direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C112⋊C2, C16⋊8D14, C4.7D56, C14⋊1SD32, C8.11D28, C28.32D8, C56.61D4, C112⋊9C22, C56.56C23, D56.6C22, C22.13D56, Dic28⋊6C22, (C2×C16)⋊7D7, C7⋊1(C2×SD32), (C2×C112)⋊11C2, (C2×D56).5C2, C2.12(C2×D56), C4.37(C2×D28), C14.10(C2×D8), (C2×C14).19D8, (C2×C4).84D28, (C2×Dic28)⋊8C2, (C2×C8).304D14, (C2×C28).381D4, C28.280(C2×D4), C8.46(C22×D7), (C2×C56).377C22, SmallGroup(448,437)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C112⋊C2
G = < a,b,c | a2=b112=c2=1, ab=ba, ac=ca, cbc=b55 >
Subgroups: 756 in 90 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C16, C2×C8, D8, Q16, C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C2×C16, SD32, C2×D8, C2×Q16, C56, Dic14, D28, C2×Dic7, C2×C28, C22×D7, C2×SD32, C112, D56, D56, Dic28, Dic28, C2×C56, C2×Dic14, C2×D28, C112⋊C2, C2×C112, C2×D56, C2×Dic28, C2×C112⋊C2
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, SD32, C2×D8, D28, C22×D7, C2×SD32, D56, C2×D28, C112⋊C2, C2×D56, C2×C112⋊C2
►On 224 points
Generators in S224
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 161)(22 162)(23 163)(24 164)(25 165)(26 166)(27 167)(28 168)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 183)(44 184)(45 185)(46 186)(47 187)(48 188)(49 189)(50 190)(51 191)(52 192)(53 193)(54 194)(55 195)(56 196)(57 197)(58 198)(59 199)(60 200)(61 201)(62 202)(63 203)(64 204)(65 205)(66 206)(67 207)(68 208)(69 209)(70 210)(71 211)(72 212)(73 213)(74 214)(75 215)(76 216)(77 217)(78 218)(79 219)(80 220)(81 221)(82 222)(83 223)(84 224)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)
(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)
(2 56)(3 111)(4 54)(5 109)(6 52)(7 107)(8 50)(9 105)(10 48)(11 103)(12 46)(13 101)(14 44)(15 99)(16 42)(17 97)(18 40)(19 95)(20 38)(21 93)(22 36)(23 91)(24 34)(25 89)(26 32)(27 87)(28 30)(29 85)(31 83)(33 81)(35 79)(37 77)(39 75)(41 73)(43 71)(45 69)(47 67)(49 65)(51 63)(53 61)(55 59)(58 112)(60 110)(62 108)(64 106)(66 104)(68 102)(70 100)(72 98)(74 96)(76 94)(78 92)(80 90)(82 88)(84 86)(113 169)(114 224)(115 167)(116 222)(117 165)(118 220)(119 163)(120 218)(121 161)(122 216)(123 159)(124 214)(125 157)(126 212)(127 155)(128 210)(129 153)(130 208)(131 151)(132 206)(133 149)(134 204)(135 147)(136 202)(137 145)(138 200)(139 143)(140 198)(142 196)(144 194)(146 192)(148 190)(150 188)(152 186)(154 184)(156 182)(158 180)(160 178)(162 176)(164 174)(166 172)(168 170)(171 223)(173 221)(175 219)(177 217)(179 215)(181 213)(183 211)(185 209)(187 207)(189 205)(191 203)(193 201)(195 199)
G:=sub<Sym(224)| (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,166)(27,167)(28,168)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,197)(58,198)(59,199)(60,200)(61,201)(62,202)(63,203)(64,204)(65,205)(66,206)(67,207)(68,208)(69,209)(70,210)(71,211)(72,212)(73,213)(74,214)(75,215)(76,216)(77,217)(78,218)(79,219)(80,220)(81,221)(82,222)(83,223)(84,224)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140), (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), (2,56)(3,111)(4,54)(5,109)(6,52)(7,107)(8,50)(9,105)(10,48)(11,103)(12,46)(13,101)(14,44)(15,99)(16,42)(17,97)(18,40)(19,95)(20,38)(21,93)(22,36)(23,91)(24,34)(25,89)(26,32)(27,87)(28,30)(29,85)(31,83)(33,81)(35,79)(37,77)(39,75)(41,73)(43,71)(45,69)(47,67)(49,65)(51,63)(53,61)(55,59)(58,112)(60,110)(62,108)(64,106)(66,104)(68,102)(70,100)(72,98)(74,96)(76,94)(78,92)(80,90)(82,88)(84,86)(113,169)(114,224)(115,167)(116,222)(117,165)(118,220)(119,163)(120,218)(121,161)(122,216)(123,159)(124,214)(125,157)(126,212)(127,155)(128,210)(129,153)(130,208)(131,151)(132,206)(133,149)(134,204)(135,147)(136,202)(137,145)(138,200)(139,143)(140,198)(142,196)(144,194)(146,192)(148,190)(150,188)(152,186)(154,184)(156,182)(158,180)(160,178)(162,176)(164,174)(166,172)(168,170)(171,223)(173,221)(175,219)(177,217)(179,215)(181,213)(183,211)(185,209)(187,207)(189,205)(191,203)(193,201)(195,199)>;
