metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D14⋊C8⋊7C2, C56⋊1C4⋊9C2, D4⋊C4⋊7D7, (C2×C8).11D14, C4⋊C4.138D14, (C2×D4).30D14, C28⋊2D4.6C2, C4.Dic14⋊7C2, C4.54(C4○D28), C14.25(C4○D8), D4⋊Dic7⋊10C2, (C2×C56).11C22, (C22×D7).16D4, C22.182(D4×D7), C28.152(C4○D4), C2.11(D8⋊3D7), C4.81(D4⋊2D7), C2.12(D56⋊C2), C14.56(C8⋊C22), (C2×C28).224C23, (C2×Dic7).144D4, (D4×C14).45C22, C7⋊3(C23.19D4), C4⋊Dic7.76C22, C2.15(D14.D4), C14.23(C22.D4), C4⋊C4⋊7D7⋊4C2, (C7×D4⋊C4)⋊7C2, (C2×C7⋊C8).22C22, (C2×C4×D7).16C22, (C2×C14).237(C2×D4), (C7×C4⋊C4).25C22, (C2×C4).331(C22×D7), SmallGroup(448,318)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊1C4⋊C2
G = < a,b,c | a56=b4=c2=1, bab-1=a-1, cac=a43b2, cbc=b-1 >
Subgroups: 564 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C7⋊C8, C56, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C23.19D4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×C7⋊D4, D4×C14, C4.Dic14, C56⋊1C4, D14⋊C8, D4⋊Dic7, C7×D4⋊C4, C4⋊C4⋊7D7, C28⋊2D4, C56⋊1C4⋊C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22.D4, C4○D8, C8⋊C22, C22×D7, C23.19D4, C4○D28, D4×D7, D4⋊2D7, D14.D4, D8⋊3D7, D56⋊C2, C56⋊1C4⋊C2
►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)
(1 209 142 98)(2 208 143 97)(3 207 144 96)(4 206 145 95)(5 205 146 94)(6 204 147 93)(7 203 148 92)(8 202 149 91)(9 201 150 90)(10 200 151 89)(11 199 152 88)(12 198 153 87)(13 197 154 86)(14 196 155 85)(15 195 156 84)(16 194 157 83)(17 193 158 82)(18 192 159 81)(19 191 160 80)(20 190 161 79)(21 189 162 78)(22 188 163 77)(23 187 164 76)(24 186 165 75)(25 185 166 74)(26 184 167 73)(27 183 168 72)(28 182 113 71)(29 181 114 70)(30 180 115 69)(31 179 116 68)(32 178 117 67)(33 177 118 66)(34 176 119 65)(35 175 120 64)(36 174 121 63)(37 173 122 62)(38 172 123 61)(39 171 124 60)(40 170 125 59)(41 169 126 58)(42 224 127 57)(43 223 128 112)(44 222 129 111)(45 221 130 110)(46 220 131 109)(47 219 132 108)(48 218 133 107)(49 217 134 106)(50 216 135 105)(51 215 136 104)(52 214 137 103)(53 213 138 102)(54 212 139 101)(55 211 140 100)(56 210 141 99)
(1 209)(2 85)(3 183)(4 59)(5 213)(6 89)(7 187)(8 63)(9 217)(10 93)(11 191)(12 67)(13 221)(14 97)(15 195)(16 71)(17 169)(18 101)(19 199)(20 75)(21 173)(22 105)(23 203)(24 79)(25 177)(26 109)(27 207)(28 83)(29 181)(30 57)(31 211)(32 87)(33 185)(34 61)(35 215)(36 91)(37 189)(38 65)(39 219)(40 95)(41 193)(42 69)(43 223)(44 99)(45 197)(46 73)(47 171)(48 103)(49 201)(50 77)(51 175)(52 107)(53 205)(54 81)(55 179)(56 111)(58 158)(60 132)(62 162)(64 136)(66 166)(68 140)(70 114)(72 144)(74 118)(76 148)(78 122)(80 152)(82 126)(84 156)(86 130)(88 160)(90 134)(92 164)(94 138)(96 168)(98 142)(100 116)(102 146)(104 120)(106 150)(108 124)(110 154)(112 128)(113 194)(115 224)(117 198)(119 172)(121 202)(123 176)(125 206)(127 180)(129 210)(131 184)(133 214)(135 188)(137 218)(139 192)(141 222)(143 196)(145 170)(147 200)(149 174)(151 204)(153 178)(155 208)(157 182)(159 212)(161 186)(163 216)(165 190)(167 220)
