metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊32D4, D14⋊C8⋊3C2, C7⋊6(C8⋊9D4), C8⋊15(C7⋊D4), Dic7⋊C8⋊3C2, D14⋊C4.3C4, C56⋊C4⋊23C2, C14.78(C4×D4), (C22×C8)⋊11D7, (C22×C56)⋊16C2, C28.437(C2×D4), (C2×C8).294D14, (C2×C14)⋊5M4(2), Dic7⋊C4.3C4, C23.35(C4×D7), C23.D7.9C4, C14.19(C8○D4), C22⋊2(C8⋊D7), C4.136(C4○D28), C28.252(C4○D4), C28.55D4⋊26C2, (C2×C28).861C23, (C2×C56).355C22, (C22×C4).402D14, C14.13(C2×M4(2)), C2.19(D28.2C4), (C22×C28).561C22, (C4×Dic7).187C22, (C2×C4).94(C4×D7), (C2×C7⋊D4).9C4, C2.23(C4×C7⋊D4), (C2×C8⋊D7)⋊23C2, C2.15(C2×C8⋊D7), (C4×C7⋊D4).15C2, C4.127(C2×C7⋊D4), C22.142(C2×C4×D7), (C2×C28).210(C2×C4), (C2×C7⋊C8).205C22, (C2×C4×D7).184C22, (C22×C14).96(C2×C4), (C2×Dic7).32(C2×C4), (C22×D7).24(C2×C4), (C2×C4).803(C22×D7), (C2×C14).131(C22×C4), SmallGroup(448,645)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊32D4
G = < a,b,c | a56=b4=c2=1, bab-1=cac=a13, cbc=b-1 >
Subgroups: 484 in 124 conjugacy classes, 55 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C7⋊C8, C56, C56, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8⋊9D4, C8⋊D7, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C2×C56, C2×C56, C2×C4×D7, C2×C7⋊D4, C22×C28, Dic7⋊C8, C56⋊C4, D14⋊C8, C28.55D4, C2×C8⋊D7, C4×C7⋊D4, C22×C56, C56⋊32D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, M4(2), C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×M4(2), C8○D4, C4×D7, C7⋊D4, C22×D7, C8⋊9D4, C8⋊D7, C2×C4×D7, C4○D28, C2×C7⋊D4, C2×C8⋊D7, D28.2C4, C4×C7⋊D4, C56⋊32D4
►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)
(1 223 131 109)(2 180 132 66)(3 193 133 79)(4 206 134 92)(5 219 135 105)(6 176 136 62)(7 189 137 75)(8 202 138 88)(9 215 139 101)(10 172 140 58)(11 185 141 71)(12 198 142 84)(13 211 143 97)(14 224 144 110)(15 181 145 67)(16 194 146 80)(17 207 147 93)(18 220 148 106)(19 177 149 63)(20 190 150 76)(21 203 151 89)(22 216 152 102)(23 173 153 59)(24 186 154 72)(25 199 155 85)(26 212 156 98)(27 169 157 111)(28 182 158 68)(29 195 159 81)(30 208 160 94)(31 221 161 107)(32 178 162 64)(33 191 163 77)(34 204 164 90)(35 217 165 103)(36 174 166 60)(37 187 167 73)(38 200 168 86)(39 213 113 99)(40 170 114 112)(41 183 115 69)(42 196 116 82)(43 209 117 95)(44 222 118 108)(45 179 119 65)(46 192 120 78)(47 205 121 91)(48 218 122 104)(49 175 123 61)(50 188 124 74)(51 201 125 87)(52 214 126 100)(53 171 127 57)(54 184 128 70)(55 197 129 83)(56 210 130 96)
(1 109)(2 66)(3 79)(4 92)(5 105)(6 62)(7 75)(8 88)(9 101)(10 58)(11 71)(12 84)(13 97)(14 110)(15 67)(16 80)(17 93)(18 106)(19 63)(20 76)(21 89)(22 102)(23 59)(24 72)(25 85)(26 98)(27 111)(28 68)(29 81)(30 94)(31 107)(32 64)(33 77)(34 90)(35 103)(36 60)(37 73)(38 86)(39 99)(40 112)(41 69)(42 82)(43 95)(44 108)(45 65)(46 78)(47 91)(48 104)(49 61)(50 74)(51 87)(52 100)(53 57)(54 70)(55 83)(56 96)(113 213)(114 170)(115 183)(116 196)(117 209)(118 222)(119 179)(120 192)(121 205)(122 218)(123 175)(124 188)(125 201)(126 214)(127 171)(128 184)(129 197)(130 210)(131 223)(132 180)(133 193)(134 206)(135 219)(136 176)(137 189)(138 202)(139 215)(140 172)(141 185)(142 198)(143 211)(144 224)(145 181)(146 194)(147 207)(148 220)(149 177)(150 190)(151 203)(152 216)(153 173)(154 186)(155 199)(156 212)(157 169)(158 182)(159 195)(160 208)(161 221)(162 178)(163 191)(164 204)(165 217)(166 174)(167 187)(168 200)
