metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.92D4, M4(2).3Dic7, D4.(C7⋊C8), Q8.(C7⋊C8), C28.9(C2×C8), C7⋊3(D4.C8), C8○D4.2D7, (C7×D4).1C8, (C7×Q8).1C8, (C2×C8).268D14, C8.34(C7⋊D4), C4○D4.1Dic7, C28.C8⋊12C2, (C7×M4(2)).5C4, (C2×C14).6M4(2), C28.96(C22⋊C4), C14.19(C22⋊C8), (C2×C56).223C22, C4.30(C23.D7), C2.8(C28.55D4), C22.1(C4.Dic7), C4.3(C2×C7⋊C8), (C2×C7⋊C16)⋊14C2, (C7×C8○D4).2C2, (C7×C4○D4).1C4, (C2×C28).67(C2×C4), (C2×C4).41(C2×Dic7), SmallGroup(448,118)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.92D4
G = < a,b,c | a56=1, b4=a14, c2=a49, bab-1=cac-1=a41, cbc-1=a35b3 >
Smallest permutation representation of C56.92D4
►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)
(1 211 36 190 15 169 50 204 29 183 8 218 43 197 22 176)(2 196 37 175 16 210 51 189 30 224 9 203 44 182 23 217)(3 181 38 216 17 195 52 174 31 209 10 188 45 223 24 202)(4 222 39 201 18 180 53 215 32 194 11 173 46 208 25 187)(5 207 40 186 19 221 54 200 33 179 12 214 47 193 26 172)(6 192 41 171 20 206 55 185 34 220 13 199 48 178 27 213)(7 177 42 212 21 191 56 170 35 205 14 184 49 219 28 198)(57 134 64 141 71 148 78 155 85 162 92 113 99 120 106 127)(58 119 65 126 72 133 79 140 86 147 93 154 100 161 107 168)(59 160 66 167 73 118 80 125 87 132 94 139 101 146 108 153)(60 145 67 152 74 159 81 166 88 117 95 124 102 131 109 138)(61 130 68 137 75 144 82 151 89 158 96 165 103 116 110 123)(62 115 69 122 76 129 83 136 90 143 97 150 104 157 111 164)(63 156 70 163 77 114 84 121 91 128 98 135 105 142 112 149)
(1 155 50 148 43 141 36 134 29 127 22 120 15 113 8 162)(2 140 51 133 44 126 37 119 30 168 23 161 16 154 9 147)(3 125 52 118 45 167 38 160 31 153 24 146 17 139 10 132)(4 166 53 159 46 152 39 145 32 138 25 131 18 124 11 117)(5 151 54 144 47 137 40 130 33 123 26 116 19 165 12 158)(6 136 55 129 48 122 41 115 34 164 27 157 20 150 13 143)(7 121 56 114 49 163 42 156 35 149 28 142 21 135 14 128)(57 190 106 183 99 176 92 169 85 218 78 211 71 204 64 197)(58 175 107 224 100 217 93 210 86 203 79 196 72 189 65 182)(59 216 108 209 101 202 94 195 87 188 80 181 73 174 66 223)(60 201 109 194 102 187 95 180 88 173 81 222 74 215 67 208)(61 186 110 179 103 172 96 221 89 214 82 207 75 200 68 193)(62 171 111 220 104 213 97 206 90 199 83 192 76 185 69 178)(63 212 112 205 105 198 98 191 91 184 84 177 77 170 70 219)
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), (1,211,36,190,15,169,50,204,29,183,8,218,43,197,22,176)(2,196,37,175,16,210,51,189,30,224,9,203,44,182,23,217)(3,181,38,216,17,195,52,174,31,209,10,188,45,223,24,202)(4,222,39,201,18,180,53,215,32,194,11,173,46,208,25,187)(5,207,40,186,19,221,54,200,33,179,12,214,47,193,26,172)(6,192,41,171,20,206,55,185,34,220,13,199,48,178,27,213)(7,177,42,212,21,191,56,170,35,205,14,184,49,219,28,198)(57,134,64,141,71,148,78,155,85,162,92,113,99,120,106,127)(58,119,65,126,72,133,79,140,86,147,93,154,100,161,107,168)(59,160,66,167,73,118,80,125,87,132,94,139,101,146,108,153)(60,145,