metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56.37D4, C7⋊C8.10D4, (C2×Q16)⋊9D7, C4.28(D4×D7), C56⋊C4⋊10C2, (C2×C8).96D14, C8.6(C7⋊D4), C7⋊5(C8.2D4), (C14×Q16)⋊10C2, C28.188(C2×D4), (C2×Q8).66D14, Dic7⋊Q8⋊6C2, (C2×Dic7).78D4, C22.281(D4×D7), C14.34(C4⋊1D4), C2.25(C28⋊D4), (C2×C56).151C22, (C2×C28).464C23, C28.23D4.6C2, (Q8×C14).93C22, C2.31(Q16⋊D7), (C2×D28).126C22, C14.81(C8.C22), (C4×Dic7).56C22, (C2×Dic14).133C22, C4.15(C2×C7⋊D4), (C2×Q8⋊D7).9C2, (C2×C7⋊Q16)⋊21C2, (C2×C56⋊C2).8C2, (C2×C14).375(C2×D4), (C2×C7⋊C8).168C22, (C2×C4).552(C22×D7), SmallGroup(448,724)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56.37D4
G = < a,b,c | a56=b4=c2=1, bab-1=a13, cac=a27, cbc=a28b-1 >
Subgroups: 676 in 124 conjugacy classes, 43 normal (31 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, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic7, C28, C28, D14, C2×C14, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16, C2×Q16, C2×Q16, C7⋊C8, C56, Dic14, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C8.2D4, C56⋊C2, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, Q8⋊D7, C7⋊Q16, C2×C56, C7×Q16, C2×Dic14, C2×D28, Q8×C14, C56⋊C4, C2×C56⋊C2, C2×Q8⋊D7, C2×C7⋊Q16, Dic7⋊Q8, C28.23D4, C14×Q16, C56.37D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C8.C22, C7⋊D4, C22×D7, C8.2D4, D4×D7, C2×C7⋊D4, Q16⋊D7, C28⋊D4, C56.37D4
►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)
(1 121 212 96)(2 134 213 109)(3 147 214 66)(4 160 215 79)(5 117 216 92)(6 130 217 105)(7 143 218 62)(8 156 219 75)(9 113 220 88)(10 126 221 101)(11 139 222 58)(12 152 223 71)(13 165 224 84)(14 122 169 97)(15 135 170 110)(16 148 171 67)(17 161 172 80)(18 118 173 93)(19 131 174 106)(20 144 175 63)(21 157 176 76)(22 114 177 89)(23 127 178 102)(24 140 179 59)(25 153 180 72)(26 166 181 85)(27 123 182 98)(28 136 183 111)(29 149 184 68)(30 162 185 81)(31 119 186 94)(32 132 187 107)(33 145 188 64)(34 158 189 77)(35 115 190 90)(36 128 191 103)(37 141 192 60)(38 154 193 73)(39 167 194 86)(40 124 195 99)(41 137 196 112)(42 150 197 69)(43 163 198 82)(44 120 199 95)(45 133 200 108)(46 146 201 65)(47 159 202 78)(48 116 203 91)(49 129 204 104)(50 142 205 61)(51 155 206 74)(52 168 207 87)(53 125 208 100)(54 138 209 57)(55 151 210 70)(56 164 211 83)
(2 28)(3 55)(4 26)(5 53)(6 24)(7 51)(8 22)(9 49)(10 20)(11 47)(12 18)(13 45)(14 16)(15 43)(17 41)(19 39)(21 37)(23 35)(25 33)(27 31)(30 56)(32 54)(34 52)(36 50)(38 48)(40 46)(42 44)(57 160)(58 131)(59 158)(60 129)(61 156)(62 127)(63 154)(64 125)(65 152)(66 123)(67 150)(68 121)(69 148)(70 119)(71 146)(72 117)(73 144)(74 115)(75 142)(76 113)(77 140)(78 167)(79 138)(80 165)(81 136)(82 163)(83 134)(84 161)(85 132)(86 159)(87 130)(88 157)(89 128)(90 155)(91 126)(92 153)(93 124)(94 151)(95 122)(96 149)(97 120)(98 147)(99 118)(100 145)(101 116)(102 143)(103 114)(104 141)(105 168)(106 139)(107 166)(108 137)(109 164)(110 135)(111 162)(112 133)(169 171)(170 198)(172 196)(173 223)(174 194)(175 221)(176 192)(177 219)(178 190)(179 217)(180 188)(181 215)(182 186)(183 213)(185 211)(187 209)(189 207)(191 205)(193 203)(195 201)(197 199)(200 224)(202 222)(204 220)(206 218)(208 216)(210 214)
