metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊12D4, (C2×D8)⋊9D7, C8⋊4(C7⋊D4), C7⋊5(C8⋊2D4), (C14×D8)⋊10C2, C28⋊2D4⋊6C2, C8⋊Dic7⋊21C2, (C2×C8).86D14, (C2×D4).66D14, C28.168(C2×D4), C28.94(C4○D4), D4⋊Dic7⋊31C2, (C2×Dic7).66D4, (C22×D7).36D4, C22.259(D4×D7), C4.29(D4⋊2D7), C2.31(D8⋊D7), C2.17(C28⋊2D4), C14.52(C8⋊C22), (C2×C56).148C22, (C2×C28).436C23, (D4×C14).85C22, C14.110(C4⋊D4), C4⋊Dic7.167C22, (C2×C8⋊D7)⋊7C2, C4.80(C2×C7⋊D4), (C2×C4×D7).46C22, (C2×C14).349(C2×D4), (C2×C7⋊C8).150C22, (C2×C4).526(C22×D7), SmallGroup(448,693)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊12D4
G = < a,b,c | a56=b4=c2=1, bab-1=a27, cac=a13, cbc=b-1 >
Subgroups: 708 in 130 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, D7, C14, C14, C14, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, Dic7, C28, D14, C2×C14, C2×C14, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, C8⋊2D4, C8⋊D7, C2×C7⋊C8, C4⋊Dic7, C23.D7, C2×C56, C7×D8, C2×C4×D7, C2×C7⋊D4, D4×C14, C8⋊Dic7, D4⋊Dic7, C2×C8⋊D7, C28⋊2D4, C14×D8, C56⋊12D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, C7⋊D4, C22×D7, C8⋊2D4, D4×D7, D4⋊2D7, C2×C7⋊D4, D8⋊D7, C28⋊2D4, C56⋊12D4
►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)
(1 147 86 174)(2 118 87 201)(3 145 88 172)(4 116 89 199)(5 143 90 170)(6 114 91 197)(7 141 92 224)(8 168 93 195)(9 139 94 222)(10 166 95 193)(11 137 96 220)(12 164 97 191)(13 135 98 218)(14 162 99 189)(15 133 100 216)(16 160 101 187)(17 131 102 214)(18 158 103 185)(19 129 104 212)(20 156 105 183)(21 127 106 210)(22 154 107 181)(23 125 108 208)(24 152 109 179)(25 123 110 206)(26 150 111 177)(27 121 112 204)(28 148 57 175)(29 119 58 202)(30 146 59 173)(31 117 60 200)(32 144 61 171)(33 115 62 198)(34 142 63 169)(35 113 64 196)(36 140 65 223)(37 167 66 194)(38 138 67 221)(39 165 68 192)(40 136 69 219)(41 163 70 190)(42 134 71 217)(43 161 72 188)(44 132 73 215)(45 159 74 186)(46 130 75 213)(47 157 76 184)(48 128 77 211)(49 155 78 182)(50 126 79 209)(51 153 80 180)(52 124 81 207)(53 151 82 178)(54 122 83 205)(55 149 84 176)(56 120 85 203)
(2 14)(3 27)(4 40)(5 53)(6 10)(7 23)(8 36)(9 49)(11 19)(12 32)(13 45)(16 28)(17 41)(18 54)(20 24)(21 37)(22 50)(25 33)(26 46)(30 42)(31 55)(34 38)(35 51)(39 47)(44 56)(48 52)(57 101)(59 71)(60 84)(61 97)(62 110)(63 67)(64 80)(65 93)(66 106)(68 76)(69 89)(70 102)(73 85)(74 98)(75 111)(77 81)(78 94)(79 107)(82 90)(83 103)(87 99)(88 112)(91 95)(92 108)(96 104)(105 109)(113 180)(114 193)(115 206)(116 219)(117 176)(118 189)(119 202)(120 215)(121 172)(122 185)(123 198)(124 211)(125 224)(126 181)(127 194)(128 207)(129 220)(130 177)(131 190)(132 203)(133 216)(134 173)(135 186)(136 199)(137 212)(138 169)(139 182)(140 195)(141 208)(142 221)(143 178)(144 191)(145 204)(146 217)(147 174)(148 187)(149 200)(150 213)(151 170)(152 183)(153 196)(154 209)(155 222)(156 179)(157 192)(158 205)(159 218)(160 175)(161 188)(162 201)(163 214)(164 171)(165 184)(166 197)(167 210)(168 223)