67,152,74,159,81,166,88,117,95,124,102,131,109,138)(61,130,68,137,75,144,82,151,89,158,96,165,103,116,110,123)(62,115,69,122,76,129,83,136,90,143,97,150,104,157,111,164)(63,156,70,163,77,114,84,121,91,128,98,135,105,142,112,149), (1,155,50,148,43,141,36,134,29,127,22,120,15,113,8,162)(2,140,51,133,44,126,37,119,30,168,23,161,16,154,9,147)(3,125,52,118,45,167,38,160,31,153,24,146,17,139,10,132)(4,166,53,159,46,152,39,145,32,138,25,131,18,124,11,117)(5,151,54,144,47,137,40,130,33,123,26,116,19,165,12,158)(6,136,55,129,48,122,41,115,34,164,27,157,20,150,13,143)(7,121,56,114,49,163,42,156,35,149,28,142,21,135,14,128)(57,190,106,183,99,176,92,169,85,218,78,211,71,204,64,197)(58,175,107,224,100,217,93,210,86,203,79,196,72,189,65,182)(59,216,108,209,101,202,94,195,87,188,80,181,73,174,66,223)(60,201,109,194,102,187,95,180,88,173,81,222,74,215,67,208)(61,186,110,179,103,172,96,221,89,214,82,207,75,200,68,193)(62,171,111,220,104,213,97,206,90,199,83,192,76,185,69,178)(63,212,112,205,105,198,98,191,91,184,84,177,77,170,70,219)>;
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), (1,211,36,190,15,169,50,204,29,183,8,218,43,197,22,176)(2,196,37,175,16,210,51,189,30,224,9,203,44,182,23,217)(3,181,38,216,17,195,52,174,31,209,10,188,45,223,24,202)(4,222,39,201,18,180,53,215,32,194,11,173,46,208,25,187)(5,207,40,186,19,221,54,200,33,179,12,214,47,193,26,172)(6,192,41,171,20,206,55,185,34,220,13,199,48,178,27,213)(7,177,42,212,21,191,56,170,35,205,14,184,49,219,28,198)(57,134,64,141,71,148,78,155,85,162,92,113,99,120,106,127)(58,119,65,126,72,133,79,140,86,147,93,154,100,161,107,168)(59,160,66,167,73,118,80,125,87,132,94,139,101,146,108,153)(60,145,67,152,74,159,81,166,88,117,95,124,102,131,109,138)(61,130,68,137,75,144,82,151,89,158,96,165,103,116,110,123)(62,115,69,122,76,129,83,136,90,143,97,150,104,157,111,164)(63,156,70,163,77,114,84,121,91,128,98,135,105,142,112,149), (1,155,50,148,43,141,36,134,29,127,22,120,15,113,8,162)(2,140,51,133,44,126,37,119,30,168,23,161,16,154,9,147)(3,125,52,118,45,167,38,160,31,153,24,146,17,139,10,132)(4,166,53,159,46,152,39,145,32,138,25,131,18,124,11,117)(5,151,54,144,47,137,40,130,33,123,26,116,19,165,12,158)(6,136,55,129,48,122,41,115,34,164,27,157,20,150,13,143)(7,121,56,114,49,163,42,156,35,149,28,142,21,135,14,128)(57,190,106,183,99,176,92,169,85,218,78,211,71,204,64,197)(58,175,107,224,100,217,93,210,86,203,79,196,72,189,65,182)(59,216,108,209,101,202,94,195,87,188,80,181,73,174,66,223)(60,201,109,194,102,187,95,180,88,173,81,222,74,215,67,208)(61,186,110,179,103,172,96,221,89,214,82,207,75,200,68,193)(62,171,111,220,104,213,97,206,90,199,83,192,76,185,69,178)(63,212,112,205,105,198,98,191,91,184,84,177,77,170,70,219) );
