metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.28D4, C7⋊C8.22D4, (C2×Q16)⋊6D7, C4.29(D4×D7), (C14×Q16)⋊6C2, (C8×Dic7)⋊7C2, (C2×D56).11C2, (C2×C8).244D14, C28.189(C2×D4), C7⋊5(C8.12D4), C8.19(C7⋊D4), (C2×Q8).67D14, C14.82(C4○D8), C28.23D4⋊6C2, (C2×C56).96C22, C22.282(D4×D7), C14.35(C4⋊1D4), C2.26(C28⋊D4), (C2×C28).465C23, (C2×Dic7).118D4, (Q8×C14).94C22, C2.19(Q8.D14), (C2×D28).127C22, (C4×Dic7).245C22, (C2×Q8⋊D7)⋊21C2, C4.16(C2×C7⋊D4), (C2×C14).376(C2×D4), (C2×C7⋊C8).279C22, (C2×C4).553(C22×D7), SmallGroup(448,725)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.28D4
G = < a,b,c | a56=b4=c2=1, bab-1=a41, cac=a-1, cbc=a28b-1 >
Subgroups: 804 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, C7⋊C8, C56, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C8.12D4, D56, C2×C7⋊C8, C4×Dic7, D14⋊C4, Q8⋊D7, C2×C56, C7×Q16, C2×D28, Q8×C14, C8×Dic7, C2×D56, C2×Q8⋊D7, C28.23D4, C14×Q16, C56.28D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C4○D8, C7⋊D4, C22×D7, C8.12D4, D4×D7, C2×C7⋊D4, Q8.D14, C28⋊D4, C56.28D4
►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)
(1 187 104 151)(2 172 105 136)(3 213 106 121)(4 198 107 162)(5 183 108 147)(6 224 109 132)(7 209 110 117)(8 194 111 158)(9 179 112 143)(10 220 57 128)(11 205 58 113)(12 190 59 154)(13 175 60 139)(14 216 61 124)(15 201 62 165)(16 186 63 150)(17 171 64 135)(18 212 65 120)(19 197 66 161)(20 182 67 146)(21 223 68 131)(22 208 69 116)(23 193 70 157)(24 178 71 142)(25 219 72 127)(26 204 73 168)(27 189 74 153)(28 174 75 138)(29 215 76 123)(30 200 77 164)(31 185 78 149)(32 170 79 134)(33 211 80 119)(34 196 81 160)(35 181 82 145)(36 222 83 130)(37 207 84 115)(38 192 85 156)(39 177 86 141)(40 218 87 126)(41 203 88 167)(42 188 89 152)(43 173 90 137)(44 214 91 122)(45 199 92 163)(46 184 93 148)(47 169 94 133)(48 210 95 118)(49 195 96 159)(50 180 97 144)(51 221 98 129)(52 206 99 114)(53 191 100 155)(54 176 101 140)(55 217 102 125)(56 202 103 166)
(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(57 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(96 112)(97 111)(98 110)(99 109)(100 108)(101 107)(102 106)(103 105)(113 197)(114 196)(115 195)(116 194)(117 193)(118 192)(119 191)(120 190)(121 189)(122 188)(123 187)(124 186)(125 185)(126 184)(127 183)(128 182)(129 181)(130 180)(131 179)(132 178)(133 177)(134 176)(135 175)(136 174)(137 173)(138 172)(139 171)(140 170)(141 169)(142 224)(143 223)(144 222)(145 221)(146 220)(147 219)(148 218)(149 217)(150 216)(151 215)(152 214)(153 213)(154 212)(155 211)(156 210)(157 209)(158 208)(159 207)(160 206)(161 205)(162 204)(163 203)(164 202)(165 201)(166 200)(167 199)(168 198)
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), (1,187,104,151)(2,172,105,136)(3,213,106,121)(4,198,107,162)(5,183,108,147)(6,224,109,132)(7,209,110,117)(8,194,111,158)(9,179,112,143)(10,220,57,128)(11,205,58,113)(12,190,59,154)(13,175,60,139)(14,216,61,124)(15,201,62,165)(16,186,63,150)(17,171,64,135)(18,212,65,120)(19,197,66,161)(20,182,67,146)(21,223,68,131)(22,208,69,116)(23,193,70,157)(24,178,71,142)(25,219,72,127)(26,204,73,168)(27,189,74,153)(28,174,75,138)(29,215,76,123)(30,200,77,164)(31,185,78,149)(32,170,79,134)(33,211,80,119)(34,196,81,160)(35,181,82,145)(36,222,83,130)(37,207,84,115)(38,192,85,156)(39,177,86,141)(40,218,87,126)(41,203,88,167)(42,188,89,152)(43,173,90,137)(44,214,91,122)(45,199,92,163)(46,184,93,148)(47,169,94,133)(48,210,95,118)(49,195,96,159)(50,180,97,144)(51,221,98,129)(52,206,99,114)(53,191,100,155)(54,176,101,140)(55,217,102,125)(56,202,103,166), (2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,197)(114,196)(115,195)(116,194)(117,193)(118,192)(119,191)(120,190)(121,189)(122,188)(123,187)(124,186)(125,185)(126,184)(127,183)(128,182)(129,181)(130,180)(131,179)(132,178)(133,177)(134,176)(135,175)(136,174)(137,173)(138,172)(139,171)(140,170)(141,169)(142,224)(143,223)(144,222)(145,221)(146,220)(147,219)(148,218)(149,217)(150,216)(151,215)(152,214)(153,213)(154,212)(155,211)(156,210)(157,209)(158,208)(159,207)(160,206)(161,205)(162,204)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198)>;
