metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊5D4, Dic7⋊1D8, C7⋊C8⋊14D4, (C2×D8)⋊3D7, (C14×D8)⋊5C2, C8⋊7(C7⋊D4), C7⋊3(C8⋊4D4), C2.27(D7×D8), C4.20(D4×D7), (C2×D56)⋊18C2, C28⋊D4⋊4C2, (C8×Dic7)⋊5C2, C28.45(C2×D4), C14.44(C2×D8), (C2×D4).60D14, (C2×C8).236D14, (C2×C56).88C22, C22.252(D4×D7), C2.17(C28⋊D4), C14.26(C4⋊1D4), (C2×C28).428C23, (C2×Dic7).110D4, (D4×C14).78C22, (C2×D28).114C22, (C4×Dic7).238C22, C4.4(C2×C7⋊D4), (C2×D4⋊D7)⋊17C2, (C2×C14).341(C2×D4), (C2×C7⋊C8).269C22, (C2×C4).518(C22×D7), SmallGroup(448,685)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊5D4
G = < a,b,c | a56=b4=c2=1, bab-1=a41, cac=a-1, cbc=b-1 >
Subgroups: 1124 in 162 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, D7, C14, C14, C14, C42, C2×C8, C2×C8, D8, C2×D4, C2×D4, Dic7, C28, D14, C2×C14, C2×C14, C4×C8, C4⋊1D4, C2×D8, C2×D8, C7⋊C8, C56, D28, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, C8⋊4D4, D56, C2×C7⋊C8, C4×Dic7, D4⋊D7, C2×C56, C7×D8, C2×D28, C2×C7⋊D4, D4×C14, C8×Dic7, C2×D56, C2×D4⋊D7, C28⋊D4, C14×D8, C56⋊5D4
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, C4⋊1D4, C2×D8, C7⋊D4, C22×D7, C8⋊4D4, D4×D7, C2×C7⋊D4, D7×D8, C28⋊D4, C56⋊5D4
►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 84 123 169)(2 69 124 210)(3 110 125 195)(4 95 126 180)(5 80 127 221)(6 65 128 206)(7 106 129 191)(8 91 130 176)(9 76 131 217)(10 61 132 202)(11 102 133 187)(12 87 134 172)(13 72 135 213)(14 57 136 198)(15 98 137 183)(16 83 138 224)(17 68 139 209)(18 109 140 194)(19 94 141 179)(20 79 142 220)(21 64 143 205)(22 105 144 190)(23 90 145 175)(24 75 146 216)(25 60 147 201)(26 101 148 186)(27 86 149 171)(28 71 150 212)(29 112 151 197)(30 97 152 182)(31 82 153 223)(32 67 154 208)(33 108 155 193)(34 93 156 178)(35 78 157 219)(36 63 158 204)(37 104 159 189)(38 89 160 174)(39 74 161 215)(40 59 162 200)(41 100 163 185)(42 85 164 170)(43 70 165 211)(44 111 166 196)(45 96 167 181)(46 81 168 222)(47 66 113 207)(48 107 114 192)(49 92 115 177)(50 77 116 218)(51 62 117 203)(52 103 118 188)(53 88 119 173)(54 73 120 214)(55 58 121 199)(56 99 122 184)
(1 50)(2 49)(3 48)(4 47)(5 46)(6 45)(7 44)(8 43)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(51 56)(52 55)(53 54)(57 189)(58 188)(59 187)(60 186)(61 185)(62 184)(63 183)(64 182)(65 181)(66 180)(67 179)(68 178)(69 177)(70 176)(71 175)(72 174)(73 173)(74 172)(75 171)(76 170)(77 169)(78 224)(79 223)(80 222)(81 221)(82 220)(83 219)(84 218)(85 217)(86 216)(87 215)(88 214)(89 213)(90 212)(91 211)(92 210)(93 209)(94 208)(95 207)(96 206)(97 205)(98 204)(99 203)(100 202)(101 201)(102 200)(103 199)(104 198)(105 197)(106 196)(107 195)(108 194)(109 193)(110 192)(111 191)(112 190)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 168)(128 167)(129 166)(130 165)(131 164)(132 163)(133 162)(134 161)(135 160)(136 159)(137 158)(138 157)(139 156)(140 155)(141 154)(142 153)(143 152)(144 151)(145 150)(146 149)(147 148)
