metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊14D4, D14⋊4SD16, C7⋊7(C8⋊8D4), C8⋊11(C7⋊D4), C8⋊Dic7⋊27C2, D14⋊3Q8⋊4C2, (C14×SD16)⋊8C2, (C2×SD16)⋊12D7, (C2×D4).73D14, (C2×C8).263D14, C28.176(C2×D4), C28⋊2D4.8C2, (C2×Q8).54D14, C2.30(D7×SD16), C14.63(C4○D8), D4⋊Dic7⋊34C2, Q8⋊Dic7⋊29C2, C14.47(C2×SD16), (C22×D7).59D4, C22.268(D4×D7), C28.101(C4○D4), C4.32(D4⋊2D7), C2.19(C28⋊2D4), (C2×C28).448C23, (C2×C56).164C22, (C2×Dic7).114D4, (D4×C14).97C22, (Q8×C14).77C22, C14.116(C4⋊D4), C4⋊Dic7.175C22, C2.29(SD16⋊3D7), (D7×C2×C8)⋊8C2, C4.82(C2×C7⋊D4), (C2×C14).360(C2×D4), (C2×C7⋊C8).274C22, (C2×C4×D7).240C22, (C2×C4).537(C22×D7), SmallGroup(448,705)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊14D4
G = < a,b,c | a56=b4=c2=1, bab-1=a27, cac=a41, cbc=b-1 >
Subgroups: 612 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, C8⋊8D4, C8×D7, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C2×C56, C7×SD16, C2×C4×D7, C2×C7⋊D4, D4×C14, Q8×C14, C8⋊Dic7, D4⋊Dic7, Q8⋊Dic7, D7×C2×C8, C28⋊2D4, D14⋊3Q8, C14×SD16, C56⋊14D4
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C4⋊D4, C2×SD16, C4○D8, C7⋊D4, C22×D7, C8⋊8D4, D4×D7, D4⋊2D7, C2×C7⋊D4, D7×SD16, SD16⋊3D7, C28⋊2D4, C56⋊14D4
►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)
(1 155 61 179)(2 126 62 206)(3 153 63 177)(4 124 64 204)(5 151 65 175)(6 122 66 202)(7 149 67 173)(8 120 68 200)(9 147 69 171)(10 118 70 198)(11 145 71 169)(12 116 72 196)(13 143 73 223)(14 114 74 194)(15 141 75 221)(16 168 76 192)(17 139 77 219)(18 166 78 190)(19 137 79 217)(20 164 80 188)(21 135 81 215)(22 162 82 186)(23 133 83 213)(24 160 84 184)(25 131 85 211)(26 158 86 182)(27 129 87 209)(28 156 88 180)(29 127 89 207)(30 154 90 178)(31 125 91 205)(32 152 92 176)(33 123 93 203)(34 150 94 174)(35 121 95 201)(36 148 96 172)(37 119 97 199)(38 146 98 170)(39 117 99 197)(40 144 100 224)(41 115 101 195)(42 142 102 222)(43 113 103 193)(44 140 104 220)(45 167 105 191)(46 138 106 218)(47 165 107 189)(48 136 108 216)(49 163 109 187)(50 134 110 214)(51 161 111 185)(52 132 112 212)(53 159 57 183)(54 130 58 210)(55 157 59 181)(56 128 60 208)
(2 42)(3 27)(4 12)(5 53)(6 38)(7 23)(9 49)(10 34)(11 19)(13 45)(14 30)(16 56)(17 41)(18 26)(20 52)(21 37)(24 48)(25 33)(28 44)(31 55)(32 40)(35 51)(39 47)(46 54)(57 65)(58 106)(59 91)(60 76)(62 102)(63 87)(64 72)(66 98)(67 83)(69 109)(70 94)(71 79)(73 105)(74 90)(77 101)(78 86)(80 112)(81 97)(84 108)(85 93)(88 104)(92 100)(95 111)(99 107)(113 193)(114 178)(115 219)(116 204)(117 189)(118 174)(119 215)(120 200)(121 185)(122 170)(123 211)(124 196)(125 181)(126 222)(127 207)(128 192)(129 177)(130 218)(131 203)(132 188)(133 173)(134 214)(135 199)(136 184)(137 169)(138 210)(139 195)(140 180)(141 221)(142 206)(143 191)(144 176)(145 217)(146 202)(147 187)(148 172)(149 213)(150 198)(151 183)(152 224)(153 209)(154 194)(155 179)(156 220)(157 205)(158 190)(159 175)(160 216)(161 201)(162 186)(163 171)(164 212)(165 197)(166 182)(167 223)(168 208)
