metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊11D4, C7⋊C8⋊9D4, (C2×D8)⋊8D7, (C14×D8)⋊9C2, C8⋊3(C7⋊D4), C7⋊4(C8⋊3D4), C56⋊C4⋊9C2, C4.21(D4×D7), C28⋊D4⋊5C2, (C2×C8).85D14, (C2×D4).62D14, C28.164(C2×D4), C28.17D4⋊4C2, (C2×Dic7).64D4, C22.254(D4×D7), C2.28(D8⋊D7), C14.27(C4⋊1D4), C2.18(C28⋊D4), C14.49(C8⋊C22), (C2×C28).431C23, (C2×C56).147C22, (D4×C14).81C22, (C2×D28).115C22, (C4×Dic7).45C22, (C2×Dic14).120C22, C4.5(C2×C7⋊D4), (C2×D4⋊D7)⋊18C2, (C2×C56⋊C2)⋊23C2, (C2×D4.D7)⋊17C2, (C2×C14).344(C2×D4), (C2×C7⋊C8).148C22, (C2×C4).521(C22×D7), SmallGroup(448,688)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊11D4
G = < a,b,c | a56=b4=c2=1, bab-1=a13, cac=a27, cbc=b-1 >
Subgroups: 868 in 144 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, C14, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, C2×D4, C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C2×C14, C8⋊C4, C4.4D4, C4⋊1D4, C2×D8, C2×D8, C2×SD16, C7⋊C8, C56, Dic14, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×C14, C8⋊3D4, C56⋊C2, C2×C7⋊C8, C4×Dic7, D4⋊D7, D4.D7, C23.D7, C2×C56, C7×D8, C2×Dic14, C2×D28, C2×C7⋊D4, D4×C14, C56⋊C4, C2×C56⋊C2, C2×D4⋊D7, C2×D4.D7, C28.17D4, C28⋊D4, C14×D8, C56⋊11D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C4⋊1D4, C8⋊C22, C7⋊D4, C22×D7, C8⋊3D4, D4×D7, C2×C7⋊D4, D8⋊D7, C28⋊D4, C56⋊11D4
►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)
(1 159 213 61)(2 116 214 74)(3 129 215 87)(4 142 216 100)(5 155 217 57)(6 168 218 70)(7 125 219 83)(8 138 220 96)(9 151 221 109)(10 164 222 66)(11 121 223 79)(12 134 224 92)(13 147 169 105)(14 160 170 62)(15 117 171 75)(16 130 172 88)(17 143 173 101)(18 156 174 58)(19 113 175 71)(20 126 176 84)(21 139 177 97)(22 152 178 110)(23 165 179 67)(24 122 180 80)(25 135 181 93)(26 148 182 106)(27 161 183 63)(28 118 184 76)(29 131 185 89)(30 144 186 102)(31 157 187 59)(32 114 188 72)(33 127 189 85)(34 140 190 98)(35 153 191 111)(36 166 192 68)(37 123 193 81)(38 136 194 94)(39 149 195 107)(40 162 196 64)(41 119 197 77)(42 132 198 90)(43 145 199 103)(44 158 200 60)(45 115 201 73)(46 128 202 86)(47 141 203 99)(48 154 204 112)(49 167 205 69)(50 124 206 82)(51 137 207 95)(52 150 208 108)(53 163 209 65)(54 120 210 78)(55 133 211 91)(56 146 212 104)
(1 61)(2 88)(3 59)(4 86)(5 57)(6 84)(7 111)(8 82)(9 109)(10 80)(11 107)(12 78)(13 105)(14 76)(15 103)(16 74)(17 101)(18 72)(19 99)(20 70)(21 97)(22 68)(23 95)(24 66)(25 93)(26 64)(27 91)(28 62)(29 89)(30 60)(31 87)(32 58)(33 85)(34 112)(35 83)(36 110)(37 81)(38 108)(39 79)(40 106)(41 77)(42 104)(43 75)(44 102)(45 73)(46 100)(47 71)(48 98)(49 69)(50 96)(51 67)(52 94)(53 65)(54 92)(55 63)(56 90)(113 203)(114 174)(115 201)(116 172)(117 199)(118 170)(119 197)(120 224)(121 195)(122 222)(123 193)(124 220)(125 191)(126 218)(127 189)(128 216)(129 187)(130 214)(131 185)(132 212)(133 183)(134 210)(135 181)(136 208)(137 179)(138 206)(139 177)(140 204)(141 175)(142 202)(143 173)(144 200)(145 171)(146 198)(147 169)(148 196)(149 223)(150 194)(151 221)(152 192)(153 219)(154 190)(155 217)(156 188)(157 215)(158 186)(159 213)(160 184)(161 211)(162 182)(163 209)(164 180)(165 207)(166 178)(167 205)(168 176)