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), (1,121,212,96)(2,134,213,109)(3,147,214,66)(4,160,215,79)(5,117,216,92)(6,130,217,105)(7,143,218,62)(8,156,219,75)(9,113,220,88)(10,126,221,101)(11,139,222,58)(12,152,223,71)(13,165,224,84)(14,122,169,97)(15,135,170,110)(16,148,171,67)(17,161,172,80)(18,118,173,93)(19,131,174,106)(20,144,175,63)(21,157,176,76)(22,114,177,89)(23,127,178,102)(24,140,179,59)(25,153,180,72)(26,166,181,85)(27,123,182,98)(28,136,183,111)(29,149,184,68)(30,162,185,81)(31,119,186,94)(32,132,187,107)(33,145,188,64)(34,158,189,77)(35,115,190,90)(36,128,191,103)(37,141,192,60)(38,154,193,73)(39,167,194,86)(40,124,195,99)(41,137,196,112)(42,150,197,69)(43,163,198,82)(44,120,199,95)(45,133,200,108)(46,146,201,65)(47,159,202,78)(48,116,203,91)(49,129,204,104)(50,142,205,61)(51,155,206,74)(52,168,207,87)(53,125,208,100)(54,138,209,57)(55,151,210,70)(56,164,211,83), (2,28)(3,55)(4,26)(5,53)(6,24)(7,51)(8,22)(9,49)(10,20)(11,47)(12,18)(13,45)(14,16)(15,43)(17,41)(19,39)(21,37)(23,35)(25,33)(27,31)(30,56)(32,54)(34,52)(36,50)(38,48)(40,46)(42,44)(57,160)(58,131)(59,158)(60,129)(61,156)(62,127)(63,154)(64,125)(65,152)(66,123)(67,150)(68,121)(69,148)(70,119)(71,146)(72,117)(73,144)(74,115)(75,142)(76,113)(77,140)(78,167)(79,138)(80,165)(81,136)(82,163)(83,134)(84,161)(85,132)(86,159)(87,130)(88,157)(89,128)(90,155)(91,126)(92,153)(93,124)(94,151)(95,122)(96,149)(97,120)(98,147)(99,118)(100,145)(101,116)(102,143)(103,114)(104,141)(105,168)(106,139)(107,166)(108,137)(109,164)(110,135)(111,162)(112,133)(169,171)(170,198)(172,196)(173,223)(174,194)(175,221)(176,192)(177,219)(178,190)(179,217)(180,188)(181,215)(182,186)(183,213)(185,211)(187,209)(189,207)(191,205)(193,203)(195,201)(197,199)(200,224)(202,222)(204,220)(206,218)(208,216)(210,214)>;
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), (1,121,212,96)(2,134,213,109)(3,147,214,66)(4,160,215,79)(5,117,216,92)(6,130,217,105)(7,143,218,62)(8,156,219,75)(9,113,220,88)(10,126,221,101)(11,139,222,58)(12,152,223,71)(13,165,224,84)(14,122,169,97)(15,135,170,110)(16,148,171,67)(17,161,172,80)(18,118,173,93)(19,131,174,106)(20,144,175,63)(21,157,176,76)(22,114,177,89)(23,127,178,102)(24,140,179,59)(25,153,180,72)(26,166,181,85)(27,123,182,98)(28,136,183,111)(29,149,184,68)(30,162,185,81)(31,119,186,94)(32,132,187,107)(33,145,188,64)(34,158,189,77)(35,115,190,90)(36,128,191,103)(37,141,192,60)(38,154,193,73)(39,167,194,86)(40,124,195,99)(41,137,196,112)(42,150,197,69)(43,163,198,82)(44,120,199,95)(45,133,200,108)(46,146,201,65)(47,159,202,78)(48,116,203,91)(49,129,204,104)(50,142,205,61)(51,155,206,74)(52,168,207,87)(53,125,208,100)(54,138,209,57)(55,151,210,70)(56,164,211,83), (2,28)(3,55)(4,26)(5,53)(6,24)(7,51)(8,22)(9,49)(10,20)(11,47)(12,18)(13,45)(14,16)(15,43)(17,41)(19,39)(21,37)(23,35)(25,33)(27,31)(30,56)(32,54)(34,52)(36,50)(38,48)(40,46)(42,44)(57,160)(58,131)(59,158)(60,129)(61,156)(62,127)(63,154)(64,125)(65,152)(66,123)(67,150)(68,121)(69,148)(70,119)(71,146)(72,117)(73,144)(74,115)(75,142)(76,113)(77,140)(78,167)(79,138)(80,165)(81,136)(82,163)(83,134)(84,161)(85,132)(86,159)(87,130)(88,157)(89,128)(90,155)(91,126)(92,153)(93,124)(94,151)(95,122)(96,149)(97,120)(98,147)(99,118)(100,145)(101,116)(102,143)(103,114)(104,141)(105,168)(106,139)(107,166)(108,137)(109,164)(110,135)(111,162)(112,133)(169,171)(170,198)(172,196)(173,223)(174,194)(175,221)(176,192)(177,219)(178,190)(179,217)(180,188)(181,215)(182,186)(183,213)(185,211)(187,209)(189,207)(191,205)(193,203)(195,201)(197,199)(200,224)(202,222)(204,220)(206,218)(208,216)(210,214) );
