metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.91D4, C14.6M5(2), C28.29M4(2), C22⋊(C7⋊C16), (C2×C14)⋊2C16, (C2×C28).5C8, C7⋊2(C22⋊C16), (C2×C56).11C4, C23.2(C7⋊C8), (C2×C8).6Dic7, (C22×C8).1D7, C14.10(C2×C16), (C2×C8).336D14, C8.52(C7⋊D4), (C22×C14).5C8, (C22×C56).20C2, (C22×C28).29C4, C2.3(C28.C8), (C22×C4).8Dic7, C14.17(C22⋊C8), C28.92(C22⋊C4), (C2×C56).403C22, C4.11(C4.Dic7), C4.24(C23.D7), C2.2(C28.55D4), C2.5(C2×C7⋊C16), (C2×C7⋊C16)⋊10C2, (C2×C4).4(C7⋊C8), C22.11(C2×C7⋊C8), (C2×C14).31(C2×C8), (C2×C28).312(C2×C4), (C2×C4).93(C2×Dic7), SmallGroup(448,106)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.91D4
G = < a,b,c | a56=1, b4=a42, c2=a49, bab-1=cac-1=a41, cbc-1=a7b3 >
Subgroups: 132 in 66 conjugacy classes, 39 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C14, C14, C16, C2×C8, C2×C8, C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C2×C16, C22×C8, C56, C56, C2×C28, C2×C28, C22×C14, C22⋊C16, C7⋊C16, C2×C56, C2×C56, C22×C28, C2×C7⋊C16, C22×C56, C56.91D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D7, C16, C22⋊C4, C2×C8, M4(2), Dic7, D14, C22⋊C8, C2×C16, M5(2), C7⋊C8, C2×Dic7, C7⋊D4, C22⋊C16, C7⋊C16, C2×C7⋊C8, C4.Dic7, C23.D7, C2×C7⋊C16, C28.C8, C28.55D4, C56.91D4
►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)
(1 149 184 58 43 135 170 100 29 121 212 86 15 163 198 72)(2 134 185 99 44 120 171 85 30 162 213 71 16 148 199 57)(3 119 186 84 45 161 172 70 31 147 214 112 17 133 200 98)(4 160 187 69 46 146 173 111 32 132 215 97 18 118 201 83)(5 145 188 110 47 131 174 96 33 117 216 82 19 159 202 68)(6 130 189 95 48 116 175 81 34 158 217 67 20 144 203 109)(7 115 190 80 49 157 176 66 35 143 218 108 21 129 204 94)(8 156 191 65 50 142 177 107 36 128 219 93 22 114 205 79)(9 141 192 106 51 127 178 92 37 113 220 78 23 155 206 64)(10 126 193 91 52 168 179 77 38 154 221 63 24 140 207 105)(11 167 194 76 53 153 180 62 39 139 222 104 25 125 208 90)(12 152 195 61 54 138 181 103 40 124 223 89 26 166 209 75)(13 137 196 102 55 123 182 88 41 165 224 74 27 151 210 60)(14 122 197 87 56 164 183 73 42 150 169 59 28 136 211 101)
(1 65 50 58 43 107 36 100 29 93 22 86 15 79 8 72)(2 106 51 99 44 92 37 85 30 78 23 71 16 64 9 57)(3 91 52 84 45 77 38 70 31 63 24 112 17 105 10 98)(4 76 53 69 46 62 39 111 32 104 25 97 18 90 11 83)(5 61 54 110 47 103 40 96 33 89 26 82 19 75 12 68)(6 102 55 95 48 88 41 81 34 74 27 67 20 60 13 109)(7 87 56 80 49 73 42 66 35 59 28 108 21 101 14 94)(113 220 162 213 155 206 148 199 141 192 134 185 127 178 120 171)(114 205 163 198 156 191 149 184 142 177 135 170 128 219 121 212)(115 190 164 183 157 176 150 169 143 218 136 211 129 204 122 197)(116 175 165 224 158 217 151 210 144 203 137 196 130 189 123 182)(117 216 166 209 159 202 152 195 145 188 138 181 131 174 124 223)(118 201 167 194 160 187 153 180 146 173 139 222 132 215 125 208)(119 186 168 179 161 172 154 221 147 214 140 207 133 200 126 193)
