metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊30D4, C23.24D28, C7⋊6(C8⋊8D4), (C22×C8)⋊9D7, C8⋊14(C7⋊D4), C8⋊Dic7⋊18C2, (C2×C14)⋊8SD16, (C2×C4).67D28, C2.D56⋊2C2, (C22×C56)⋊13C2, C28⋊7D4.5C2, (C2×C8).320D14, C28.412(C2×D4), (C2×C28).355D4, C14.17(C4○D8), C28.48D4⋊2C2, C28.44D4⋊2C2, C22⋊1(C56⋊C2), C14.17(C2×SD16), C4.111(C4○D28), C28.227(C4○D4), C2.18(C28⋊7D4), C14.70(C4⋊D4), (C2×C56).393C22, (C2×C28).768C23, (C2×D28).18C22, (C22×C4).430D14, C22.131(C2×D28), (C22×C14).140D4, C4⋊Dic7.23C22, C2.17(D56⋊7C2), (C22×C28).518C22, (C2×Dic14).17C22, (C2×C56⋊C2)⋊21C2, C2.17(C2×C56⋊C2), C4.105(C2×C7⋊D4), (C2×C14).158(C2×D4), (C2×C4).716(C22×D7), SmallGroup(448,648)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊30D4
G = < a,b,c | a56=b4=c2=1, bab-1=cac=a27, cbc=b-1 >
Subgroups: 708 in 124 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C56, C56, Dic14, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8⋊8D4, C56⋊C2, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C2×C56, C2×C56, C2×Dic14, C2×D28, C2×C7⋊D4, C22×C28, C28.44D4, C8⋊Dic7, C2.D56, C2×C56⋊C2, C28.48D4, C28⋊7D4, C22×C56, C56⋊30D4
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C4⋊D4, C2×SD16, C4○D8, D28, C7⋊D4, C22×D7, C8⋊8D4, C56⋊C2, C2×D28, C4○D28, C2×C7⋊D4, C2×C56⋊C2, D56⋊7C2, C28⋊7D4, C56⋊30D4
►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 88 194 160)(2 59 195 131)(3 86 196 158)(4 57 197 129)(5 84 198 156)(6 111 199 127)(7 82 200 154)(8 109 201 125)(9 80 202 152)(10 107 203 123)(11 78 204 150)(12 105 205 121)(13 76 206 148)(14 103 207 119)(15 74 208 146)(16 101 209 117)(17 72 210 144)(18 99 211 115)(19 70 212 142)(20 97 213 113)(21 68 214 140)(22 95 215 167)(23 66 216 138)(24 93 217 165)(25 64 218 136)(26 91 219 163)(27 62 220 134)(28 89 221 161)(29 60 222 132)(30 87 223 159)(31 58 224 130)(32 85 169 157)(33 112 170 128)(34 83 171 155)(35 110 172 126)(36 81 173 153)(37 108 174 124)(38 79 175 151)(39 106 176 122)(40 77 177 149)(41 104 178 120)(42 75 179 147)(43 102 180 118)(44 73 181 145)(45 100 182 116)(46 71 183 143)(47 98 184 114)(48 69 185 141)(49 96 186 168)(50 67 187 139)(51 94 188 166)(52 65 189 137)(53 92 190 164)(54 63 191 135)(55 90 192 162)(56 61 193 133)
(1 160)(2 131)(3 158)(4 129)(5 156)(6 127)(7 154)(8 125)(9 152)(10 123)(11 150)(12 121)(13 148)(14 119)(15 146)(16 117)(17 144)(18 115)(19 142)(20 113)(21 140)(22 167)(23 138)(24 165)(25 136)(26 163)(27 134)(28 161)(29 132)(30 159)(31 130)(32 157)(33 128)(34 155)(35 126)(36 153)(37 124)(38 151)(39 122)(40 149)(41 120)(42 147)(43 118)(44 145)(45 116)(46 143)(47 114)(48 141)(49 168)(50 139)(51 166)(52 137)(53 164)(54 135)(55 162)(56 133)(57 197)(58 224)(59 195)(60 222)(61 193)(62 220)(63 191)(64 218)(65 189)(66 216)(67 187)(68 214)(69 185)(70 212)(71 183)(72 210)(73 181)(74 208)(75 179)(76 206)(77 177)(78 204)(79 175)(80 202)(81 173)(82 200)(83 171)(84 198)(85 169)(86 196)(87 223)(88 194)(89 221)(90 192)(91 219)(92 190)(93 217)(94 188)(95 215)(96 186)(97 213)(98 184)(99 211)(100 182)(101 209)(102 180)(103 207)(104 178)(105 205)(106 176)(107 203)(108 174)(109 201)(110 172)(111 199)(112 170)