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), (1,84,123,169)(2,69,124,210)(3,110,125,195)(4,95,126,180)(5,80,127,221)(6,65,128,206)(7,106,129,191)(8,91,130,176)(9,76,131,217)(10,61,132,202)(11,102,133,187)(12,87,134,172)(13,72,135,213)(14,57,136,198)(15,98,137,183)(16,83,138,224)(17,68,139,209)(18,109,140,194)(19,94,141,179)(20,79,142,220)(21,64,143,205)(22,105,144,190)(23,90,145,175)(24,75,146,216)(25,60,147,201)(26,101,148,186)(27,86,149,171)(28,71,150,212)(29,112,151,197)(30,97,152,182)(31,82,153,223)(32,67,154,208)(33,108,155,193)(34,93,156,178)(35,78,157,219)(36,63,158,204)(37,104,159,189)(38,89,160,174)(39,74,161,215)(40,59,162,200)(41,100,163,185)(42,85,164,170)(43,70,165,211)(44,111,166,196)(45,96,167,181)(46,81,168,222)(47,66,113,207)(48,107,114,192)(49,92,115,177)(50,77,116,218)(51,62,117,203)(52,103,118,188)(53,88,119,173)(54,73,120,214)(55,58,121,199)(56,99,122,184), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,56)(52,55)(53,54)(57,189)(58,188)(59,187)(60,186)(61,185)(62,184)(63,183)(64,182)(65,181)(66,180)(67,179)(68,178)(69,177)(70,176)(71,175)(72,174)(73,173)(74,172)(75,171)(76,170)(77,169)(78,224)(79,223)(80,222)(81,221)(82,220)(83,219)(84,218)(85,217)(86,216)(87,215)(88,214)(89,213)(90,212)(91,211)(92,210)(93,209)(94,208)(95,207)(96,206)(97,205)(98,204)(99,203)(100,202)(101,201)(102,200)(103,199)(104,198)(105,197)(106,196)(107,195)(108,194)(109,193)(110,192)(111,191)(112,190)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,168)(128,167)(129,166)(130,165)(131,164)(132,163)(133,162)(134,161)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148)>;
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), (1,84,123,169)(2,69,124,210)(3,110,125,195)(4,95,126,180)(5,80,127,221)(6,65,128,206)(7,106,129,191)(8,91,130,176)(9,76,131,217)(10,61,132,202)(11,102,133,187)(12,87,134,172)(13,72,135,213)(14,57,136,198)(15,98,137,183)(16,83,138,224)(17,68,139,209)(18,109,140,194)(19,94,141,179)(20,79,142,220)(21,64,143,205)(22,105,144,190)(23,90,145,175)(24,75,146,216)(25,60,147,201)(26,101,148,186)(27,86,149,171)(28,71,150,212)(29,112,151,197)(30,97,152,182)(31,82,153,223)(32,67,154,208)(33,108,155,193)(34,93,156,178)(35,78,157,219)(36,63,158,204)(37,104,159,189)(38,89,160,174)(39,74,161,215)(40,59,162,200)(41,100,163,185)(42,85,164,170)(43,70,165,211)(44,111,166,196)(45,96,167,181)(46,81,168,222)(47,66,113,207)(48,107,114,192)(49,92,115,177)(50,77,116,218)(51,62,117,203)(52,103,118,188)(53,88,119,173)(54,73,120,214)(55,58,121,199)(56,99,122,184), (1,50)(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,56)(52,55)(53,54)(57,189)(58,188)(59,187)(60,186)(61,185)(62,184)(63,183)(64,182)(65,181)(66,180)(67,179)(68,178)(69,177)(70,176)(71,175)(72,174)(73,173)(74,172)(75,171)(76,170)(77,169)(78,224)(79,223)(80,222)(81,221)(82,220)(83,219)(84,218)(85,217)(86,216)(87,215)(88,214)(89,213)(90,212)(91,211)(92,210)(93,209)(94,208)(95,207)(96,206)(97,205)(98,204)(99,203)(100,202)(101,201)(102,200)(103,199)(104,198)(105,197)(106,196)(107,195)(108,194)(109,193)(110,192)(111,191)(112,190)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,168)(128,167)(129,166)(130,165)(131,164)(132,163)(133,162)(134,161)(135,160)(136,159)(137,158)(138,157)(139,156)(140,155)(141,154)(142,153)(143,152)(144,151)(145,150)(146,149)(147,148) );
