metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.4D4, C23.19D28, C8⋊Dic7⋊5C2, (C2×C4).54D28, (C2×C8).79D14, C8.1(C7⋊D4), C7⋊4(C8.D4), (C2×C28).299D4, C28.423(C2×D4), (C2×Dic28)⋊12C2, (C2×C56).65C22, (C2×M4(2)).3D7, C28.232(C4○D4), C4.116(C4○D28), C28.44D4⋊42C2, C14.75(C4⋊D4), C2.23(C28⋊7D4), (C2×C28).777C23, (C22×C4).145D14, C22.136(C2×D28), (C22×C14).105D4, (C14×M4(2)).3C2, C4⋊Dic7.28C22, C28.48D4.17C2, C2.23(C8.D14), C14.23(C8.C22), (C22×C28).306C22, (C2×Dic14).20C22, C4.116(C2×C7⋊D4), (C2×C14).167(C2×D4), (C2×C4).726(C22×D7), SmallGroup(448,671)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.4D4
G = < a,b,c | a56=b4=1, c2=a28, bab-1=a27, cac-1=a-1, cbc-1=a28b-1 >
Subgroups: 516 in 110 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C56, C56, Dic14, C2×Dic7, C2×C28, C2×C28, C22×C14, C8.D4, Dic28, Dic7⋊C4, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C2×Dic14, C22×C28, C28.44D4, C8⋊Dic7, C2×Dic28, C28.48D4, C14×M4(2), C56.4D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8.C22, D28, C7⋊D4, C22×D7, C8.D4, C2×D28, C4○D28, C2×C7⋊D4, C8.D14, C28⋊7D4, C56.4D4
►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)
(1 103 143 174)(2 74 144 201)(3 101 145 172)(4 72 146 199)(5 99 147 170)(6 70 148 197)(7 97 149 224)(8 68 150 195)(9 95 151 222)(10 66 152 193)(11 93 153 220)(12 64 154 191)(13 91 155 218)(14 62 156 189)(15 89 157 216)(16 60 158 187)(17 87 159 214)(18 58 160 185)(19 85 161 212)(20 112 162 183)(21 83 163 210)(22 110 164 181)(23 81 165 208)(24 108 166 179)(25 79 167 206)(26 106 168 177)(27 77 113 204)(28 104 114 175)(29 75 115 202)(30 102 116 173)(31 73 117 200)(32 100 118 171)(33 71 119 198)(34 98 120 169)(35 69 121 196)(36 96 122 223)(37 67 123 194)(38 94 124 221)(39 65 125 192)(40 92 126 219)(41 63 127 190)(42 90 128 217)(43 61 129 188)(44 88 130 215)(45 59 131 186)(46 86 132 213)(47 57 133 184)(48 84 134 211)(49 111 135 182)(50 82 136 209)(51 109 137 180)(52 80 138 207)(53 107 139 178)(54 78 140 205)(55 105 141 176)(56 76 142 203)
(1 202 29 174)(2 201 30 173)(3 200 31 172)(4 199 32 171)(5 198 33 170)(6 197 34 169)(7 196 35 224)(8 195 36 223)(9 194 37 222)(10 193 38 221)(11 192 39 220)(12 191 40 219)(13 190 41 218)(14 189 42 217)(15 188 43 216)(16 187 44 215)(17 186 45 214)(18 185 46 213)(19 184 47 212)(20 183 48 211)(21 182 49 210)(22 181 50 209)(23 180 51 208)(24 179 52 207)(25 178 53 206)(26 177 54 205)(27 176 55 204)(28 175 56 203)(57 133 85 161)(58 132 86 160)(59 131 87 159)(60 130 88 158)(61 129 89 157)(62 128 90 156)(63 127 91 155)(64 126 92 154)(65 125 93 153)(66 124 94 152)(67 123 95 151)(68 122 96 150)(69 121 97 149)(70 120 98 148)(71 119 99 147)(72 118 100 146)(73 117 101 145)(74 116 102 144)(75 115 103 143)(76 114 104 142)(77 113 105 141)(78 168 106 140)(79 167 107 139)(80 166 108 138)(81 165 109 137)(82 164 110 136)(83 163 111 135)(84 162 112 134)