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), (1,187,104,151)(2,172,105,136)(3,213,106,121)(4,198,107,162)(5,183,108,147)(6,224,109,132)(7,209,110,117)(8,194,111,158)(9,179,112,143)(10,220,57,128)(11,205,58,113)(12,190,59,154)(13,175,60,139)(14,216,61,124)(15,201,62,165)(16,186,63,150)(17,171,64,135)(18,212,65,120)(19,197,66,161)(20,182,67,146)(21,223,68,131)(22,208,69,116)(23,193,70,157)(24,178,71,142)(25,219,72,127)(26,204,73,168)(27,189,74,153)(28,174,75,138)(29,215,76,123)(30,200,77,164)(31,185,78,149)(32,170,79,134)(33,211,80,119)(34,196,81,160)(35,181,82,145)(36,222,83,130)(37,207,84,115)(38,192,85,156)(39,177,86,141)(40,218,87,126)(41,203,88,167)(42,188,89,152)(43,173,90,137)(44,214,91,122)(45,199,92,163)(46,184,93,148)(47,169,94,133)(48,210,95,118)(49,195,96,159)(50,180,97,144)(51,221,98,129)(52,206,99,114)(53,191,100,155)(54,176,101,140)(55,217,102,125)(56,202,103,166), (2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(96,112)(97,111)(98,110)(99,109)(100,108)(101,107)(102,106)(103,105)(113,197)(114,196)(115,195)(116,194)(117,193)(118,192)(119,191)(120,190)(121,189)(122,188)(123,187)(124,186)(125,185)(126,184)(127,183)(128,182)(129,181)(130,180)(131,179)(132,178)(133,177)(134,176)(135,175)(136,174)(137,173)(138,172)(139,171)(140,170)(141,169)(142,224)(143,223)(144,222)(145,221)(146,220)(147,219)(148,218)(149,217)(150,216)(151,215)(152,214)(153,213)(154,212)(155,211)(156,210)(157,209)(158,208)(159,207)(160,206)(161,205)(162,204)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198) );
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)], [(1,187,104,151),(2,172,105,136),(3,213,106,121),(4,198,107,162),(5,183,108,147),(6,224,109,132),(7,209,110,117),(8,194,111,158),(9,179,112,143),(10,220,57,128),(11,205,58,113),(12,190,59,154),(13,175,60,139),(14,216,61,124),(15,201,62,165),(16,186,63,150),(17,171,64,135),(18,212,65,120),(19,197,66,161),(20,182,67,146),(21,223,68,131),(22,208,69,116),(23,193,70,157),(24,178,71,142),(25,219,72,127),(26,204,73,168),(27,189,74,153),(28,174,75,138),(29,215,76,123),(30,200,77,164),(31,185,78,149),(32,170,79,134),(33,211,80,119),(34,196,81,160),(35,181,82,145),(36,222,83,130),(37,207,84,115),(38,192,85,156),(39,177,86,141),(40,218,87,126),(41,203,88,167),(42,188,89,152),(43,173,90,137),(44,214,91,122),(45,199,92,163),(46,184,93,148),(47,169,94,133),(48,210,95,118),(49,195,96,159),(50,180,97,144),(51,221,98,129),(52,206,99,114),(53,191,100,155),(54,176,101,140),(55,217,102,125),(56,202,103,166)], [(2,56),(3,55),(4,54),(5,53),(6,52),(7,51),(8,50),(9,49),(10,48),(11,47),(12,46),(13,45),(14,44),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(57,95),(58,94),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77),(96,112),(97,111),(98,110),(99,109),(100,108),(101,107),(102,106),(103,105),(113,197),(114,196),(115,195),(116,194),(117,193),(118,192),(119,191),(120,190),(121,189),(122,188),(123,187),(124,186),(125,185),(126,184),(127,183),(128,182),(129,181),(130,180),(131,179),(132,178),(133,177),(134,176),(135,175),(136,174),(137,173),(138,172),(139,171),(140,170),(141,169),(142,224),(143,223),(144,222),(145,221),(146,220),(147,219),(148,218),(149,217),(150,216),(151,215),(152,214),(153,213),(154,212),(155,211),(156,210),(157,209),(158,208),(159,207),(160,206),(161,205),(162,204),(163,203),(164,202),(165,201),(166,200),(167,199),(168,198)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 56 | 56 | 2 | 2 | 8 | 8 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | C4○D8 | C7⋊D4 | D4×D7 | D4×D7 | Q8.D14 |
| kernel | C56.28D4 | C8×Dic7 | C2×D56 | C2×Q8⋊D7 | C28.23D4 | C14×Q16 | C7⋊C8 | C56 | C2×Dic7 | C2×Q16 | C2×C8 | C2×Q8 | C14 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 8 | 12 | 3 | 3 | 12 |
Matrix representation of C56.28D4 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 1 | 0 | 0 |
| 0 | 0 | 102 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 25 | 62 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 1 | 0 | 0 |
| 0 | 0 | 14 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 1 | 0 | 0 |
| 0 | 0 | 14 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 32 | 112 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,102,0,0,0,0,1,10,0,0,0,0,0,0,0,25,0,0,0,0,9,62],[0,1,0,0,0,0,112,0,0,0,0,0,0,0,103,14,0,0,0,0,1,10,0,0,0,0,0,0,98,0,0,0,0,0,0,98],[1,0,0,0,0,0,0,112,0,0,0,0,0,0,103,14,0,0,0,0,1,10,0,0,0,0,0,0,1,32,0,0,0,0,0,112] >;
C56.28D4 in GAP, Magma, Sage, TeX
C_{56}._{28}D_4 % in TeX
G:=Group("C56.28D4"); // GroupNames label
G:=SmallGroup(448,725);
// by ID
G=gap.SmallGroup(448,725);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,344,254,555,184,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^41,c*a*c=a^-1,c*b*c=a^28*b^-1>;
// generators/relations