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), (1,149,184,58,43,135,170,100,29,121,212,86,15,163,198,72)(2,134,185,99,44,120,171,85,30,162,213,71,16,148,199,57)(3,119,186,84,45,161,172,70,31,147,214,112,17,133,200,98)(4,160,187,69,46,146,173,111,32,132,215,97,18,118,201,83)(5,145,188,110,47,131,174,96,33,117,216,82,19,159,202,68)(6,130,189,95,48,116,175,81,34,158,217,67,20,144,203,109)(7,115,190,80,49,157,176,66,35,143,218,108,21,129,204,94)(8,156,191,65,50,142,177,107,36,128,219,93,22,114,205,79)(9,141,192,106,51,127,178,92,37,113,220,78,23,155,206,64)(10,126,193,91,52,168,179,77,38,154,221,63,24,140,207,105)(11,167,194,76,53,153,180,62,39,139,222,104,25,125,208,90)(12,152,195,61,54,138,181,103,40,124,223,89,26,166,209,75)(13,137,196,102,55,123,182,88,41,165,224,74,27,151,210,60)(14,122,197,87,56,164,183,73,42,150,169,59,28,136,211,101), (1,65,50,58,43,107,36,100,29,93,22,86,15,79,8,72)(2,106,51,99,44,92,37,85,30,78,23,71,16,64,9,57)(3,91,52,84,45,77,38,70,31,63,24,112,17,105,10,98)(4,76,53,69,46,62,39,111,32,104,25,97,18,90,11,83)(5,61,54,110,47,103,40,96,33,89,26,82,19,75,12,68)(6,102,55,95,48,88,41,81,34,74,27,67,20,60,13,109)(7,87,56,80,49,73,42,66,35,59,28,108,21,101,14,94)(113,220,162,213,155,206,148,199,141,192,134,185,127,178,120,171)(114,205,163,198,156,191,149,184,142,177,135,170,128,219,121,212)(115,190,164,183,157,176,150,169,143,218,136,211,129,204,122,197)(116,175,165,224,158,217,151,210,144,203,137,196,130,189,123,182)(117,216,166,209,159,202,152,195,145,188,138,181,131,174,124,223)(118,201,167,194,160,187,153,180,146,173,139,222,132,215,125,208)(119,186,168,179,161,172,154,221,147,214,140,207,133,200,126,193)>;
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), (1,149,184,58,43,135,170,100,29,121,212,86,15,163,198,72)(2,134,185,99,44,120,171,85,30,162,213,71,16,148,199,57)(3,119,186,84,45,161,172,70,31,147,214,112,17,133,200,98)(4,160,187,69,46,146,173,111,32,132,215,97,18,118,201,83)(5,145,188,110,47,131,174,96,33,117,216,82,19,159,202,68)(6,130,189,95,48,116,175,81,34,158,217,67,20,144,203,109)(7,115,190,80,49,157,176,66,35,143,218,108,21,129,204,94)(8,156,191,65,50,142,177,107,36,128,219,93,22,114,205,79)(9,141,192,106,51,127,178,92,37,113,220,78,23,155,206,64)(10,126,193,91,52,168,179,77,38,154,221,63,24,140,207,105)(11,167,194,76,53,153,180,62,39,139,222,104,25,125,208,90)(12,152,195,61,54,138,181,103,40,124,223,89,26,166,209,75)(13,137,196,102,55,123,182,88,41,165,224,74,27,151,210,60)(14,122,197,87,56,164,183,73,42,150,169,59,28,136,211,101), (1,65,50,58,43,107,36,100,29,93,22,86,15,79,8,72)(2,106,51,99,44,92,37,85,30,78,23,71,16,64,9,57)(3,91,52,84,45,77,38,70,31,63,24,112,17,105,10,98)(4,76,53,69,46,62,39,111,32,104,25,97,18,90,11,83)(5,61,54,110,47,103,40,96,33,89,26,82,19,75,12,68)(6,102,55,95,48,88,41,81,34,74,27,67,20,60,13,109)(7,87,56,80,49,73,42,66,35,59,28,108,21,101,14,94)(113,220,162,213,155,206,148,199,141,192,134,185,127,178,120,171)(114,205,163,198,156,191,149,184,142,177,135,170,128,219,121,212)(115,190,164,183,157,176,150,169,143,218,136,211,129,204,122,197)(116,175,165,224,158,217,151,210,144,203,137,196,130,189,123,182)(117,216,166,209,159,202,152,195,145,188,138,181,131,174,124,223)(118,201,167,194,160,187,153,180,146,173,139,222,132,215,125,208)(119,186,168,179,161,172,154,221,147,214,140,207,133,200,126,193) );