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), (1,147,86,174)(2,118,87,201)(3,145,88,172)(4,116,89,199)(5,143,90,170)(6,114,91,197)(7,141,92,224)(8,168,93,195)(9,139,94,222)(10,166,95,193)(11,137,96,220)(12,164,97,191)(13,135,98,218)(14,162,99,189)(15,133,100,216)(16,160,101,187)(17,131,102,214)(18,158,103,185)(19,129,104,212)(20,156,105,183)(21,127,106,210)(22,154,107,181)(23,125,108,208)(24,152,109,179)(25,123,110,206)(26,150,111,177)(27,121,112,204)(28,148,57,175)(29,119,58,202)(30,146,59,173)(31,117,60,200)(32,144,61,171)(33,115,62,198)(34,142,63,169)(35,113,64,196)(36,140,65,223)(37,167,66,194)(38,138,67,221)(39,165,68,192)(40,136,69,219)(41,163,70,190)(42,134,71,217)(43,161,72,188)(44,132,73,215)(45,159,74,186)(46,130,75,213)(47,157,76,184)(48,128,77,211)(49,155,78,182)(50,126,79,209)(51,153,80,180)(52,124,81,207)(53,151,82,178)(54,122,83,205)(55,149,84,176)(56,120,85,203), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,101)(59,71)(60,84)(61,97)(62,110)(63,67)(64,80)(65,93)(66,106)(68,76)(69,89)(70,102)(73,85)(74,98)(75,111)(77,81)(78,94)(79,107)(82,90)(83,103)(87,99)(88,112)(91,95)(92,108)(96,104)(105,109)(113,180)(114,193)(115,206)(116,219)(117,176)(118,189)(119,202)(120,215)(121,172)(122,185)(123,198)(124,211)(125,224)(126,181)(127,194)(128,207)(129,220)(130,177)(131,190)(132,203)(133,216)(134,173)(135,186)(136,199)(137,212)(138,169)(139,182)(140,195)(141,208)(142,221)(143,178)(144,191)(145,204)(146,217)(147,174)(148,187)(149,200)(150,213)(151,170)(152,183)(153,196)(154,209)(155,222)(156,179)(157,192)(158,205)(159,218)(160,175)(161,188)(162,201)(163,214)(164,171)(165,184)(166,197)(167,210)(168,223)>;
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), (1,147,86,174)(2,118,87,201)(3,145,88,172)(4,116,89,199)(5,143,90,170)(6,114,91,197)(7,141,92,224)(8,168,93,195)(9,139,94,222)(10,166,95,193)(11,137,96,220)(12,164,97,191)(13,135,98,218)(14,162,99,189)(15,133,100,216)(16,160,101,187)(17,131,102,214)(18,158,103,185)(19,129,104,212)(20,156,105,183)(21,127,106,210)(22,154,107,181)(23,125,108,208)(24,152,109,179)(25,123,110,206)(26,150,111,177)(27,121,112,204)(28,148,57,175)(29,119,58,202)(30,146,59,173)(31,117,60,200)(32,144,61,171)(33,115,62,198)(34,142,63,169)(35,113,64,196)(36,140,65,223)(37,167,66,194)(38,138,67,221)(39,165,68,192)(40,136,69,219)(41,163,70,190)(42,134,71,217)(43,161,72,188)(44,132,73,215)(45,159,74,186)(46,130,75,213)(47,157,76,184)(48,128,77,211)(49,155,78,182)(50,126,79,209)(51,153,80,180)(52,124,81,207)(53,151,82,178)(54,122,83,205)(55,149,84,176)(56,120,85,203), (2,14)(3,27)(4,40)(5,53)(6,10)(7,23)(8,36)(9,49)(11,19)(12,32)(13,45)(16,28)(17,41)(18,54)(20,24)(21,37)(22,50)(25,33)(26,46)(30,42)(31,55)(34,38)(35,51)(39,47)(44,56)(48,52)(57,101)(59,71)(60,84)(61,97)(62,110)(63,67)(64,80)(65,93)(66,106)(68,76)(69,89)(70,102)(73,85)(74,98)(75,111)(77,81)(78,94)(79,107)(82,90)(83,103)(87,99)(88,112)(91,95)(92,108)(96,104)(105,109)(113,180)(114,193)(115,206)(116,219)(117,176)(118,189)(119,202)(120,215)(121,172)(122,185)(123,198)(124,211)(125,224)(126,181)(127,194)(128,207)(129,220)(130,177)(131,190)(132,203)(133,216)(134,173)(135,186)(136,199)(137,212)(138,169)(139,182)(140,195)(141,208)(142,221)(143,178)(144,191)(145,204)(146,217)(147,174)(148,187)(149,200)(150,213)(151,170)(152,183)(153,196)(154,209)(155,222)(156,179)(157,192)(158,205)(159,218)(160,175)(161,188)(162,201)(163,214)(164,171)(165,184)(166,197)(167,210)(168,223) );