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), (1,223,131,109)(2,180,132,66)(3,193,133,79)(4,206,134,92)(5,219,135,105)(6,176,136,62)(7,189,137,75)(8,202,138,88)(9,215,139,101)(10,172,140,58)(11,185,141,71)(12,198,142,84)(13,211,143,97)(14,224,144,110)(15,181,145,67)(16,194,146,80)(17,207,147,93)(18,220,148,106)(19,177,149,63)(20,190,150,76)(21,203,151,89)(22,216,152,102)(23,173,153,59)(24,186,154,72)(25,199,155,85)(26,212,156,98)(27,169,157,111)(28,182,158,68)(29,195,159,81)(30,208,160,94)(31,221,161,107)(32,178,162,64)(33,191,163,77)(34,204,164,90)(35,217,165,103)(36,174,166,60)(37,187,167,73)(38,200,168,86)(39,213,113,99)(40,170,114,112)(41,183,115,69)(42,196,116,82)(43,209,117,95)(44,222,118,108)(45,179,119,65)(46,192,120,78)(47,205,121,91)(48,218,122,104)(49,175,123,61)(50,188,124,74)(51,201,125,87)(52,214,126,100)(53,171,127,57)(54,184,128,70)(55,197,129,83)(56,210,130,96), (1,109)(2,66)(3,79)(4,92)(5,105)(6,62)(7,75)(8,88)(9,101)(10,58)(11,71)(12,84)(13,97)(14,110)(15,67)(16,80)(17,93)(18,106)(19,63)(20,76)(21,89)(22,102)(23,59)(24,72)(25,85)(26,98)(27,111)(28,68)(29,81)(30,94)(31,107)(32,64)(33,77)(34,90)(35,103)(36,60)(37,73)(38,86)(39,99)(40,112)(41,69)(42,82)(43,95)(44,108)(45,65)(46,78)(47,91)(48,104)(49,61)(50,74)(51,87)(52,100)(53,57)(54,70)(55,83)(56,96)(113,213)(114,170)(115,183)(116,196)(117,209)(118,222)(119,179)(120,192)(121,205)(122,218)(123,175)(124,188)(125,201)(126,214)(127,171)(128,184)(129,197)(130,210)(131,223)(132,180)(133,193)(134,206)(135,219)(136,176)(137,189)(138,202)(139,215)(140,172)(141,185)(142,198)(143,211)(144,224)(145,181)(146,194)(147,207)(148,220)(149,177)(150,190)(151,203)(152,216)(153,173)(154,186)(155,199)(156,212)(157,169)(158,182)(159,195)(160,208)(161,221)(162,178)(163,191)(164,204)(165,217)(166,174)(167,187)(168,200)>;
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), (1,223,131,109)(2,180,132,66)(3,193,133,79)(4,206,134,92)(5,219,135,105)(6,176,136,62)(7,189,137,75)(8,202,138,88)(9,215,139,101)(10,172,140,58)(11,185,141,71)(12,198,142,84)(13,211,143,97)(14,224,144,110)(15,181,145,67)(16,194,146,80)(17,207,147,93)(18,220,148,106)(19,177,149,63)(20,190,150,76)(21,203,151,89)(22,216,152,102)(23,173,153,59)(24,186,154,72)(25,199,155,85)(26,212,156,98)(27,169,157,111)(28,182,158,68)(29,195,159,81)(30,208,160,94)(31,221,161,107)(32,178,162,64)(33,191,163,77)(34,204,164,90)(35,217,165,103)(36,174,166,60)(37,187,167,73)(38,200,168,86)(39,213,113,99)(40,170,114,112)(41,183,115,69)(42,196,116,82)(43,209,117,95)(44,222,118,108)(45,179,119,65)(46,192,120,78)(47,205,121,91)(48,218,122,104)(49,175,123,61)(50,188,124,74)(51,201,125,87)(52,214,126,100)(53,171,127,57)(54,184,128,70)(55,197,129,83)(56,210,130,96), (1,109)(2,66)(3,79)(4,92)(5,105)(6,62)(7,75)(8,88)(9,101)(10,58)(11,71)(12,84)(13,97)(14,110)(15,67)(16,80)(17,93)(18,106)(19,63)(20,76)(21,89)(22,102)(23,59)(24,72)(25,85)(26,98)(27,111)(28,68)(29,81)(30,94)(31,107)(32,64)(33,77)(34,90)(35,103)(36,60)(37,73)(38,86)(39,99)(40,112)(41,69)(42,82)(43,95)(44,108)(45,65)(46,78)(47,91)(48,104)(49,61)(50,74)(51,87)(52,100)(53,57)(54,70)(55,83)(56,96)(113,213)(114,170)(115,183)(116,196)(117,209)(118,222)(119,179)(120,192)(121,205)(122,218)(123,175)(124,188)(125,201)(126,214)(127,171)(128,184)(129,197)(130,210)(131,223)(132,180)(133,193)(134,206)(135,219)(136,176)(137,189)(138,202)(139,215)(140,172)(141,185)(142,198)(143,211)(144,224)(145,181)(146,194)(147,207)(148,220)(149,177)(150,190)(151,203)(152,216)(153,173)(154,186)(155,199)(156,212)(157,169)(158,182)(159,195)(160,208)(161,221)(162,178)(163,191)(164,204)(165,217)(166,174)(167,187)(168,200) );