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), (1,209,142,98)(2,208,143,97)(3,207,144,96)(4,206,145,95)(5,205,146,94)(6,204,147,93)(7,203,148,92)(8,202,149,91)(9,201,150,90)(10,200,151,89)(11,199,152,88)(12,198,153,87)(13,197,154,86)(14,196,155,85)(15,195,156,84)(16,194,157,83)(17,193,158,82)(18,192,159,81)(19,191,160,80)(20,190,161,79)(21,189,162,78)(22,188,163,77)(23,187,164,76)(24,186,165,75)(25,185,166,74)(26,184,167,73)(27,183,168,72)(28,182,113,71)(29,181,114,70)(30,180,115,69)(31,179,116,68)(32,178,117,67)(33,177,118,66)(34,176,119,65)(35,175,120,64)(36,174,121,63)(37,173,122,62)(38,172,123,61)(39,171,124,60)(40,170,125,59)(41,169,126,58)(42,224,127,57)(43,223,128,112)(44,222,129,111)(45,221,130,110)(46,220,131,109)(47,219,132,108)(48,218,133,107)(49,217,134,106)(50,216,135,105)(51,215,136,104)(52,214,137,103)(53,213,138,102)(54,212,139,101)(55,211,140,100)(56,210,141,99), (1,209)(2,85)(3,183)(4,59)(5,213)(6,89)(7,187)(8,63)(9,217)(10,93)(11,191)(12,67)(13,221)(14,97)(15,195)(16,71)(17,169)(18,101)(19,199)(20,75)(21,173)(22,105)(23,203)(24,79)(25,177)(26,109)(27,207)(28,83)(29,181)(30,57)(31,211)(32,87)(33,185)(34,61)(35,215)(36,91)(37,189)(38,65)(39,219)(40,95)(41,193)(42,69)(43,223)(44,99)(45,197)(46,73)(47,171)(48,103)(49,201)(50,77)(51,175)(52,107)(53,205)(54,81)(55,179)(56,111)(58,158)(60,132)(62,162)(64,136)(66,166)(68,140)(70,114)(72,144)(74,118)(76,148)(78,122)(80,152)(82,126)(84,156)(86,130)(88,160)(90,134)(92,164)(94,138)(96,168)(98,142)(100,116)(102,146)(104,120)(106,150)(108,124)(110,154)(112,128)(113,194)(115,224)(117,198)(119,172)(121,202)(123,176)(125,206)(127,180)(129,210)(131,184)(133,214)(135,188)(137,218)(139,192)(141,222)(143,196)(145,170)(147,200)(149,174)(151,204)(153,178)(155,208)(157,182)(159,212)(161,186)(163,216)(165,190)(167,220)>;
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), (1,209,142,98)(2,208,143,97)(3,207,144,96)(4,206,145,95)(5,205,146,94)(6,204,147,93)(7,203,148,92)(8,202,149,91)(9,201,150,90)(10,200,151,89)(11,199,152,88)(12,198,153,87)(13,197,154,86)(14,196,155,85)(15,195,156,84)(16,194,157,83)(17,193,158,82)(18,192,159,81)(19,191,160,80)(20,190,161,79)(21,189,162,78)(22,188,163,77)(23,187,164,76)(24,186,165,75)(25,185,166,74)(26,184,167,73)(27,183,168,72)(28,182,113,71)(29,181,114,70)(30,180,115,69)(31,179,116,68)(32,178,117,67)(33,177,118,66)(34,176,119,65)(35,175,120,64)(36,174,121,63)(37,173,122,62)(38,172,123,61)(39,171,124,60)(40,170,125,59)(41,169,126,58)(42,224,127,57)(43,223,128,112)(44,222,129,111)(45,221,130,110)(46,220,131,109)(47,219,132,108)(48,218,133,107)(49,217,134,106)(50,216,135,105)(51,215,136,104)(52,214,137,103)(53,213,138,102)(54,212,139,101)(55,211,140,100)(56,210,141,99), (1,209)(2,85)(3,183)(4,59)(5,213)(6,89)(7,187)(8,63)(9,217)(10,93)(11,191)(12,67)(13,221)(14,97)(15,195)(16,71)(17,169)(18,101)(19,199)(20,75)(21,173)(22,105)(23,203)(24,79)(25,177)(26,109)(27,207)(28,83)(29,181)(30,57)(31,211)(32,87)(33,185)(34,61)(35,215)(36,91)(37,189)(38,65)(39,219)(40,95)(41,193)(42,69)(43,223)(44,99)(45,197)(46,73)(47,171)(48,103)(49,201)(50,77)(51,175)(52,107)(53,205)(54,81)(55,179)(56,111)(58,158)(60,132)(62,162)(64,136)(66,166)(68,140)(70,114)(72,144)(74,118)(76,148)(78,122)(80,152)(82,126)(84,156)(86,130)(88,160)(90,134)(92,164)(94,138)(96,168)(98,142)(100,116)(102,146)(104,120)(106,150)(108,124)(110,154)(112,128)(113,194)(115,224)(117,198)(119,172)(121,202)(123,176)(125,206)(127,180)(129,210)(131,184)(133,214)(135,188)(137,218)(139,192)(141,222)(143,196)(145,170)(147,200)(149,174)(151,204)(153,178)(155,208)(157,182)(159,212)(161,186)(163,216)(165,190)(167,220) );
