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## G = C2×D112order 448 = 26·7

### Direct product of C2 and D112

Series: Derived Chief Lower central Upper central

 Derived series C1 — C56 — C2×D112
 Chief series C1 — C7 — C14 — C28 — C56 — D56 — C2×D56 — C2×D112
 Lower central C7 — C14 — C28 — C56 — C2×D112
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16

Generators and relations for C2×D112
G = < a,b,c | a2=b112=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 1012 in 98 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C2, C4, C22, C22, C7, C8, C2×C4, D4, C23, D7, C14, C14, C16, C2×C8, D8, C2×D4, C28, D14, C2×C14, C2×C16, D16, C2×D8, C56, D28, C2×C28, C22×D7, C2×D16, C112, D56, D56, C2×C56, C2×D28, D112, C2×C112, C2×D56, C2×D112
Quotients: C1, C2, C22, D4, C23, D7, D8, C2×D4, D14, D16, C2×D8, D28, C22×D7, C2×D16, D56, C2×D28, D112, C2×D56, C2×D112

Smallest permutation representation of C2×D112
On 224 points
Generators in S224
(1 169)(2 170)(3 171)(4 172)(5 173)(6 174)(7 175)(8 176)(9 177)(10 178)(11 179)(12 180)(13 181)(14 182)(15 183)(16 184)(17 185)(18 186)(19 187)(20 188)(21 189)(22 190)(23 191)(24 192)(25 193)(26 194)(27 195)(28 196)(29 197)(30 198)(31 199)(32 200)(33 201)(34 202)(35 203)(36 204)(37 205)(38 206)(39 207)(40 208)(41 209)(42 210)(43 211)(44 212)(45 213)(46 214)(47 215)(48 216)(49 217)(50 218)(51 219)(52 220)(53 221)(54 222)(55 223)(56 224)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 129)(74 130)(75 131)(76 132)(77 133)(78 134)(79 135)(80 136)(81 137)(82 138)(83 139)(84 140)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)(101 157)(102 158)(103 159)(104 160)(105 161)(106 162)(107 163)(108 164)(109 165)(110 166)(111 167)(112 168)
(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)
(1 168)(2 167)(3 166)(4 165)(5 164)(6 163)(7 162)(8 161)(9 160)(10 159)(11 158)(12 157)(13 156)(14 155)(15 154)(16 153)(17 152)(18 151)(19 150)(20 149)(21 148)(22 147)(23 146)(24 145)(25 144)(26 143)(27 142)(28 141)(29 140)(30 139)(31 138)(32 137)(33 136)(34 135)(35 134)(36 133)(37 132)(38 131)(39 130)(40 129)(41 128)(42 127)(43 126)(44 125)(45 124)(46 123)(47 122)(48 121)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 224)(58 223)(59 222)(60 221)(61 220)(62 219)(63 218)(64 217)(65 216)(66 215)(67 214)(68 213)(69 212)(70 211)(71 210)(72 209)(73 208)(74 207)(75 206)(76 205)(77 204)(78 203)(79 202)(80 201)(81 200)(82 199)(83 198)(84 197)(85 196)(86 195)(87 194)(88 193)(89 192)(90 191)(91 190)(92 189)(93 188)(94 187)(95 186)(96 185)(97 184)(98 183)(99 182)(100 181)(101 180)(102 179)(103 178)(104 177)(105 176)(106 175)(107 174)(108 173)(109 172)(110 171)(111 170)(112 169)

G:=sub<Sym(224)| (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,193)(26,194)(27,195)(28,196)(29,197)(30,198)(31,199)(32,200)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(41,209)(42,210)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,218)(51,219)(52,220)(53,221)(54,222)(55,223)(56,224)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156)(101,157)(102,158)(103,159)(104,160)(105,161)(106,162)(107,163)(108,164)(109,165)(110,166)(111,167)(112,168), (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), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,224)(58,223)(59,222)(60,221)(61,220)(62,219)(63,218)(64,217)(65,216)(66,215)(67,214)(68,213)(69,212)(70,211)(71,210)(72,209)(73,208)(74,207)(75,206)(76,205)(77,204)(78,203)(79,202)(80,201)(81,200)(82,199)(83,198)(84,197)(85,196)(86,195)(87,194)(88,193)(89,192)(90,191)(91,190)(92,189)(93,188)(94,187)(95,186)(96,185)(97,184)(98,183)(99,182)(100,181)(101,180)(102,179)(103,178)(104,177)(105,176)(106,175)(107,174)(108,173)(109,172)(110,171)(111,170)(112,169)>;