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), (1,159,213,61)(2,116,214,74)(3,129,215,87)(4,142,216,100)(5,155,217,57)(6,168,218,70)(7,125,219,83)(8,138,220,96)(9,151,221,109)(10,164,222,66)(11,121,223,79)(12,134,224,92)(13,147,169,105)(14,160,170,62)(15,117,171,75)(16,130,172,88)(17,143,173,101)(18,156,174,58)(19,113,175,71)(20,126,176,84)(21,139,177,97)(22,152,178,110)(23,165,179,67)(24,122,180,80)(25,135,181,93)(26,148,182,106)(27,161,183,63)(28,118,184,76)(29,131,185,89)(30,144,186,102)(31,157,187,59)(32,114,188,72)(33,127,189,85)(34,140,190,98)(35,153,191,111)(36,166,192,68)(37,123,193,81)(38,136,194,94)(39,149,195,107)(40,162,196,64)(41,119,197,77)(42,132,198,90)(43,145,199,103)(44,158,200,60)(45,115,201,73)(46,128,202,86)(47,141,203,99)(48,154,204,112)(49,167,205,69)(50,124,206,82)(51,137,207,95)(52,150,208,108)(53,163,209,65)(54,120,210,78)(55,133,211,91)(56,146,212,104), (1,61)(2,88)(3,59)(4,86)(5,57)(6,84)(7,111)(8,82)(9,109)(10,80)(11,107)(12,78)(13,105)(14,76)(15,103)(16,74)(17,101)(18,72)(19,99)(20,70)(21,97)(22,68)(23,95)(24,66)(25,93)(26,64)(27,91)(28,62)(29,89)(30,60)(31,87)(32,58)(33,85)(34,112)(35,83)(36,110)(37,81)(38,108)(39,79)(40,106)(41,77)(42,104)(43,75)(44,102)(45,73)(46,100)(47,71)(48,98)(49,69)(50,96)(51,67)(52,94)(53,65)(54,92)(55,63)(56,90)(113,203)(114,174)(115,201)(116,172)(117,199)(118,170)(119,197)(120,224)(121,195)(122,222)(123,193)(124,220)(125,191)(126,218)(127,189)(128,216)(129,187)(130,214)(131,185)(132,212)(133,183)(134,210)(135,181)(136,208)(137,179)(138,206)(139,177)(140,204)(141,175)(142,202)(143,173)(144,200)(145,171)(146,198)(147,169)(148,196)(149,223)(150,194)(151,221)(152,192)(153,219)(154,190)(155,217)(156,188)(157,215)(158,186)(159,213)(160,184)(161,211)(162,182)(163,209)(164,180)(165,207)(166,178)(167,205)(168,176)>;
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), (1,159,213,61)(2,116,214,74)(3,129,215,87)(4,142,216,100)(5,155,217,57)(6,168,218,70)(7,125,219,83)(8,138,220,96)(9,151,221,109)(10,164,222,66)(11,121,223,79)(12,134,224,92)(13,147,169,105)(14,160,170,62)(15,117,171,75)(16,130,172,88)(17,143,173,101)(18,156,174,58)(19,113,175,71)(20,126,176,84)(21,139,177,97)(22,152,178,110)(23,165,179,67)(24,122,180,80)(25,135,181,93)(26,148,182,106)(27,161,183,63)(28,118,184,76)(29,131,185,89)(30,144,186,102)(31,157,187,59)(32,114,188,72)(33,127,189,85)(34,140,190,98)(35,153,191,111)(36,166,192,68)(37,123,193,81)(38,136,194,94)(39,149,195,107)(40,162,196,64)(41,119,197,77)(42,132,198,90)(43,145,199,103)(44,158,200,60)(45,115,201,73)(46,128,202,86)(47,141,203,99)(48,154,204,112)(49,167,205,69)(50,124,206,82)(51,137,207,95)(52,150,208,108)(53,163,209,65)(54,120,210,78)(55,133,211,91)(56,146,212,104), (1,61)(2,88)(3,59)(4,86)(5,57)(6,84)(7,111)(8,82)(9,109)(10,80)(11,107)(12,78)(13,105)(14,76)(15,103)(16,74)(17,101)(18,72)(19,99)(20,70)(21,97)(22,68)(23,95)(24,66)(25,93)(26,64)(27,91)(28,62)(29,89)(30,60)(31,87)(32,58)(33,85)(34,112)(35,83)(36,110)(37,81)(38,108)(39,79)(40,106)(41,77)(42,104)(43,75)(44,102)(45,73)(46,100)(47,71)(48,98)(49,69)(50,96)(51,67)(52,94)(53,65)(54,92)(55,63)(56,90)(113,203)(114,174)(115,201)(116,172)(117,199)(118,170)(119,197)(120,224)(121,195)(122,222)(123,193)(124,220)(125,191)(126,218)(127,189)(128,216)(129,187)(130,214)(131,185)(132,212)(133,183)(134,210)(135,181)(136,208)(137,179)(138,206)(139,177)(140,204)(141,175)(142,202)(143,173)(144,200)(145,171)(146,198)(147,169)(148,196)(149,223)(150,194)(151,221)(152,192)(153,219)(154,190)(155,217)(156,188)(157,215)(158,186)(159,213)(160,184)(161,211)(162,182)(163,209)(164,180)(165,207)(166,178)(167,205)(168,176) );