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)], [(1,149,184,58,43,135,170,100,29,121,212,86,15,163,198,72),(2,134,185,99,44,120,171,85,30,162,213,71,16,148,199,57),(3,119,186,84,45,161,172,70,31,147,214,112,17,133,200,98),(4,160,187,69,46,146,173,111,32,132,215,97,18,118,201,83),(5,145,188,110,47,131,174,96,33,117,216,82,19,159,202,68),(6,130,189,95,48,116,175,81,34,158,217,67,20,144,203,109),(7,115,190,80,49,157,176,66,35,143,218,108,21,129,204,94),(8,156,191,65,50,142,177,107,36,128,219,93,22,114,205,79),(9,141,192,106,51,127,178,92,37,113,220,78,23,155,206,64),(10,126,193,91,52,168,179,77,38,154,221,63,24,140,207,105),(11,167,194,76,53,153,180,62,39,139,222,104,25,125,208,90),(12,152,195,61,54,138,181,103,40,124,223,89,26,166,209,75),(13,137,196,102,55,123,182,88,41,165,224,74,27,151,210,60),(14,122,197,87,56,164,183,73,42,150,169,59,28,136,211,101)], [(1,65,50,58,43,107,36,100,29,93,22,86,15,79,8,72),(2,106,51,99,44,92,37,85,30,78,23,71,16,64,9,57),(3,91,52,84,45,77,38,70,31,63,24,112,17,105,10,98),(4,76,53,69,46,62,39,111,32,104,25,97,18,90,11,83),(5,61,54,110,47,103,40,96,33,89,26,82,19,75,12,68),(6,102,55,95,48,88,41,81,34,74,27,67,20,60,13,109),(7,87,56,80,49,73,42,66,35,59,28,108,21,101,14,94),(113,220,162,213,155,206,148,199,141,192,134,185,127,178,120,171),(114,205,163,198,156,191,149,184,142,177,135,170,128,219,121,212),(115,190,164,183,157,176,150,169,143,218,136,211,129,204,122,197),(116,175,165,224,158,217,151,210,144,203,137,196,130,189,123,182),(117,216,166,209,159,202,152,195,145,188,138,181,131,174,124,223),(118,201,167,194,160,187,153,180,146,173,139,222,132,215,125,208),(119,186,168,179,161,172,154,221,147,214,140,207,133,200,126,193)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14U | 16A | ··· | 16P | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | C16 | D4 | D7 | M4(2) | Dic7 | D14 | Dic7 | M5(2) | C7⋊D4 | C7⋊C8 | C7⋊C8 | C4.Dic7 | C7⋊C16 | C28.C8 |
| kernel | C56.91D4 | C2×C7⋊C16 | C22×C56 | C2×C56 | C22×C28 | C2×C28 | C22×C14 | C2×C14 | C56 | C22×C8 | C28 | C2×C8 | C2×C8 | C22×C4 | C14 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 16 | 2 | 3 | 2 | 3 | 3 | 3 | 4 | 12 | 6 | 6 | 12 | 24 | 24 |
Matrix representation of C56.91D4 ►in GL3(𝔽113) generated by
| 44 | 0 | 0 |
| 0 | 106 | 0 |
| 0 | 76 | 16 |
| 65 | 0 | 0 |
| 0 | 69 | 64 |
| 0 | 28 | 44 |
| 48 | 0 | 0 |
| 0 | 69 | 64 |
| 0 | 81 | 44 |
G:=sub<GL(3,GF(113))| [44,0,0,0,106,76,0,0,16],[65,0,0,0,69,28,0,64,44],[48,0,0,0,69,81,0,64,44] >;
C56.91D4 in GAP, Magma, Sage, TeX
C_{56}._{91}D_4 % in TeX
G:=Group("C56.91D4"); // GroupNames label
G:=SmallGroup(448,106);
// by ID
G=gap.SmallGroup(448,106);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,100,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=1,b^4=a^42,c^2=a^49,b*a*b^-1=c*a*c^-1=a^41,c*b*c^-1=a^7*b^3>;
// generators/relations