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), (1,103,143,174)(2,74,144,201)(3,101,145,172)(4,72,146,199)(5,99,147,170)(6,70,148,197)(7,97,149,224)(8,68,150,195)(9,95,151,222)(10,66,152,193)(11,93,153,220)(12,64,154,191)(13,91,155,218)(14,62,156,189)(15,89,157,216)(16,60,158,187)(17,87,159,214)(18,58,160,185)(19,85,161,212)(20,112,162,183)(21,83,163,210)(22,110,164,181)(23,81,165,208)(24,108,166,179)(25,79,167,206)(26,106,168,177)(27,77,113,204)(28,104,114,175)(29,75,115,202)(30,102,116,173)(31,73,117,200)(32,100,118,171)(33,71,119,198)(34,98,120,169)(35,69,121,196)(36,96,122,223)(37,67,123,194)(38,94,124,221)(39,65,125,192)(40,92,126,219)(41,63,127,190)(42,90,128,217)(43,61,129,188)(44,88,130,215)(45,59,131,186)(46,86,132,213)(47,57,133,184)(48,84,134,211)(49,111,135,182)(50,82,136,209)(51,109,137,180)(52,80,138,207)(53,107,139,178)(54,78,140,205)(55,105,141,176)(56,76,142,203), (1,202,29,174)(2,201,30,173)(3,200,31,172)(4,199,32,171)(5,198,33,170)(6,197,34,169)(7,196,35,224)(8,195,36,223)(9,194,37,222)(10,193,38,221)(11,192,39,220)(12,191,40,219)(13,190,41,218)(14,189,42,217)(15,188,43,216)(16,187,44,215)(17,186,45,214)(18,185,46,213)(19,184,47,212)(20,183,48,211)(21,182,49,210)(22,181,50,209)(23,180,51,208)(24,179,52,207)(25,178,53,206)(26,177,54,205)(27,176,55,204)(28,175,56,203)(57,133,85,161)(58,132,86,160)(59,131,87,159)(60,130,88,158)(61,129,89,157)(62,128,90,156)(63,127,91,155)(64,126,92,154)(65,125,93,153)(66,124,94,152)(67,123,95,151)(68,122,96,150)(69,121,97,149)(70,120,98,148)(71,119,99,147)(72,118,100,146)(73,117,101,145)(74,116,102,144)(75,115,103,143)(76,114,104,142)(77,113,105,141)(78,168,106,140)(79,167,107,139)(80,166,108,138)(81,165,109,137)(82,164,110,136)(83,163,111,135)(84,162,112,134)>;
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), (1,103,143,174)(2,74,144,201)(3,101,145,172)(4,72,146,199)(5,99,147,170)(6,70,148,197)(7,97,149,224)(8,68,150,195)(9,95,151,222)(10,66,152,193)(11,93,153,220)(12,64,154,191)(13,91,155,218)(14,62,156,189)(15,89,157,216)(16,60,158,187)(17,87,159,214)(18,58,160,185)(19,85,161,212)(20,112,162,183)(21,83,163,210)(22,110,164,181)(23,81,165,208)(24,108,166,179)(25,79,167,206)(26,106,168,177)(27,77,113,204)(28,104,114,175)(29,75,115,202)(30,102,116,173)(31,73,117,200)(32,100,118,171)(33,71,119,198)(34,98,120,169)(35,69,121,196)(36,96,122,223)(37,67,123,194)(38,94,124,221)(39,65,125,192)(40,92,126,219)(41,63,127,190)(42,90,128,217)(43,61,129,188)(44,88,130,215)(45,59,131,186)(46,86,132,213)(47,57,133,184)(48,84,134,211)(49,111,135,182)(50,82,136,209)(51,109,137,180)(52,80,138,207)(53,107,139,178)(54,78,140,205)(55,105,141,176)(56,76,142,203), (1,202,29,174)(2,201,30,173)(3,200,31,172)(4,199,32,171)(5,198,33,170)(6,197,34,169)(7,196,35,224)(8,195,36,223)(9,194,37,222)(10,193,38,221)(11,192,39,220)(12,191,40,219)(13,190,41,218)(14,189,42,217)(15,188,43,216)(16,187,44,215)(17,186,45,214)(18,185,46,213)(19,184,47,212)(20,183,48,211)(21,182,49,210)(22,181,50,209)(23,180,51,208)(24,179,52,207)(25,178,53,206)(26,177,54,205)(27,176,55,204)(28,175,56,203)(57,133,85,161)(58,132,86,160)(59,131,87,159)(60,130,88,158)(61,129,89,157)(62,128,90,156)(63,127,91,155)(64,126,92,154)(65,125,93,153)(66,124,94,152)(67,123,95,151)(68,122,96,150)(69,121,97,149)(70,120,98,148)(71,119,99,147)(72,118,100,146)(73,117,101,145)(74,116,102,144)(75,115,103,143)(76,114,104,142)(77,113,105,141)(78,168,106,140)(79,167,107,139)(80,166,108,138)(81,165,109,137)(82,164,110,136)(83,163,111,135)(84,162,112,134) );