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), (1,155,61,179)(2,126,62,206)(3,153,63,177)(4,124,64,204)(5,151,65,175)(6,122,66,202)(7,149,67,173)(8,120,68,200)(9,147,69,171)(10,118,70,198)(11,145,71,169)(12,116,72,196)(13,143,73,223)(14,114,74,194)(15,141,75,221)(16,168,76,192)(17,139,77,219)(18,166,78,190)(19,137,79,217)(20,164,80,188)(21,135,81,215)(22,162,82,186)(23,133,83,213)(24,160,84,184)(25,131,85,211)(26,158,86,182)(27,129,87,209)(28,156,88,180)(29,127,89,207)(30,154,90,178)(31,125,91,205)(32,152,92,176)(33,123,93,203)(34,150,94,174)(35,121,95,201)(36,148,96,172)(37,119,97,199)(38,146,98,170)(39,117,99,197)(40,144,100,224)(41,115,101,195)(42,142,102,222)(43,113,103,193)(44,140,104,220)(45,167,105,191)(46,138,106,218)(47,165,107,189)(48,136,108,216)(49,163,109,187)(50,134,110,214)(51,161,111,185)(52,132,112,212)(53,159,57,183)(54,130,58,210)(55,157,59,181)(56,128,60,208), (2,42)(3,27)(4,12)(5,53)(6,38)(7,23)(9,49)(10,34)(11,19)(13,45)(14,30)(16,56)(17,41)(18,26)(20,52)(21,37)(24,48)(25,33)(28,44)(31,55)(32,40)(35,51)(39,47)(46,54)(57,65)(58,106)(59,91)(60,76)(62,102)(63,87)(64,72)(66,98)(67,83)(69,109)(70,94)(71,79)(73,105)(74,90)(77,101)(78,86)(80,112)(81,97)(84,108)(85,93)(88,104)(92,100)(95,111)(99,107)(113,193)(114,178)(115,219)(116,204)(117,189)(118,174)(119,215)(120,200)(121,185)(122,170)(123,211)(124,196)(125,181)(126,222)(127,207)(128,192)(129,177)(130,218)(131,203)(132,188)(133,173)(134,214)(135,199)(136,184)(137,169)(138,210)(139,195)(140,180)(141,221)(142,206)(143,191)(144,176)(145,217)(146,202)(147,187)(148,172)(149,213)(150,198)(151,183)(152,224)(153,209)(154,194)(155,179)(156,220)(157,205)(158,190)(159,175)(160,216)(161,201)(162,186)(163,171)(164,212)(165,197)(166,182)(167,223)(168,208)>;
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), (1,155,61,179)(2,126,62,206)(3,153,63,177)(4,124,64,204)(5,151,65,175)(6,122,66,202)(7,149,67,173)(8,120,68,200)(9,147,69,171)(10,118,70,198)(11,145,71,169)(12,116,72,196)(13,143,73,223)(14,114,74,194)(15,141,75,221)(16,168,76,192)(17,139,77,219)(18,166,78,190)(19,137,79,217)(20,164,80,188)(21,135,81,215)(22,162,82,186)(23,133,83,213)(24,160,84,184)(25,131,85,211)(26,158,86,182)(27,129,87,209)(28,156,88,180)(29,127,89,207)(30,154,90,178)(31,125,91,205)(32,152,92,176)(33,123,93,203)(34,150,94,174)(35,121,95,201)(36,148,96,172)(37,119,97,199)(38,146,98,170)(39,117,99,197)(40,144,100,224)(41,115,101,195)(42,142,102,222)(43,113,103,193)(44,140,104,220)(45,167,105,191)(46,138,106,218)(47,165,107,189)(48,136,108,216)(49,163,109,187)(50,134,110,214)(51,161,111,185)(52,132,112,212)(53,159,57,183)(54,130,58,210)(55,157,59,181)(56,128,60,208), (2,42)(3,27)(4,12)(5,53)(6,38)(7,23)(9,49)(10,34)(11,19)(13,45)(14,30)(16,56)(17,41)(18,26)(20,52)(21,37)(24,48)(25,33)(28,44)(31,55)(32,40)(35,51)(39,47)(46,54)(57,65)(58,106)(59,91)(60,76)(62,102)(63,87)(64,72)(66,98)(67,83)(69,109)(70,94)(71,79)(73,105)(74,90)(77,101)(78,86)(80,112)(81,97)(84,108)(85,93)(88,104)(92,100)(95,111)(99,107)(113,193)(114,178)(115,219)(116,204)(117,189)(118,174)(119,215)(120,200)(121,185)(122,170)(123,211)(124,196)(125,181)(126,222)(127,207)(128,192)(129,177)(130,218)(131,203)(132,188)(133,173)(134,214)(135,199)(136,184)(137,169)(138,210)(139,195)(140,180)(141,221)(142,206)(143,191)(144,176)(145,217)(146,202)(147,187)(148,172)(149,213)(150,198)(151,183)(152,224)(153,209)(154,194)(155,179)(156,220)(157,205)(158,190)(159,175)(160,216)(161,201)(162,186)(163,171)(164,212)(165,197)(166,182)(167,223)(168,208) );