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)], [(1,211,36,190,15,169,50,204,29,183,8,218,43,197,22,176),(2,196,37,175,16,210,51,189,30,224,9,203,44,182,23,217),(3,181,38,216,17,195,52,174,31,209,10,188,45,223,24,202),(4,222,39,201,18,180,53,215,32,194,11,173,46,208,25,187),(5,207,40,186,19,221,54,200,33,179,12,214,47,193,26,172),(6,192,41,171,20,206,55,185,34,220,13,199,48,178,27,213),(7,177,42,212,21,191,56,170,35,205,14,184,49,219,28,198),(57,134,64,141,71,148,78,155,85,162,92,113,99,120,106,127),(58,119,65,126,72,133,79,140,86,147,93,154,100,161,107,168),(59,160,66,167,73,118,80,125,87,132,94,139,101,146,108,153),(60,145,67,152,74,159,81,166,88,117,95,124,102,131,109,138),(61,130,68,137,75,144,82,151,89,158,96,165,103,116,110,123),(62,115,69,122,76,129,83,136,90,143,97,150,104,157,111,164),(63,156,70,163,77,114,84,121,91,128,98,135,105,142,112,149)], [(1,155,50,148,43,141,36,134,29,127,22,120,15,113,8,162),(2,140,51,133,44,126,37,119,30,168,23,161,16,154,9,147),(3,125,52,118,45,167,38,160,31,153,24,146,17,139,10,132),(4,166,53,159,46,152,39,145,32,138,25,131,18,124,11,117),(5,151,54,144,47,137,40,130,33,123,26,116,19,165,12,158),(6,136,55,129,48,122,41,115,34,164,27,157,20,150,13,143),(7,121,56,114,49,163,42,156,35,149,28,142,21,135,14,128),(57,190,106,183,99,176,92,169,85,218,78,211,71,204,64,197),(58,175,107,224,100,217,93,210,86,203,79,196,72,189,65,182),(59,216,108,209,101,202,94,195,87,188,80,181,73,174,66,223),(60,201,109,194,102,187,95,180,88,173,81,222,74,215,67,208),(61,186,110,179,103,172,96,221,89,214,82,207,75,200,68,193),(62,171,111,220,104,213,97,206,90,199,83,192,76,185,69,178),(63,212,112,205,105,198,98,191,91,184,84,177,77,170,70,219)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | 14B | 14C | 14D | ··· | 14L | 16A | ··· | 16H | 16I | 16J | 16K | 16L | 28A | ··· | 28F | 28G | ··· | 28O | 56A | ··· | 56L | 56M | ··· | 56AD |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 16 | ··· | 16 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | D7 | M4(2) | D14 | Dic7 | Dic7 | C7⋊D4 | C7⋊C8 | C7⋊C8 | D4.C8 | C4.Dic7 | C56.92D4 |
| kernel | C56.92D4 | C2×C7⋊C16 | C28.C8 | C7×C8○D4 | C7×M4(2) | C7×C4○D4 | C7×D4 | C7×Q8 | C56 | C8○D4 | C2×C14 | C2×C8 | M4(2) | C4○D4 | C8 | D4 | Q8 | C7 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 3 | 2 | 3 | 3 | 3 | 12 | 6 | 6 | 8 | 12 | 12 |
Matrix representation of C56.92D4 ►in GL4(𝔽113) generated by
| 98 | 15 | 0 | 0 |
| 62 | 36 | 0 | 0 |
| 0 | 0 | 44 | 0 |
| 0 | 0 | 0 | 44 |
| 67 | 101 | 0 | 0 |
| 74 | 46 | 0 | 0 |
| 0 | 0 | 78 | 0 |
| 0 | 0 | 67 | 73 |
| 42 | 55 | 0 | 0 |
| 36 | 71 | 0 | 0 |
| 0 | 0 | 73 | 89 |
| 0 | 0 | 46 | 40 |
G:=sub<GL(4,GF(113))| [98,62,0,0,15,36,0,0,0,0,44,0,0,0,0,44],[67,74,0,0,101,46,0,0,0,0,78,67,0,0,0,73],[42,36,0,0,55,71,0,0,0,0,73,46,0,0,89,40] >;
C56.92D4 in GAP, Magma, Sage, TeX
C_{56}._{92}D_4 % in TeX
G:=Group("C56.92D4"); // GroupNames label
G:=SmallGroup(448,118);
// by ID
G=gap.SmallGroup(448,118);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,100,1123,570,136,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=1,b^4=a^14,c^2=a^49,b*a*b^-1=c*a*c^-1=a^41,c*b*c^-1=a^35*b^3>;
// generators/relations
Export