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), (1,88,194,160)(2,59,195,131)(3,86,196,158)(4,57,197,129)(5,84,198,156)(6,111,199,127)(7,82,200,154)(8,109,201,125)(9,80,202,152)(10,107,203,123)(11,78,204,150)(12,105,205,121)(13,76,206,148)(14,103,207,119)(15,74,208,146)(16,101,209,117)(17,72,210,144)(18,99,211,115)(19,70,212,142)(20,97,213,113)(21,68,214,140)(22,95,215,167)(23,66,216,138)(24,93,217,165)(25,64,218,136)(26,91,219,163)(27,62,220,134)(28,89,221,161)(29,60,222,132)(30,87,223,159)(31,58,224,130)(32,85,169,157)(33,112,170,128)(34,83,171,155)(35,110,172,126)(36,81,173,153)(37,108,174,124)(38,79,175,151)(39,106,176,122)(40,77,177,149)(41,104,178,120)(42,75,179,147)(43,102,180,118)(44,73,181,145)(45,100,182,116)(46,71,183,143)(47,98,184,114)(48,69,185,141)(49,96,186,168)(50,67,187,139)(51,94,188,166)(52,65,189,137)(53,92,190,164)(54,63,191,135)(55,90,192,162)(56,61,193,133), (1,160)(2,131)(3,158)(4,129)(5,156)(6,127)(7,154)(8,125)(9,152)(10,123)(11,150)(12,121)(13,148)(14,119)(15,146)(16,117)(17,144)(18,115)(19,142)(20,113)(21,140)(22,167)(23,138)(24,165)(25,136)(26,163)(27,134)(28,161)(29,132)(30,159)(31,130)(32,157)(33,128)(34,155)(35,126)(36,153)(37,124)(38,151)(39,122)(40,149)(41,120)(42,147)(43,118)(44,145)(45,116)(46,143)(47,114)(48,141)(49,168)(50,139)(51,166)(52,137)(53,164)(54,135)(55,162)(56,133)(57,197)(58,224)(59,195)(60,222)(61,193)(62,220)(63,191)(64,218)(65,189)(66,216)(67,187)(68,214)(69,185)(70,212)(71,183)(72,210)(73,181)(74,208)(75,179)(76,206)(77,177)(78,204)(79,175)(80,202)(81,173)(82,200)(83,171)(84,198)(85,169)(86,196)(87,223)(88,194)(89,221)(90,192)(91,219)(92,190)(93,217)(94,188)(95,215)(96,186)(97,213)(98,184)(99,211)(100,182)(101,209)(102,180)(103,207)(104,178)(105,205)(106,176)(107,203)(108,174)(109,201)(110,172)(111,199)(112,170)>;
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), (1,88,194,160)(2,59,195,131)(3,86,196,158)(4,57,197,129)(5,84,198,156)(6,111,199,127)(7,82,200,154)(8,109,201,125)(9,80,202,152)(10,107,203,123)(11,78,204,150)(12,105,205,121)(13,76,206,148)(14,103,207,119)(15,74,208,146)(16,101,209,117)(17,72,210,144)(18,99,211,115)(19,70,212,142)(20,97,213,113)(21,68,214,140)(22,95,215,167)(23,66,216,138)(24,93,217,165)(25,64,218,136)(26,91,219,163)(27,62,220,134)(28,89,221,161)(29,60,222,132)(30,87,223,159)(31,58,224,130)(32,85,169,157)(33,112,170,128)(34,83,171,155)(35,110,172,126)(36,81,173,153)(37,108,174,124)(38,79,175,151)(39,106,176,122)(40,77,177,149)(41,104,178,120)(42,75,179,147)(43,102,180,118)(44,73,181,145)(45,100,182,116)(46,71,183,143)(47,98,184,114)(48,69,185,141)(49,96,186,168)(50,67,187,139)(51,94,188,166)(52,65,189,137)(53,92,190,164)(54,63,191,135)(55,90,192,162)(56,61,193,133), (1,160)(2,131)(3,158)(4,129)(5,156)(6,127)(7,154)(8,125)(9,152)(10,123)(11,150)(12,121)(13,148)(14,119)(15,146)(16,117)(17,144)(18,115)(19,142)(20,113)(21,140)(22,167)(23,138)(24,165)(25,136)(26,163)(27,134)(28,161)(29,132)(30,159)(31,130)(32,157)(33,128)(34,155)(35,126)(36,153)(37,124)(38,151)(39,122)(40,149)(41,120)(42,147)(43,118)(44,145)(45,116)(46,143)(47,114)(48,141)(49,168)(50,139)(51,166)(52,137)(53,164)(54,135)(55,162)(56,133)(57,197)(58,224)(59,195)(60,222)(61,193)(62,220)(63,191)(64,218)(65,189)(66,216)(67,187)(68,214)(69,185)(70,212)(71,183)(72,210)(73,181)(74,208)(75,179)(76,206)(77,177)(78,204)(79,175)(80,202)(81,173)(82,200)(83,171)(84,198)(85,169)(86,196)(87,223)(88,194)(89,221)(90,192)(91,219)(92,190)(93,217)(94,188)(95,215)(96,186)(97,213)(98,184)(99,211)(100,182)(101,209)(102,180)(103,207)(104,178)(105,205)(106,176)(107,203)(108,174)(109,201)(110,172)(111,199)(112,170) );