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)], [(1,155,61,179),(2,126,62,206),(3,153,63,177),(4,124,64,204),(5,151,65,175),(6,122,66,202),(7,149,67,173),(8,120,68,200),(9,147,69,171),(10,118,70,198),(11,145,71,169),(12,116,72,196),(13,143,73,223),(14,114,74,194),(15,141,75,221),(16,168,76,192),(17,139,77,219),(18,166,78,190),(19,137,79,217),(20,164,80,188),(21,135,81,215),(22,162,82,186),(23,133,83,213),(24,160,84,184),(25,131,85,211),(26,158,86,182),(27,129,87,209),(28,156,88,180),(29,127,89,207),(30,154,90,178),(31,125,91,205),(32,152,92,176),(33,123,93,203),(34,150,94,174),(35,121,95,201),(36,148,96,172),(37,119,97,199),(38,146,98,170),(39,117,99,197),(40,144,100,224),(41,115,101,195),(42,142,102,222),(43,113,103,193),(44,140,104,220),(45,167,105,191),(46,138,106,218),(47,165,107,189),(48,136,108,216),(49,163,109,187),(50,134,110,214),(51,161,111,185),(52,132,112,212),(53,159,57,183),(54,130,58,210),(55,157,59,181),(56,128,60,208)], [(2,42),(3,27),(4,12),(5,53),(6,38),(7,23),(9,49),(10,34),(11,19),(13,45),(14,30),(16,56),(17,41),(18,26),(20,52),(21,37),(24,48),(25,33),(28,44),(31,55),(32,40),(35,51),(39,47),(46,54),(57,65),(58,106),(59,91),(60,76),(62,102),(63,87),(64,72),(66,98),(67,83),(69,109),(70,94),(71,79),(73,105),(74,90),(77,101),(78,86),(80,112),(81,97),(84,108),(85,93),(88,104),(92,100),(95,111),(99,107),(113,193),(114,178),(115,219),(116,204),(117,189),(118,174),(119,215),(120,200),(121,185),(122,170),(123,211),(124,196),(125,181),(126,222),(127,207),(128,192),(129,177),(130,218),(131,203),(132,188),(133,173),(134,214),(135,199),(136,184),(137,169),(138,210),(139,195),(140,180),(141,221),(142,206),(143,191),(144,176),(145,217),(146,202),(147,187),(148,172),(149,213),(150,198),(151,183),(152,224),(153,209),(154,194),(155,179),(156,220),(157,205),(158,190),(159,175),(160,216),(161,201),(162,186),(163,171),(164,212),(165,197),(166,182),(167,223),(168,208)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 14 | 14 | 2 | 2 | 8 | 14 | 14 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | SD16 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | D4⋊2D7 | D4×D7 | D7×SD16 | SD16⋊3D7 |
| kernel | C56⋊14D4 | C8⋊Dic7 | D4⋊Dic7 | Q8⋊Dic7 | D7×C2×C8 | C28⋊2D4 | D14⋊3Q8 | C14×SD16 | C56 | C2×Dic7 | C22×D7 | C2×SD16 | C28 | D14 | C2×C8 | C2×D4 | C2×Q8 | C14 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 3 | 3 | 3 | 4 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of C56⋊14D4 ►in GL4(𝔽113) generated by
| 0 | 112 | 0 | 0 |
| 1 | 34 | 0 | 0 |
| 0 | 0 | 44 | 80 |
| 0 | 0 | 0 | 95 |
| 22 | 101 | 0 | 0 |
| 31 | 91 | 0 | 0 |
| 0 | 0 | 103 | 38 |
| 0 | 0 | 36 | 10 |
| 1 | 0 | 0 | 0 |
| 79 | 112 | 0 | 0 |
| 0 | 0 | 1 | 101 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [0,1,0,0,112,34,0,0,0,0,44,0,0,0,80,95],[22,31,0,0,101,91,0,0,0,0,103,36,0,0,38,10],[1,79,0,0,0,112,0,0,0,0,1,0,0,0,101,112] >;
C56⋊14D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_{14}D_4 % in TeX
G:=Group("C56:14D4"); // GroupNames label
G:=SmallGroup(448,705);
// by ID
G=gap.SmallGroup(448,705);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,555,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^27,c*a*c=a^41,c*b*c=b^-1>;
// generators/relations