metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊15D4, Dic7⋊1SD16, C7⋊C8⋊15D4, C8⋊8(C7⋊D4), C7⋊4(C8⋊5D4), C4.25(D4×D7), C28.50(C2×D4), (C14×SD16)⋊9C2, (C2×SD16)⋊15D7, (C8×Dic7)⋊11C2, (C2×D4).76D14, (C2×C8).264D14, C28⋊D4.6C2, (C2×Q8).57D14, C2.31(D7×SD16), Dic7⋊Q8⋊5C2, C14.48(C2×SD16), C22.272(D4×D7), C14.31(C4⋊1D4), C2.22(C28⋊D4), (C2×C28).452C23, (C2×C56).165C22, (C2×Dic7).115D4, (Q8×C14).81C22, (D4×C14).101C22, (C2×D28).122C22, (C4×Dic7).242C22, (C2×Dic14).129C22, C4.9(C2×C7⋊D4), (C2×Q8⋊D7)⋊18C2, (C2×C56⋊C2)⋊30C2, (C2×D4.D7)⋊21C2, (C2×C14).364(C2×D4), (C2×C7⋊C8).275C22, (C2×C4).541(C22×D7), SmallGroup(448,709)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊15D4
G = < a,b,c | a56=b4=c2=1, bab-1=a41, cac=a27, cbc=b-1 >
Subgroups: 836 in 142 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C14, C42, C4⋊C4, C2×C8, C2×C8, SD16, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×C14, C2×C14, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C2×SD16, C7⋊C8, C56, Dic14, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×C14, C8⋊5D4, C56⋊C2, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D4.D7, Q8⋊D7, C2×C56, C7×SD16, C2×Dic14, C2×D28, C2×C7⋊D4, D4×C14, Q8×C14, C8×Dic7, C2×C56⋊C2, C2×D4.D7, C28⋊D4, C2×Q8⋊D7, Dic7⋊Q8, C14×SD16, C56⋊15D4
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C4⋊1D4, C2×SD16, C7⋊D4, C22×D7, C8⋊5D4, D4×D7, C2×C7⋊D4, D7×SD16, C28⋊D4, C56⋊15D4
►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)
(1 180 142 106)(2 221 143 91)(3 206 144 76)(4 191 145 61)(5 176 146 102)(6 217 147 87)(7 202 148 72)(8 187 149 57)(9 172 150 98)(10 213 151 83)(11 198 152 68)(12 183 153 109)(13 224 154 94)(14 209 155 79)(15 194 156 64)(16 179 157 105)(17 220 158 90)(18 205 159 75)(19 190 160 60)(20 175 161 101)(21 216 162 86)(22 201 163 71)(23 186 164 112)(24 171 165 97)(25 212 166 82)(26 197 167 67)(27 182 168 108)(28 223 113 93)(29 208 114 78)(30 193 115 63)(31 178 116 104)(32 219 117 89)(33 204 118 74)(34 189 119 59)(35 174 120 100)(36 215 121 85)(37 200 122 70)(38 185 123 111)(39 170 124 96)(40 211 125 81)(41 196 126 66)(42 181 127 107)(43 222 128 92)(44 207 129 77)(45 192 130 62)(46 177 131 103)(47 218 132 88)(48 203 133 73)(49 188 134 58)(50 173 135 99)(51 214 136 84)(52 199 137 69)(53 184 138 110)(54 169 139 95)(55 210 140 80)(56 195 141 65)
(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)(57 201)(58 172)(59 199)(60 170)(61 197)(62 224)(63 195)(64 222)(65 193)(66 220)(67 191)(68 218)(69 189)(70 216)(71 187)(72 214)(73 185)(74 212)(75 183)(76 210)(77 181)(78 208)(79 179)(80 206)(81 177)(82 204)(83 175)(84 202)(85 173)(86 200)(87 171)(88 198)(89 169)(90 196)(91 223)(92 194)(93 221)(94 192)(95 219)(96 190)(97 217)(98 188)(99 215)(100 186)(101 213)(102 184)(103 211)(104 182)(105 209)(106 180)(107 207)(108 178)(109 205)(110 176)(111 203)(112 174)(113 143)(115 141)(116 168)(117 139)(118 166)(119 137)(120 164)(121 135)(122 162)(123 133)(124 160)(125 131)(126 158)(127 129)(128 156)(130 154)(132 152)(134 150)(136 148)(138 146)(140 144)(145 167)(147 165)(149 163)(151 161)(153 159)(155 157)
