metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C32⋊2D7, C224⋊2C2, C7⋊1SD64, C8.6D28, C4.2D56, C14.2D16, C56.56D4, C2.4D112, C28.27D8, Dic56⋊1C2, D112.1C2, C16.14D14, C112.15C22, SmallGroup(448,6)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C224⋊C2
G = < a,b | a224=b2=1, bab=a111 >
Smallest permutation representation of C224⋊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)
(2 112)(3 223)(4 110)(5 221)(6 108)(7 219)(8 106)(9 217)(10 104)(11 215)(12 102)(13 213)(14 100)(15 211)(16 98)(17 209)(18 96)(19 207)(20 94)(21 205)(22 92)(23 203)(24 90)(25 201)(26 88)(27 199)(28 86)(29 197)(30 84)(31 195)(32 82)(33 193)(34 80)(35 191)(36 78)(37 189)(38 76)(39 187)(40 74)(41 185)(42 72)(43 183)(44 70)(45 181)(46 68)(47 179)(48 66)(49 177)(50 64)(51 175)(52 62)(53 173)(54 60)(55 171)(56 58)(57 169)(59 167)(61 165)(63 163)(65 161)(67 159)(69 157)(71 155)(73 153)(75 151)(77 149)(79 147)(81 145)(83 143)(85 141)(87 139)(89 137)(91 135)(93 133)(95 131)(97 129)(99 127)(101 125)(103 123)(105 121)(107 119)(109 117)(111 115)(114 224)(116 222)(118 220)(120 218)(122 216)(124 214)(126 212)(128 210)(130 208)(132 206)(134 204)(136 202)(138 200)(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)
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), (2,112)(3,223)(4,110)(5,221)(6,108)(7,219)(8,106)(9,217)(10,104)(11,215)(12,102)(13,213)(14,100)(15,211)(16,98)(17,209)(18,96)(19,207)(20,94)(21,205)(22,92)(23,203)(24,90)(25,201)(26,88)(27,199)(28,86)(29,197)(30,84)(31,195)(32,82)(33,193)(34,80)(35,191)(36,78)(37,189)(38,76)(39,187)(40,74)(41,185)(42,72)(43,183)(44,70)(45,181)(46,68)(47,179)(48,66)(49,177)(50,64)(51,175)(52,62)(53,173)(54,60)(55,171)(56,58)(57,169)(59,167)(61,165)(63,163)(65,161)(67,159)(69,157)(71,155)(73,153)(75,151)(77,149)(79,147)(81,145)(83,143)(85,141)(87,139)(89,137)(91,135)(93,133)(95,131)(97,129)(99,127)(101,125)(103,123)(105,121)(107,119)(109,117)(111,115)(114,224)(116,222)(118,220)(120,218)(122,216)(124,214)(126,212)(128,210)(130,208)(132,206)(134,204)(136,202)(138,200)(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)>;
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), (2,112)(3,223)(4,110)(5,221)(6,108)(7,219)(8,106)(9,217)(10,104)(11,215)(12,102)(13,213)(14,100)(15,211)(16,98)(17,209)(18,96)(19,207)(20,94)(21,205)(22,92)(23,203)(24,90)(25,201)(26,88)(27,199)(28,86)(29,197)(30,84)(31,195)(32,82)(33,193)(34,80)(35,191)(36,78)(37,189)(38,76)(39,187)(40,74)(41,185)(42,72)(43,183)(44,70)(45,181)(46,68)(47,179)(48,66)(49,177)(50,64)(51,175)(52,62)(53,173)(54,60)(55,171)(56,58)(57,169)(59,167)(61,165)(63,163)(65,161)(67,159)(69,157)(71,155)(73,153)(75,151)(77,149)(79,147)(81,145)(83,143)(85,141)(87,139)(89,137)(91,135)(93,133)(95,131)(97,129)(99,127)(101,125)(103,123)(105,121)(107,119)(109,117)(111,115)(114,224)(116,222)(118,220)(120,218)(122,216)(124,214)(126,212)(128,210)(130,208)(132,206)(134,204)(136,202)(138,200)(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) );
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)], [(2,112),(3,223),(4,110),(5,221),(6,108),(7,219),(8,106),(9,217),(10,104),(11,215),(12,102),(13,213),(14,100),(15,211),(16,98),(17,209),(18,96),(19,207),(20,94),(21,205),(22,92),(23,203),(24,90),(25,201),(26,88),(27,199),(28,86),(29,197),(30,84),(31,195),(32,82),(33,193),(34,80),(35,191),(36,78),(37,189),(38,76),(39,187),(40,74),(41,185),(42,72),(43,183),(44,70),(45,181),(46,68),(47,179),(48,66),(49,177),(50,64),(51,175),(52,62),(53,173),(54,60),(55,171),(56,58),(57,169),(59,167),(61,165),(63,163),(65,161),(67,159),(69,157),(71,155),(73,153),(75,151),(77,149),(79,147),(81,145),(83,143),(85,141),(87,139),(89,137),(91,135),(93,133),(95,131),(97,129),(99,127),(101,125),(103,123),(105,121),(107,119),(109,117),(111,115),(114,224),(116,222),(118,220),(120,218),(122,216),(124,214),(126,212),(128,210),(130,208),(132,206),(134,204),(136,202),(138,200),(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)]])
115 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 16A | 16B | 16C | 16D | 28A | ··· | 28F | 32A | ··· | 32H | 56A | ··· | 56L | 112A | ··· | 112X | 224A | ··· | 224AV |
| order | 1 | 2 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | ··· | 28 | 32 | ··· | 32 | 56 | ··· | 56 | 112 | ··· | 112 | 224 | ··· | 224 |
| size | 1 | 1 | 112 | 2 | 112 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
115 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D4 | D7 | D8 | D14 | D16 | D28 | SD64 | D56 | D112 | C224⋊C2 |
| kernel | C224⋊C2 | C224 | D112 | Dic56 | C56 | C32 | C28 | C16 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 8 | 12 | 24 | 48 |
Matrix representation of C224⋊C2 ►in GL2(𝔽449) generated by
| 209 | 356 |
| 93 | 167 |
| 1 | 43 |
| 0 | 448 |
G:=sub<GL(2,GF(449))| [209,93,356,167],[1,0,43,448] >;
C224⋊C2 in GAP, Magma, Sage, TeX
C_{224}\rtimes C_2 % in TeX
G:=Group("C224:C2"); // GroupNames label
G:=SmallGroup(448,6);
// by ID
G=gap.SmallGroup(448,6);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,85,92,926,142,1571,80,1684,102,18822]);
// Polycyclic
G:=Group<a,b|a^224=b^2=1,b*a*b=a^111>;
// generators/relations
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