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)], [(1,84,123,169),(2,69,124,210),(3,110,125,195),(4,95,126,180),(5,80,127,221),(6,65,128,206),(7,106,129,191),(8,91,130,176),(9,76,131,217),(10,61,132,202),(11,102,133,187),(12,87,134,172),(13,72,135,213),(14,57,136,198),(15,98,137,183),(16,83,138,224),(17,68,139,209),(18,109,140,194),(19,94,141,179),(20,79,142,220),(21,64,143,205),(22,105,144,190),(23,90,145,175),(24,75,146,216),(25,60,147,201),(26,101,148,186),(27,86,149,171),(28,71,150,212),(29,112,151,197),(30,97,152,182),(31,82,153,223),(32,67,154,208),(33,108,155,193),(34,93,156,178),(35,78,157,219),(36,63,158,204),(37,104,159,189),(38,89,160,174),(39,74,161,215),(40,59,162,200),(41,100,163,185),(42,85,164,170),(43,70,165,211),(44,111,166,196),(45,96,167,181),(46,81,168,222),(47,66,113,207),(48,107,114,192),(49,92,115,177),(50,77,116,218),(51,62,117,203),(52,103,118,188),(53,88,119,173),(54,73,120,214),(55,58,121,199),(56,99,122,184)], [(1,50),(2,49),(3,48),(4,47),(5,46),(6,45),(7,44),(8,43),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(51,56),(52,55),(53,54),(57,189),(58,188),(59,187),(60,186),(61,185),(62,184),(63,183),(64,182),(65,181),(66,180),(67,179),(68,178),(69,177),(70,176),(71,175),(72,174),(73,173),(74,172),(75,171),(76,170),(77,169),(78,224),(79,223),(80,222),(81,221),(82,220),(83,219),(84,218),(85,217),(86,216),(87,215),(88,214),(89,213),(90,212),(91,211),(92,210),(93,209),(94,208),(95,207),(96,206),(97,205),(98,204),(99,203),(100,202),(101,201),(102,200),(103,199),(104,198),(105,197),(106,196),(107,195),(108,194),(109,193),(110,192),(111,191),(112,190),(113,126),(114,125),(115,124),(116,123),(117,122),(118,121),(119,120),(127,168),(128,167),(129,166),(130,165),(131,164),(132,163),(133,162),(134,161),(135,160),(136,159),(137,158),(138,157),(139,156),(140,155),(141,154),(142,153),(143,152),(144,151),(145,150),(146,149),(147,148)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 56 | 56 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 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 | D8 | D14 | D14 | C7⋊D4 | D4×D7 | D4×D7 | D7×D8 |
| kernel | C56⋊5D4 | C8×Dic7 | C2×D56 | C2×D4⋊D7 | C28⋊D4 | C14×D8 | C7⋊C8 | C56 | C2×Dic7 | C2×D8 | Dic7 | C2×C8 | C2×D4 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 3 | 8 | 3 | 6 | 12 | 3 | 3 | 12 |
Matrix representation of C56⋊5D4 ►in GL4(𝔽113) generated by
| 82 | 31 | 0 | 0 |
| 82 | 82 | 0 | 0 |
| 0 | 0 | 112 | 1 |
| 0 | 0 | 102 | 10 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 6 | 79 |
| 0 | 0 | 31 | 107 |
| 82 | 82 | 0 | 0 |
| 82 | 31 | 0 | 0 |
| 0 | 0 | 103 | 1 |
| 0 | 0 | 14 | 10 |
G:=sub<GL(4,GF(113))| [82,82,0,0,31,82,0,0,0,0,112,102,0,0,1,10],[0,112,0,0,1,0,0,0,0,0,6,31,0,0,79,107],[82,82,0,0,82,31,0,0,0,0,103,14,0,0,1,10] >;
C56⋊5D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_5D_4 % in TeX
G:=Group("C56:5D4"); // GroupNames label
G:=SmallGroup(448,685);
// by ID
G=gap.SmallGroup(448,685);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,422,135,570,297,136,18822]);
// 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=b^-1>;
// generators/relations