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)], [(1,103,143,174),(2,74,144,201),(3,101,145,172),(4,72,146,199),(5,99,147,170),(6,70,148,197),(7,97,149,224),(8,68,150,195),(9,95,151,222),(10,66,152,193),(11,93,153,220),(12,64,154,191),(13,91,155,218),(14,62,156,189),(15,89,157,216),(16,60,158,187),(17,87,159,214),(18,58,160,185),(19,85,161,212),(20,112,162,183),(21,83,163,210),(22,110,164,181),(23,81,165,208),(24,108,166,179),(25,79,167,206),(26,106,168,177),(27,77,113,204),(28,104,114,175),(29,75,115,202),(30,102,116,173),(31,73,117,200),(32,100,118,171),(33,71,119,198),(34,98,120,169),(35,69,121,196),(36,96,122,223),(37,67,123,194),(38,94,124,221),(39,65,125,192),(40,92,126,219),(41,63,127,190),(42,90,128,217),(43,61,129,188),(44,88,130,215),(45,59,131,186),(46,86,132,213),(47,57,133,184),(48,84,134,211),(49,111,135,182),(50,82,136,209),(51,109,137,180),(52,80,138,207),(53,107,139,178),(54,78,140,205),(55,105,141,176),(56,76,142,203)], [(1,202,29,174),(2,201,30,173),(3,200,31,172),(4,199,32,171),(5,198,33,170),(6,197,34,169),(7,196,35,224),(8,195,36,223),(9,194,37,222),(10,193,38,221),(11,192,39,220),(12,191,40,219),(13,190,41,218),(14,189,42,217),(15,188,43,216),(16,187,44,215),(17,186,45,214),(18,185,46,213),(19,184,47,212),(20,183,48,211),(21,182,49,210),(22,181,50,209),(23,180,51,208),(24,179,52,207),(25,178,53,206),(26,177,54,205),(27,176,55,204),(28,175,56,203),(57,133,85,161),(58,132,86,160),(59,131,87,159),(60,130,88,158),(61,129,89,157),(62,128,90,156),(63,127,91,155),(64,126,92,154),(65,125,93,153),(66,124,94,152),(67,123,95,151),(68,122,96,150),(69,121,97,149),(70,120,98,148),(71,119,99,147),(72,118,100,146),(73,117,101,145),(74,116,102,144),(75,115,103,143),(76,114,104,142),(77,113,105,141),(78,168,106,140),(79,167,107,139),(80,166,108,138),(81,165,109,137),(82,164,110,136),(83,163,111,135),(84,162,112,134)]])
76 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
76 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | D28 | D28 | C4○D28 | C8.C22 | C8.D14 |
| kernel | C56.4D4 | C28.44D4 | C8⋊Dic7 | C2×Dic28 | C28.48D4 | C14×M4(2) | C56 | C2×C28 | C22×C14 | C2×M4(2) | C28 | C2×C8 | C22×C4 | C8 | C2×C4 | C23 | C4 | C14 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 6 | 3 | 12 | 6 | 6 | 12 | 2 | 12 |
Matrix representation of C56.4D4 ►in GL6(𝔽113)
| 98 | 0 | 0 | 0 | 0 | 0 |
| 24 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 57 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 83 | 0 |
| 92 | 2 | 0 | 0 | 0 | 0 |
| 5 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 92 | 2 | 0 | 0 | 0 | 0 |
| 6 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
G:=sub<GL(6,GF(113))| [98,24,0,0,0,0,0,15,0,0,0,0,0,0,0,57,0,0,0,0,64,0,0,0,0,0,0,0,0,83,0,0,0,0,2,0],[92,5,0,0,0,0,2,21,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,1,0,0,0,0,0,0,112,0,0],[92,6,0,0,0,0,2,21,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,1,0,0,0,0,0,0,1,0,0] >;
C56.4D4 in GAP, Magma, Sage, TeX
C_{56}._4D_4 % in TeX
G:=Group("C56.4D4"); // GroupNames label
G:=SmallGroup(448,671);
// by ID
G=gap.SmallGroup(448,671);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,344,254,387,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=1,c^2=a^28,b*a*b^-1=a^27,c*a*c^-1=a^-1,c*b*c^-1=a^28*b^-1>;
// generators/relations