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)], [(1,209,142,98),(2,208,143,97),(3,207,144,96),(4,206,145,95),(5,205,146,94),(6,204,147,93),(7,203,148,92),(8,202,149,91),(9,201,150,90),(10,200,151,89),(11,199,152,88),(12,198,153,87),(13,197,154,86),(14,196,155,85),(15,195,156,84),(16,194,157,83),(17,193,158,82),(18,192,159,81),(19,191,160,80),(20,190,161,79),(21,189,162,78),(22,188,163,77),(23,187,164,76),(24,186,165,75),(25,185,166,74),(26,184,167,73),(27,183,168,72),(28,182,113,71),(29,181,114,70),(30,180,115,69),(31,179,116,68),(32,178,117,67),(33,177,118,66),(34,176,119,65),(35,175,120,64),(36,174,121,63),(37,173,122,62),(38,172,123,61),(39,171,124,60),(40,170,125,59),(41,169,126,58),(42,224,127,57),(43,223,128,112),(44,222,129,111),(45,221,130,110),(46,220,131,109),(47,219,132,108),(48,218,133,107),(49,217,134,106),(50,216,135,105),(51,215,136,104),(52,214,137,103),(53,213,138,102),(54,212,139,101),(55,211,140,100),(56,210,141,99)], [(1,209),(2,85),(3,183),(4,59),(5,213),(6,89),(7,187),(8,63),(9,217),(10,93),(11,191),(12,67),(13,221),(14,97),(15,195),(16,71),(17,169),(18,101),(19,199),(20,75),(21,173),(22,105),(23,203),(24,79),(25,177),(26,109),(27,207),(28,83),(29,181),(30,57),(31,211),(32,87),(33,185),(34,61),(35,215),(36,91),(37,189),(38,65),(39,219),(40,95),(41,193),(42,69),(43,223),(44,99),(45,197),(46,73),(47,171),(48,103),(49,201),(50,77),(51,175),(52,107),(53,205),(54,81),(55,179),(56,111),(58,158),(60,132),(62,162),(64,136),(66,166),(68,140),(70,114),(72,144),(74,118),(76,148),(78,122),(80,152),(82,126),(84,156),(86,130),(88,160),(90,134),(92,164),(94,138),(96,168),(98,142),(100,116),(102,146),(104,120),(106,150),(108,124),(110,154),(112,128),(113,194),(115,224),(117,198),(119,172),(121,202),(123,176),(125,206),(127,180),(129,210),(131,184),(133,214),(135,188),(137,218),(139,192),(141,222),(143,196),(145,170),(147,200),(149,174),(151,204),(153,178),(155,208),(157,182),(159,212),(161,186),(163,216),(165,190),(167,220)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 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 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 28 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D8 | C4○D28 | C8⋊C22 | D4⋊2D7 | D4×D7 | D8⋊3D7 | D56⋊C2 |
| kernel | C56⋊1C4⋊C2 | C4.Dic14 | C56⋊1C4 | D14⋊C8 | D4⋊Dic7 | C7×D4⋊C4 | C4⋊C4⋊7D7 | C28⋊2D4 | C2×Dic7 | C22×D7 | D4⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×D4 | C14 | C4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 3 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C56⋊1C4⋊C2 ►in GL4(𝔽113) generated by
| 99 | 104 | 0 | 0 |
| 18 | 68 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 38 | 44 |
| 65 | 54 | 0 | 0 |
| 18 | 48 | 0 | 0 |
| 0 | 0 | 84 | 111 |
| 0 | 0 | 82 | 29 |
| 17 | 12 | 0 | 0 |
| 89 | 96 | 0 | 0 |
| 0 | 0 | 29 | 2 |
| 0 | 0 | 32 | 84 |
G:=sub<GL(4,GF(113))| [99,18,0,0,104,68,0,0,0,0,18,38,0,0,0,44],[65,18,0,0,54,48,0,0,0,0,84,82,0,0,111,29],[17,89,0,0,12,96,0,0,0,0,29,32,0,0,2,84] >;
C56⋊1C4⋊C2 in GAP, Magma, Sage, TeX
C_{56}\rtimes_1C_4\rtimes C_2 % in TeX
G:=Group("C56:1C4:C2"); // GroupNames label
G:=SmallGroup(448,318);
// by ID
G=gap.SmallGroup(448,318);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,64,926,219,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^43*b^2,c*b*c=b^-1>;
// generators/relations