95,190,142),(6,127,191,80,20,141,205,94,34,155,219,108,48,113,177,66),(7,107,178,154,49,93,220,140,35,79,206,126,21,65,192,168),(8,153,193,106,22,167,207,64,36,125,221,78,50,139,179,92),(9,77,180,124,51,63,222,166,37,105,208,152,23,91,194,138),(10,123,195,76,24,137,209,90,38,151,223,104,52,165,181,62),(11,103,182,150,53,89,224,136,39,75,210,122,25,61,196,164),(12,149,197,102,26,163,211,60,40,121,169,74,54,135,183,88),(13,73,184,120,55,59,170,162,41,101,212,148,27,87,198,134),(14,119,199,72,28,133,213,86,42,147,171,100,56,161,185,58)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | 14B | 14C | 14D | 14E | 14F | 16A | ··· | 16H | 28A | ··· | 28F | 28G | 28H | 28I | 56A | ··· | 56L | 56M | ··· | 56X |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | 14 | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 28 | 28 | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | Q8 | D4 | D7 | D8 | Q16 | D14 | Dic14 | C4×D7 | C7⋊D4 | C8.4Q8 | D4⋊D7 | C7⋊Q16 | C8.7Dic14 |
| kernel | C8.7Dic14 | C2×C7⋊C16 | C56.C4 | C7×C8.C4 | C7⋊C16 | C56 | C2×C28 | C8.C4 | C28 | C2×C14 | C2×C8 | C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 3 | 6 | 6 | 6 | 8 | 3 | 3 | 12 |
Matrix representation of C8.7Dic14 ►in GL4(𝔽113) generated by
| 18 | 0 | 0 | 0 |
| 69 | 69 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 63 | 22 | 0 | 0 |
| 36 | 50 | 0 | 0 |
| 0 | 0 | 34 | 91 |
| 0 | 0 | 87 | 60 |
| 42 | 0 | 0 | 0 |
| 57 | 35 | 0 | 0 |
| 0 | 0 | 32 | 35 |
| 0 | 0 | 3 | 81 |
G:=sub<GL(4,GF(113))| [18,69,0,0,0,69,0,0,0,0,112,0,0,0,0,112],[63,36,0,0,22,50,0,0,0,0,34,87,0,0,91,60],[42,57,0,0,0,35,0,0,0,0,32,3,0,0,35,81] >;
C8.7Dic14 in GAP, Magma, Sage, TeX
C_8._7{\rm Dic}_{14} % in TeX
G:=Group("C8.7Dic14"); // GroupNames label
G:=SmallGroup(448,50);
// by ID
G=gap.SmallGroup(448,50);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,141,36,184,346,192,851,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^8=1,b^28=a^4,c^2=a*b^14,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=a^5*b^27>;
// generators/relations
Export