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)], [(1,223,131,109),(2,180,132,66),(3,193,133,79),(4,206,134,92),(5,219,135,105),(6,176,136,62),(7,189,137,75),(8,202,138,88),(9,215,139,101),(10,172,140,58),(11,185,141,71),(12,198,142,84),(13,211,143,97),(14,224,144,110),(15,181,145,67),(16,194,146,80),(17,207,147,93),(18,220,148,106),(19,177,149,63),(20,190,150,76),(21,203,151,89),(22,216,152,102),(23,173,153,59),(24,186,154,72),(25,199,155,85),(26,212,156,98),(27,169,157,111),(28,182,158,68),(29,195,159,81),(30,208,160,94),(31,221,161,107),(32,178,162,64),(33,191,163,77),(34,204,164,90),(35,217,165,103),(36,174,166,60),(37,187,167,73),(38,200,168,86),(39,213,113,99),(40,170,114,112),(41,183,115,69),(42,196,116,82),(43,209,117,95),(44,222,118,108),(45,179,119,65),(46,192,120,78),(47,205,121,91),(48,218,122,104),(49,175,123,61),(50,188,124,74),(51,201,125,87),(52,214,126,100),(53,171,127,57),(54,184,128,70),(55,197,129,83),(56,210,130,96)], [(1,109),(2,66),(3,79),(4,92),(5,105),(6,62),(7,75),(8,88),(9,101),(10,58),(11,71),(12,84),(13,97),(14,110),(15,67),(16,80),(17,93),(18,106),(19,63),(20,76),(21,89),(22,102),(23,59),(24,72),(25,85),(26,98),(27,111),(28,68),(29,81),(30,94),(31,107),(32,64),(33,77),(34,90),(35,103),(36,60),(37,73),(38,86),(39,99),(40,112),(41,69),(42,82),(43,95),(44,108),(45,65),(46,78),(47,91),(48,104),(49,61),(50,74),(51,87),(52,100),(53,57),(54,70),(55,83),(56,96),(113,213),(114,170),(115,183),(116,196),(117,209),(118,222),(119,179),(120,192),(121,205),(122,218),(123,175),(124,188),(125,201),(126,214),(127,171),(128,184),(129,197),(130,210),(131,223),(132,180),(133,193),(134,206),(135,219),(136,176),(137,189),(138,202),(139,215),(140,172),(141,185),(142,198),(143,211),(144,224),(145,181),(146,194),(147,207),(148,220),(149,177),(150,190),(151,203),(152,216),(153,173),(154,186),(155,199),(156,212),(157,169),(158,182),(159,195),(160,208),(161,221),(162,178),(163,191),(164,204),(165,217),(166,174),(167,187),(168,200)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D7 | C4○D4 | M4(2) | D14 | D14 | C8○D4 | C7⋊D4 | C4×D7 | C4×D7 | C4○D28 | C8⋊D7 | D28.2C4 |
| kernel | C56⋊32D4 | Dic7⋊C8 | C56⋊C4 | D14⋊C8 | C28.55D4 | C2×C8⋊D7 | C4×C7⋊D4 | C22×C56 | Dic7⋊C4 | D14⋊C4 | C23.D7 | C2×C7⋊D4 | C56 | 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 | 2 | 2 | 2 | 2 | 3 | 2 | 4 | 6 | 3 | 4 | 12 | 6 | 6 | 12 | 24 | 24 |
Matrix representation of C56⋊32D4 ►in GL4(𝔽113) generated by
| 61 | 0 | 0 | 0 |
| 0 | 41 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 27 | 73 |
| 0 | 0 | 103 | 86 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 86 | 108 |
| 0 | 0 | 10 | 27 |
G:=sub<GL(4,GF(113))| [61,0,0,0,0,41,0,0,0,0,15,0,0,0,0,15],[0,112,0,0,1,0,0,0,0,0,27,103,0,0,73,86],[0,1,0,0,1,0,0,0,0,0,86,10,0,0,108,27] >;
C56⋊32D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_{32}D_4 % in TeX
G:=Group("C56:32D4"); // GroupNames label
G:=SmallGroup(448,645);
// by ID
G=gap.SmallGroup(448,645);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,758,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=c*a*c=a^13,c*b*c=b^-1>;
// generators/relations