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), (1,180,142,106)(2,221,143,91)(3,206,144,76)(4,191,145,61)(5,176,146,102)(6,217,147,87)(7,202,148,72)(8,187,149,57)(9,172,150,98)(10,213,151,83)(11,198,152,68)(12,183,153,109)(13,224,154,94)(14,209,155,79)(15,194,156,64)(16,179,157,105)(17,220,158,90)(18,205,159,75)(19,190,160,60)(20,175,161,101)(21,216,162,86)(22,201,163,71)(23,186,164,112)(24,171,165,97)(25,212,166,82)(26,197,167,67)(27,182,168,108)(28,223,113,93)(29,208,114,78)(30,193,115,63)(31,178,116,104)(32,219,117,89)(33,204,118,74)(34,189,119,59)(35,174,120,100)(36,215,121,85)(37,200,122,70)(38,185,123,111)(39,170,124,96)(40,211,125,81)(41,196,126,66)(42,181,127,107)(43,222,128,92)(44,207,129,77)(45,192,130,62)(46,177,131,103)(47,218,132,88)(48,203,133,73)(49,188,134,58)(50,173,135,99)(51,214,136,84)(52,199,137,69)(53,184,138,110)(54,169,139,95)(55,210,140,80)(56,195,141,65), (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)(57,201)(58,172)(59,199)(60,170)(61,197)(62,224)(63,195)(64,222)(65,193)(66,220)(67,191)(68,218)(69,189)(70,216)(71,187)(72,214)(73,185)(74,212)(75,183)(76,210)(77,181)(78,208)(79,179)(80,206)(81,177)(82,204)(83,175)(84,202)(85,173)(86,200)(87,171)(88,198)(89,169)(90,196)(91,223)(92,194)(93,221)(94,192)(95,219)(96,190)(97,217)(98,188)(99,215)(100,186)(101,213)(102,184)(103,211)(104,182)(105,209)(106,180)(107,207)(108,178)(109,205)(110,176)(111,203)(112,174)(113,143)(115,141)(116,168)(117,139)(118,166)(119,137)(120,164)(121,135)(122,162)(123,133)(124,160)(125,131)(126,158)(127,129)(128,156)(130,154)(132,152)(134,150)(136,148)(138,146)(140,144)(145,167)(147,165)(149,163)(151,161)(153,159)(155,157)>;
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), (1,180,142,106)(2,221,143,91)(3,206,144,76)(4,191,145,61)(5,176,146,102)(6,217,147,87)(7,202,148,72)(8,187,149,57)(9,172,150,98)(10,213,151,83)(11,198,152,68)(12,183,153,109)(13,224,154,94)(14,209,155,79)(15,194,156,64)(16,179,157,105)(17,220,158,90)(18,205,159,75)(19,190,160,60)(20,175,161,101)(21,216,162,86)(22,201,163,71)(23,186,164,112)(24,171,165,97)(25,212,166,82)(26,197,167,67)(27,182,168,108)(28,223,113,93)(29,208,114,78)(30,193,115,63)(31,178,116,104)(32,219,117,89)(33,204,118,74)(34,189,119,59)(35,174,120,100)(36,215,121,85)(37,200,122,70)(38,185,123,111)(39,170,124,96)(40,211,125,81)(41,196,126,66)(42,181,127,107)(43,222,128,92)(44,207,129,77)(45,192,130,62)(46,177,131,103)(47,218,132,88)(48,203,133,73)(49,188,134,58)(50,173,135,99)(51,214,136,84)(52,199,137,69)(53,184,138,110)(54,169,139,95)(55,210,140,80)(56,195,141,65), (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)(57,201)(58,172)(59,199)(60,170)(61,197)(62,224)(63,195)(64,222)(65,193)(66,220)(67,191)(68,218)(69,189)(70,216)(71,187)(72,214)(73,185)(74,212)(75,183)(76,210)(77,181)(78,208)(79,179)(80,206)(81,177)(82,204)(83,175)(84,202)(85,173)(86,200)(87,171)(88,198)(89,169)(90,196)(91,223)(92,194)(93,221)(94,192)(95,219)(96,190)(97,217)(98,188)(99,215)(100,186)(101,213)(102,184)(103,211)(104,182)(105,209)(106,180)(107,207)(108,178)(109,205)(110,176)(111,203)(112,174)(113,143)(115,141)(116,168)(117,139)(118,166)(119,137)(120,164)(121,135)(122,162)(123,133)(124,160)(125,131)(126,158)(127,129)(128,156)(130,154)(132,152)(134,150)(136,148)(138,146)(140,144)(145,167)(147,165)(149,163)(151,161)(153,159)(155,157) );