G:=Group( (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,165)(26,166)(27,167)(28,168)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,183)(44,184)(45,185)(46,186)(47,187)(48,188)(49,189)(50,190)(51,191)(52,192)(53,193)(54,194)(55,195)(56,196)(57,197)(58,198)(59,199)(60,200)(61,201)(62,202)(63,203)(64,204)(65,205)(66,206)(67,207)(68,208)(69,209)(70,210)(71,211)(72,212)(73,213)(74,214)(75,215)(76,216)(77,217)(78,218)(79,219)(80,220)(81,221)(82,222)(83,223)(84,224)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)(112,140), (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), (2,56)(3,111)(4,54)(5,109)(6,52)(7,107)(8,50)(9,105)(10,48)(11,103)(12,46)(13,101)(14,44)(15,99)(16,42)(17,97)(18,40)(19,95)(20,38)(21,93)(22,36)(23,91)(24,34)(25,89)(26,32)(27,87)(28,30)(29,85)(31,83)(33,81)(35,79)(37,77)(39,75)(41,73)(43,71)(45,69)(47,67)(49,65)(51,63)(53,61)(55,59)(58,112)(60,110)(62,108)(64,106)(66,104)(68,102)(70,100)(72,98)(74,96)(76,94)(78,92)(80,90)(82,88)(84,86)(113,169)(114,224)(115,167)(116,222)(117,165)(118,220)(119,163)(120,218)(121,161)(122,216)(123,159)(124,214)(125,157)(126,212)(127,155)(128,210)(129,153)(130,208)(131,151)(132,206)(133,149)(134,204)(135,147)(136,202)(137,145)(138,200)(139,143)(140,198)(142,196)(144,194)(146,192)(148,190)(150,188)(152,186)(154,184)(156,182)(158,180)(160,178)(162,176)(164,174)(166,172)(168,170)(171,223)(173,221)(175,219)(177,217)(179,215)(181,213)(183,211)(185,209)(187,207)(189,205)(191,203)(193,201)(195,199) );
G=PermutationGroup([[(1,141),(2,142),(3,143),(4,144),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(21,161),(22,162),(23,163),(24,164),(25,165),(26,166),(27,167),(28,168),(29,169),(30,170),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,177),(38,178),(39,179),(40,180),(41,181),(42,182),(43,183),(44,184),(45,185),(46,186),(47,187),(48,188),(49,189),(50,190),(51,191),(52,192),(53,193),(54,194),(55,195),(56,196),(57,197),(58,198),(59,199),(60,200),(61,201),(62,202),(63,203),(64,204),(65,205),(66,206),(67,207),(68,208),(69,209),(70,210),(71,211),(72,212),(73,213),(74,214),(75,215),(76,216),(77,217),(78,218),(79,219),(80,220),(81,221),(82,222),(83,223),(84,224),(85,113),(86,114),(87,115),(88,116),(89,117),(90,118),(91,119),(92,120),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128),(101,129),(102,130),(103,131),(104,132),(105,133),(106,134),(107,135),(108,136),(109,137),(110,138),(111,139),(112,140)], [(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)], [(2,56),(3,111),(4,54),(5,109),(6,52),(7,107),(8,50),(9,105),(10,48),(11,103),(12,46),(13,101),(14,44),(15,99),(16,42),(17,97),(18,40),(19,95),(20,38),(21,93),(22,36),(23,91),(24,34),(25,89),(26,32),(27,87),(28,30),(29,85),(31,83),(33,81),(35,79),(37,77),(39,75),(41,73),(43,71),(45,69),(47,67),(49,65),(51,63),(53,61),(55,59),(58,112),(60,110),(62,108),(64,106),(66,104),(68,102),(70,100),(72,98),(74,96),(76,94),(78,92),(80,90),(82,88),(84,86),(113,169),(114,224),(115,167),(116,222),(117,165),(118,220),(119,163),(120,218),(121,161),(122,216),(123,159),(124,214),(125,157),(126,212),(127,155),(128,210),(129,153),(130,208),(131,151),(132,206),(133,149),(134,204),(135,147),(136,202),(137,145),(138,200),(139,143),(140,198),(142,196),(144,194),(146,192),(148,190),(150,188),(152,186),(154,184),(156,182),(158,180),(160,178),(162,176),(164,174),(166,172),(168,170),(171,223),(173,221),(175,219),(177,217),(179,215),(181,213),(183,211),(185,209),(187,207),(189,205),(191,203),(193,201),(195,199)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 56 | 56 | 2 | 2 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 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 | D4 | D4 | D7 | D8 | D8 | D14 | D14 | SD32 | D28 | D28 | D56 | D56 | C112⋊C2 |
| kernel | C2×C112⋊C2 | C112⋊C2 | C2×C112 | C2×D56 | C2×Dic28 | C56 | C2×C28 | C2×C16 | C28 | C2×C14 | C16 | C2×C8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 2 | 6 | 3 | 8 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C2×C112⋊C2 ►in GL5(𝔽113)
| 112 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 60 | 69 | 0 | 0 |
| 0 | 44 | 60 | 0 | 0 |
| 0 | 0 | 0 | 28 | 73 |
| 0 | 0 | 0 | 76 | 65 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 33 | 1 |
| 0 | 0 | 0 | 42 | 80 |
G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,60,44,0,0,0,69,60,0,0,0,0,0,28,76,0,0,0,73,65],[112,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,33,42,0,0,0,1,80] >;
C2×C112⋊C2 in GAP, Magma, Sage, TeX
C_2\times C_{112}\rtimes C_2 % in TeX
G:=Group("C2xC112:C2"); // GroupNames label
G:=SmallGroup(448,437);
// by ID
G=gap.SmallGroup(448,437);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,142,1571,80,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c|a^2=b^112=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^55>;
// generators/relations