G:=Group( (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,193)(26,194)(27,195)(28,196)(29,197)(30,198)(31,199)(32,200)(33,201)(34,202)(35,203)(36,204)(37,205)(38,206)(39,207)(40,208)(41,209)(42,210)(43,211)(44,212)(45,213)(46,214)(47,215)(48,216)(49,217)(50,218)(51,219)(52,220)(53,221)(54,222)(55,223)(56,224)(57,113)(58,114)(59,115)(60,116)(61,117)(62,118)(63,119)(64,120)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,129)(74,130)(75,131)(76,132)(77,133)(78,134)(79,135)(80,136)(81,137)(82,138)(83,139)(84,140)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156)(101,157)(102,158)(103,159)(104,160)(105,161)(106,162)(107,163)(108,164)(109,165)(110,166)(111,167)(112,168), (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), (1,168)(2,167)(3,166)(4,165)(5,164)(6,163)(7,162)(8,161)(9,160)(10,159)(11,158)(12,157)(13,156)(14,155)(15,154)(16,153)(17,152)(18,151)(19,150)(20,149)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,140)(30,139)(31,138)(32,137)(33,136)(34,135)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,224)(58,223)(59,222)(60,221)(61,220)(62,219)(63,218)(64,217)(65,216)(66,215)(67,214)(68,213)(69,212)(70,211)(71,210)(72,209)(73,208)(74,207)(75,206)(76,205)(77,204)(78,203)(79,202)(80,201)(81,200)(82,199)(83,198)(84,197)(85,196)(86,195)(87,194)(88,193)(89,192)(90,191)(91,190)(92,189)(93,188)(94,187)(95,186)(96,185)(97,184)(98,183)(99,182)(100,181)(101,180)(102,179)(103,178)(104,177)(105,176)(106,175)(107,174)(108,173)(109,172)(110,171)(111,170)(112,169) );

G=PermutationGroup([[(1,169),(2,170),(3,171),(4,172),(5,173),(6,174),(7,175),(8,176),(9,177),(10,178),(11,179),(12,180),(13,181),(14,182),(15,183),(16,184),(17,185),(18,186),(19,187),(20,188),(21,189),(22,190),(23,191),(24,192),(25,193),(26,194),(27,195),(28,196),(29,197),(30,198),(31,199),(32,200),(33,201),(34,202),(35,203),(36,204),(37,205),(38,206),(39,207),(40,208),(41,209),(42,210),(43,211),(44,212),(45,213),(46,214),(47,215),(48,216),(49,217),(50,218),(51,219),(52,220),(53,221),(54,222),(55,223),(56,224),(57,113),(58,114),(59,115),(60,116),(61,117),(62,118),(63,119),(64,120),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,127),(72,128),(73,129),(74,130),(75,131),(76,132),(77,133),(78,134),(79,135),(80,136),(81,137),(82,138),(83,139),(84,140),(85,141),(86,142),(87,143),(88,144),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,153),(98,154),(99,155),(100,156),(101,157),(102,158),(103,159),(104,160),(105,161),(106,162),(107,163),(108,164),(109,165),(110,166),(111,167),(112,168)], [(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)], [(1,168),(2,167),(3,166),(4,165),(5,164),(6,163),(7,162),(8,161),(9,160),(10,159),(11,158),(12,157),(13,156),(14,155),(15,154),(16,153),(17,152),(18,151),(19,150),(20,149),(21,148),(22,147),(23,146),(24,145),(25,144),(26,143),(27,142),(28,141),(29,140),(30,139),(31,138),(32,137),(33,136),(34,135),(35,134),(36,133),(37,132),(38,131),(39,130),(40,129),(41,128),(42,127),(43,126),(44,125),(45,124),(46,123),(47,122),(48,121),(49,120),(50,119),(51,118),(52,117),(53,116),(54,115),(55,114),(56,113),(57,224),(58,223),(59,222),(60,221),(61,220),(62,219),(63,218),(64,217),(65,216),(66,215),(67,214),(68,213),(69,212),(70,211),(71,210),(72,209),(73,208),(74,207),(75,206),(76,205),(77,204),(78,203),(79,202),(80,201),(81,200),(82,199),(83,198),(84,197),(85,196),(86,195),(87,194),(88,193),(89,192),(90,191),(91,190),(92,189),(93,188),(94,187),(95,186),(96,185),(97,184),(98,183),(99,182),(100,181),(101,180),(102,179),(103,178),(104,177),(105,176),(106,175),(107,174),(108,173),(109,172),(110,171),(111,170),(112,169)]])

118 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 16A ··· 16H 28A ··· 28L 56A ··· 56X 112A ··· 112AV order 1 2 2 2 2 2 2 2 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 56 56 2 2 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 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 D4 D4 D7 D8 D8 D14 D14 D16 D28 D28 D56 D56 D112 kernel C2×D112 D112 C2×C112 C2×D56 C56 C2×C28 C2×C16 C28 C2×C14 C16 C2×C8 C14 C8 C2×C4 C4 C22 C2 # reps 1 4 1 2 1 1 3 2 2 6 3 8 6 6 12 12 48

Matrix representation of C2×D112 in GL3(𝔽113) generated by

 112 0 0 0 1 0 0 0 1
,
 112 0 0 0 108 74 0 39 78
,
 112 0 0 0 108 74 0 18 5
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[112,0,0,0,108,39,0,74,78],[112,0,0,0,108,18,0,74,5] >;

C2×D112 in GAP, Magma, Sage, TeX

C_2\times D_{112}
% in TeX

G:=Group("C2xD112");
// GroupNames label

G:=SmallGroup(448,436);
// by ID

G=gap.SmallGroup(448,436);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,142,675,192,1684,102,18822]);
// 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^-1>;
// generators/relations

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