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)], [(1,180,142,106),(2,221,143,91),(3,206,144,76),(4,191,145,61),(5,176,146,102),(6,217,147,87),(7,202,148,72),(8,187,149,57),(9,172,150,98),(10,213,151,83),(11,198,152,68),(12,183,153,109),(13,224,154,94),(14,209,155,79),(15,194,156,64),(16,179,157,105),(17,220,158,90),(18,205,159,75),(19,190,160,60),(20,175,161,101),(21,216,162,86),(22,201,163,71),(23,186,164,112),(24,171,165,97),(25,212,166,82),(26,197,167,67),(27,182,168,108),(28,223,113,93),(29,208,114,78),(30,193,115,63),(31,178,116,104),(32,219,117,89),(33,204,118,74),(34,189,119,59),(35,174,120,100),(36,215,121,85),(37,200,122,70),(38,185,123,111),(39,170,124,96),(40,211,125,81),(41,196,126,66),(42,181,127,107),(43,222,128,92),(44,207,129,77),(45,192,130,62),(46,177,131,103),(47,218,132,88),(48,203,133,73),(49,188,134,58),(50,173,135,99),(51,214,136,84),(52,199,137,69),(53,184,138,110),(54,169,139,95),(55,210,140,80),(56,195,141,65)], [(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),(57,201),(58,172),(59,199),(60,170),(61,197),(62,224),(63,195),(64,222),(65,193),(66,220),(67,191),(68,218),(69,189),(70,216),(71,187),(72,214),(73,185),(74,212),(75,183),(76,210),(77,181),(78,208),(79,179),(80,206),(81,177),(82,204),(83,175),(84,202),(85,173),(86,200),(87,171),(88,198),(89,169),(90,196),(91,223),(92,194),(93,221),(94,192),(95,219),(96,190),(97,217),(98,188),(99,215),(100,186),(101,213),(102,184),(103,211),(104,182),(105,209),(106,180),(107,207),(108,178),(109,205),(110,176),(111,203),(112,174),(113,143),(115,141),(116,168),(117,139),(118,166),(119,137),(120,164),(121,135),(122,162),(123,133),(124,160),(125,131),(126,158),(127,129),(128,156),(130,154),(132,152),(134,150),(136,148),(138,146),(140,144),(145,167),(147,165),(149,163),(151,161),(153,159),(155,157)]])
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 | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 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 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 56 | 2 | 2 | 8 | 14 | 14 | 14 | 14 | 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 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | SD16 | D14 | D14 | D14 | C7⋊D4 | D4×D7 | D4×D7 | D7×SD16 |
| kernel | C56⋊15D4 | C8×Dic7 | C2×C56⋊C2 | C2×D4.D7 | C28⋊D4 | C2×Q8⋊D7 | Dic7⋊Q8 | C14×SD16 | C7⋊C8 | C56 | C2×Dic7 | C2×SD16 | Dic7 | C2×C8 | C2×D4 | C2×Q8 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 8 | 3 | 3 | 3 | 12 | 3 | 3 | 12 |
Matrix representation of C56⋊15D4 ►in GL4(𝔽113) generated by
| 34 | 1 | 0 | 0 |
| 111 | 103 | 0 | 0 |
| 0 | 0 | 0 | 22 |
| 0 | 0 | 36 | 26 |
| 53 | 25 | 0 | 0 |
| 91 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 98 | 89 | 0 | 0 |
| 47 | 15 | 0 | 0 |
| 0 | 0 | 1 | 7 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [34,111,0,0,1,103,0,0,0,0,0,36,0,0,22,26],[53,91,0,0,25,60,0,0,0,0,1,0,0,0,0,1],[98,47,0,0,89,15,0,0,0,0,1,0,0,0,7,112] >;
C56⋊15D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_{15}D_4 % in TeX
G:=Group("C56:15D4"); // GroupNames label
G:=SmallGroup(448,709);
// by ID
G=gap.SmallGroup(448,709);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,254,555,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^41,c*a*c=a^27,c*b*c=b^-1>;
// generators/relations