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)], [(1,147,86,174),(2,118,87,201),(3,145,88,172),(4,116,89,199),(5,143,90,170),(6,114,91,197),(7,141,92,224),(8,168,93,195),(9,139,94,222),(10,166,95,193),(11,137,96,220),(12,164,97,191),(13,135,98,218),(14,162,99,189),(15,133,100,216),(16,160,101,187),(17,131,102,214),(18,158,103,185),(19,129,104,212),(20,156,105,183),(21,127,106,210),(22,154,107,181),(23,125,108,208),(24,152,109,179),(25,123,110,206),(26,150,111,177),(27,121,112,204),(28,148,57,175),(29,119,58,202),(30,146,59,173),(31,117,60,200),(32,144,61,171),(33,115,62,198),(34,142,63,169),(35,113,64,196),(36,140,65,223),(37,167,66,194),(38,138,67,221),(39,165,68,192),(40,136,69,219),(41,163,70,190),(42,134,71,217),(43,161,72,188),(44,132,73,215),(45,159,74,186),(46,130,75,213),(47,157,76,184),(48,128,77,211),(49,155,78,182),(50,126,79,209),(51,153,80,180),(52,124,81,207),(53,151,82,178),(54,122,83,205),(55,149,84,176),(56,120,85,203)], [(2,14),(3,27),(4,40),(5,53),(6,10),(7,23),(8,36),(9,49),(11,19),(12,32),(13,45),(16,28),(17,41),(18,54),(20,24),(21,37),(22,50),(25,33),(26,46),(30,42),(31,55),(34,38),(35,51),(39,47),(44,56),(48,52),(57,101),(59,71),(60,84),(61,97),(62,110),(63,67),(64,80),(65,93),(66,106),(68,76),(69,89),(70,102),(73,85),(74,98),(75,111),(77,81),(78,94),(79,107),(82,90),(83,103),(87,99),(88,112),(91,95),(92,108),(96,104),(105,109),(113,180),(114,193),(115,206),(116,219),(117,176),(118,189),(119,202),(120,215),(121,172),(122,185),(123,198),(124,211),(125,224),(126,181),(127,194),(128,207),(129,220),(130,177),(131,190),(132,203),(133,216),(134,173),(135,186),(136,199),(137,212),(138,169),(139,182),(140,195),(141,208),(142,221),(143,178),(144,191),(145,204),(146,217),(147,174),(148,187),(149,200),(150,213),(151,170),(152,183),(153,196),(154,209),(155,222),(156,179),(157,192),(158,205),(159,218),(160,175),(161,188),(162,201),(163,214),(164,171),(165,184),(166,197),(167,210),(168,223)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 28 | 2 | 2 | 28 | 56 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | C8⋊C22 | D4⋊2D7 | D4×D7 | D8⋊D7 |
| kernel | C56⋊12D4 | C8⋊Dic7 | D4⋊Dic7 | C2×C8⋊D7 | C28⋊2D4 | C14×D8 | C56 | C2×Dic7 | C22×D7 | C2×D8 | C28 | C2×C8 | C2×D4 | C8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 6 | 12 | 2 | 3 | 3 | 12 |
Matrix representation of C56⋊12D4 ►in GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 33 |
| 0 | 0 | 0 | 0 | 39 | 41 |
| 0 | 0 | 84 | 60 | 39 | 8 |
| 0 | 0 | 24 | 25 | 30 | 9 |
| 51 | 36 | 0 | 0 | 0 | 0 |
| 47 | 62 | 0 | 0 | 0 | 0 |
| 0 | 0 | 105 | 0 | 36 | 21 |
| 0 | 0 | 46 | 8 | 61 | 56 |
| 0 | 0 | 63 | 47 | 38 | 8 |
| 0 | 0 | 2 | 97 | 110 | 75 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 97 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 79 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 1 |
| 0 | 0 | 0 | 0 | 42 | 80 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,84,24,0,0,0,0,60,25,0,0,0,39,39,30,0,0,33,41,8,9],[51,47,0,0,0,0,36,62,0,0,0,0,0,0,105,46,63,2,0,0,0,8,47,97,0,0,36,61,38,110,0,0,21,56,8,75],[112,97,0,0,0,0,0,1,0,0,0,0,0,0,1,79,0,0,0,0,0,112,0,0,0,0,0,0,33,42,0,0,0,0,1,80] >;
C56⋊12D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_{12}D_4 % in TeX
G:=Group("C56:12D4"); // GroupNames label
G:=SmallGroup(448,693);
// by ID
G=gap.SmallGroup(448,693);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,1094,135,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^27,c*a*c=a^13,c*b*c=b^-1>;
// generators/relations