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)], [(1,121,212,96),(2,134,213,109),(3,147,214,66),(4,160,215,79),(5,117,216,92),(6,130,217,105),(7,143,218,62),(8,156,219,75),(9,113,220,88),(10,126,221,101),(11,139,222,58),(12,152,223,71),(13,165,224,84),(14,122,169,97),(15,135,170,110),(16,148,171,67),(17,161,172,80),(18,118,173,93),(19,131,174,106),(20,144,175,63),(21,157,176,76),(22,114,177,89),(23,127,178,102),(24,140,179,59),(25,153,180,72),(26,166,181,85),(27,123,182,98),(28,136,183,111),(29,149,184,68),(30,162,185,81),(31,119,186,94),(32,132,187,107),(33,145,188,64),(34,158,189,77),(35,115,190,90),(36,128,191,103),(37,141,192,60),(38,154,193,73),(39,167,194,86),(40,124,195,99),(41,137,196,112),(42,150,197,69),(43,163,198,82),(44,120,199,95),(45,133,200,108),(46,146,201,65),(47,159,202,78),(48,116,203,91),(49,129,204,104),(50,142,205,61),(51,155,206,74),(52,168,207,87),(53,125,208,100),(54,138,209,57),(55,151,210,70),(56,164,211,83)], [(2,28),(3,55),(4,26),(5,53),(6,24),(7,51),(8,22),(9,49),(10,20),(11,47),(12,18),(13,45),(14,16),(15,43),(17,41),(19,39),(21,37),(23,35),(25,33),(27,31),(30,56),(32,54),(34,52),(36,50),(38,48),(40,46),(42,44),(57,160),(58,131),(59,158),(60,129),(61,156),(62,127),(63,154),(64,125),(65,152),(66,123),(67,150),(68,121),(69,148),(70,119),(71,146),(72,117),(73,144),(74,115),(75,142),(76,113),(77,140),(78,167),(79,138),(80,165),(81,136),(82,163),(83,134),(84,161),(85,132),(86,159),(87,130),(88,157),(89,128),(90,155),(91,126),(92,153),(93,124),(94,151),(95,122),(96,149),(97,120),(98,147),(99,118),(100,145),(101,116),(102,143),(103,114),(104,141),(105,168),(106,139),(107,166),(108,137),(109,164),(110,135),(111,162),(112,133),(169,171),(170,198),(172,196),(173,223),(174,194),(175,221),(176,192),(177,219),(178,190),(179,217),(180,188),(181,215),(182,186),(183,213),(185,211),(187,209),(189,207),(191,205),(193,203),(195,201),(197,199),(200,224),(202,222),(204,220),(206,218),(208,216),(210,214)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 56 | 2 | 2 | 8 | 8 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D14 | D14 | C7⋊D4 | C8.C22 | D4×D7 | D4×D7 | Q16⋊D7 |
| kernel | C56.37D4 | C56⋊C4 | C2×C56⋊C2 | C2×Q8⋊D7 | C2×C7⋊Q16 | Dic7⋊Q8 | C28.23D4 | C14×Q16 | C7⋊C8 | C56 | C2×Dic7 | C2×Q16 | C2×C8 | C2×Q8 | C8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 6 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56.37D4 ►in GL6(𝔽113)
| 0 | 112 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 39 | 84 | 74 |
| 0 | 0 | 74 | 41 | 39 | 72 |
| 0 | 0 | 71 | 76 | 0 | 0 |
| 0 | 0 | 37 | 77 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 55 | 38 | 0 | 0 |
| 0 | 0 | 81 | 58 | 0 | 0 |
| 0 | 0 | 0 | 0 | 55 | 38 |
| 0 | 0 | 0 | 0 | 81 | 58 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 | 112 | 0 |
| 0 | 0 | 9 | 112 | 104 | 1 |
G:=sub<GL(6,GF(113))| [0,1,0,0,0,0,112,0,0,0,0,0,0,0,29,74,71,37,0,0,39,41,76,77,0,0,84,39,0,0,0,0,74,72,0,0],[0,1,0,0,0,0,112,0,0,0,0,0,0,0,55,81,0,0,0,0,38,58,0,0,0,0,0,0,55,81,0,0,0,0,38,58],[112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,9,1,9,0,0,0,112,0,112,0,0,0,0,112,104,0,0,0,0,0,1] >;
C56.37D4 in GAP, Magma, Sage, TeX
C_{56}._{37}D_4 % in TeX
G:=Group("C56.37D4"); // GroupNames label
G:=SmallGroup(448,724);
// by ID
G=gap.SmallGroup(448,724);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,219,184,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^13,c*a*c=a^27,c*b*c=a^28*b^-1>;
// generators/relations