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)], [(1,88,194,160),(2,59,195,131),(3,86,196,158),(4,57,197,129),(5,84,198,156),(6,111,199,127),(7,82,200,154),(8,109,201,125),(9,80,202,152),(10,107,203,123),(11,78,204,150),(12,105,205,121),(13,76,206,148),(14,103,207,119),(15,74,208,146),(16,101,209,117),(17,72,210,144),(18,99,211,115),(19,70,212,142),(20,97,213,113),(21,68,214,140),(22,95,215,167),(23,66,216,138),(24,93,217,165),(25,64,218,136),(26,91,219,163),(27,62,220,134),(28,89,221,161),(29,60,222,132),(30,87,223,159),(31,58,224,130),(32,85,169,157),(33,112,170,128),(34,83,171,155),(35,110,172,126),(36,81,173,153),(37,108,174,124),(38,79,175,151),(39,106,176,122),(40,77,177,149),(41,104,178,120),(42,75,179,147),(43,102,180,118),(44,73,181,145),(45,100,182,116),(46,71,183,143),(47,98,184,114),(48,69,185,141),(49,96,186,168),(50,67,187,139),(51,94,188,166),(52,65,189,137),(53,92,190,164),(54,63,191,135),(55,90,192,162),(56,61,193,133)], [(1,160),(2,131),(3,158),(4,129),(5,156),(6,127),(7,154),(8,125),(9,152),(10,123),(11,150),(12,121),(13,148),(14,119),(15,146),(16,117),(17,144),(18,115),(19,142),(20,113),(21,140),(22,167),(23,138),(24,165),(25,136),(26,163),(27,134),(28,161),(29,132),(30,159),(31,130),(32,157),(33,128),(34,155),(35,126),(36,153),(37,124),(38,151),(39,122),(40,149),(41,120),(42,147),(43,118),(44,145),(45,116),(46,143),(47,114),(48,141),(49,168),(50,139),(51,166),(52,137),(53,164),(54,135),(55,162),(56,133),(57,197),(58,224),(59,195),(60,222),(61,193),(62,220),(63,191),(64,218),(65,189),(66,216),(67,187),(68,214),(69,185),(70,212),(71,183),(72,210),(73,181),(74,208),(75,179),(76,206),(77,177),(78,204),(79,175),(80,202),(81,173),(82,200),(83,171),(84,198),(85,169),(86,196),(87,223),(88,194),(89,221),(90,192),(91,219),(92,190),(93,217),(94,188),(95,215),(96,186),(97,213),(98,184),(99,211),(100,182),(101,209),(102,180),(103,207),(104,178),(105,205),(106,176),(107,203),(108,174),(109,201),(110,172),(111,199),(112,170)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 56 | 2 | 2 | 2 | 2 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | SD16 | D14 | D14 | C4○D8 | C7⋊D4 | D28 | D28 | C4○D28 | C56⋊C2 | D56⋊7C2 |
| kernel | C56⋊30D4 | C28.44D4 | C8⋊Dic7 | C2.D56 | C2×C56⋊C2 | C28.48D4 | C28⋊7D4 | C22×C56 | C56 | C2×C28 | C22×C14 | C22×C8 | C28 | C2×C14 | C2×C8 | C22×C4 | C14 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 4 | 6 | 3 | 4 | 12 | 6 | 6 | 12 | 24 | 24 |
Matrix representation of C56⋊30D4 ►in GL4(𝔽113) generated by
| 0 | 87 | 0 | 0 |
| 13 | 87 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 4 | 25 |
| 112 | 0 | 0 | 0 |
| 112 | 1 | 0 | 0 |
| 0 | 0 | 56 | 111 |
| 0 | 0 | 43 | 57 |
| 112 | 0 | 0 | 0 |
| 112 | 1 | 0 | 0 |
| 0 | 0 | 56 | 111 |
| 0 | 0 | 42 | 57 |
G:=sub<GL(4,GF(113))| [0,13,0,0,87,87,0,0,0,0,9,4,0,0,0,25],[112,112,0,0,0,1,0,0,0,0,56,43,0,0,111,57],[112,112,0,0,0,1,0,0,0,0,56,42,0,0,111,57] >;
C56⋊30D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_{30}D_4 % in TeX
G:=Group("C56:30D4"); // GroupNames label
G:=SmallGroup(448,648);
// by ID
G=gap.SmallGroup(448,648);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=c*a*c=a^27,c*b*c=b^-1>;
// generators/relations