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)], [(1,159,213,61),(2,116,214,74),(3,129,215,87),(4,142,216,100),(5,155,217,57),(6,168,218,70),(7,125,219,83),(8,138,220,96),(9,151,221,109),(10,164,222,66),(11,121,223,79),(12,134,224,92),(13,147,169,105),(14,160,170,62),(15,117,171,75),(16,130,172,88),(17,143,173,101),(18,156,174,58),(19,113,175,71),(20,126,176,84),(21,139,177,97),(22,152,178,110),(23,165,179,67),(24,122,180,80),(25,135,181,93),(26,148,182,106),(27,161,183,63),(28,118,184,76),(29,131,185,89),(30,144,186,102),(31,157,187,59),(32,114,188,72),(33,127,189,85),(34,140,190,98),(35,153,191,111),(36,166,192,68),(37,123,193,81),(38,136,194,94),(39,149,195,107),(40,162,196,64),(41,119,197,77),(42,132,198,90),(43,145,199,103),(44,158,200,60),(45,115,201,73),(46,128,202,86),(47,141,203,99),(48,154,204,112),(49,167,205,69),(50,124,206,82),(51,137,207,95),(52,150,208,108),(53,163,209,65),(54,120,210,78),(55,133,211,91),(56,146,212,104)], [(1,61),(2,88),(3,59),(4,86),(5,57),(6,84),(7,111),(8,82),(9,109),(10,80),(11,107),(12,78),(13,105),(14,76),(15,103),(16,74),(17,101),(18,72),(19,99),(20,70),(21,97),(22,68),(23,95),(24,66),(25,93),(26,64),(27,91),(28,62),(29,89),(30,60),(31,87),(32,58),(33,85),(34,112),(35,83),(36,110),(37,81),(38,108),(39,79),(40,106),(41,77),(42,104),(43,75),(44,102),(45,73),(46,100),(47,71),(48,98),(49,69),(50,96),(51,67),(52,94),(53,65),(54,92),(55,63),(56,90),(113,203),(114,174),(115,201),(116,172),(117,199),(118,170),(119,197),(120,224),(121,195),(122,222),(123,193),(124,220),(125,191),(126,218),(127,189),(128,216),(129,187),(130,214),(131,185),(132,212),(133,183),(134,210),(135,181),(136,208),(137,179),(138,206),(139,177),(140,204),(141,175),(142,202),(143,173),(144,200),(145,171),(146,198),(147,169),(148,196),(149,223),(150,194),(151,221),(152,192),(153,219),(154,190),(155,217),(156,188),(157,215),(158,186),(159,213),(160,184),(161,211),(162,182),(163,209),(164,180),(165,207),(166,178),(167,205),(168,176)]])
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 | 56 | 2 | 2 | 28 | 28 | 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 | 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 | D8⋊D7 |
| kernel | C56⋊11D4 | C56⋊C4 | C2×C56⋊C2 | C2×D4⋊D7 | C2×D4.D7 | C28.17D4 | C28⋊D4 | C14×D8 | C7⋊C8 | C56 | C2×Dic7 | C2×D8 | C2×C8 | C2×D4 | 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⋊11D4 ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 62 | 3 | 48 | 14 |
| 0 | 0 | 23 | 55 | 102 | 38 |
| 0 | 0 | 30 | 67 | 23 | 13 |
| 0 | 0 | 48 | 109 | 53 | 36 |
| 53 | 36 | 0 | 0 | 0 | 0 |
| 104 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 7 | 0 | 0 |
| 0 | 0 | 44 | 91 | 0 | 0 |
| 0 | 0 | 0 | 112 | 108 | 69 |
| 0 | 0 | 112 | 34 | 93 | 5 |
| 53 | 36 | 0 | 0 | 0 | 0 |
| 35 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 7 | 0 | 0 |
| 0 | 0 | 44 | 91 | 0 | 0 |
| 0 | 0 | 6 | 111 | 5 | 44 |
| 0 | 0 | 24 | 101 | 20 | 108 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,62,23,30,48,0,0,3,55,67,109,0,0,48,102,23,53,0,0,14,38,13,36],[53,104,0,0,0,0,36,60,0,0,0,0,0,0,22,44,0,112,0,0,7,91,112,34,0,0,0,0,108,93,0,0,0,0,69,5],[53,35,0,0,0,0,36,60,0,0,0,0,0,0,22,44,6,24,0,0,7,91,111,101,0,0,0,0,5,20,0,0,0,0,44,108] >;
C56⋊11D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_{11}D_4 % in TeX
G:=Group("C56:11D4"); // GroupNames label
G:=SmallGroup(448,688);
// by ID
G=gap.SmallGroup(448,688);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,253,1094,135,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^13,c*a*c=a^27,c*b*c=b^-1>;
// generators/relations