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## G = C2×Dic28order 224 = 25·7

### Direct product of C2 and Dic28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×Dic28
 Chief series C1 — C7 — C14 — C28 — Dic14 — C2×Dic14 — C2×Dic28
 Lower central C7 — C14 — C28 — C2×Dic28
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×Dic28
G = < a,b,c | a2=b56=1, c2=b28, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 238 in 60 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C2×C4, Q8, C14, C14, C2×C8, Q16, C2×Q8, Dic7, C28, C2×C14, C2×Q16, C56, Dic14, Dic14, C2×Dic7, C2×C28, Dic28, C2×C56, C2×Dic14, C2×Dic28
Quotients: C1, C2, C22, D4, C23, D7, Q16, C2×D4, D14, C2×Q16, D28, C22×D7, Dic28, C2×D28, C2×Dic28

Smallest permutation representation of C2×Dic28
Regular action on 224 points
Generators in S224
(1 156)(2 157)(3 158)(4 159)(5 160)(6 161)(7 162)(8 163)(9 164)(10 165)(11 166)(12 167)(13 168)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 120)(22 121)(23 122)(24 123)(25 124)(26 125)(27 126)(28 127)(29 128)(30 129)(31 130)(32 131)(33 132)(34 133)(35 134)(36 135)(37 136)(38 137)(39 138)(40 139)(41 140)(42 141)(43 142)(44 143)(45 144)(46 145)(47 146)(48 147)(49 148)(50 149)(51 150)(52 151)(53 152)(54 153)(55 154)(56 155)(57 174)(58 175)(59 176)(60 177)(61 178)(62 179)(63 180)(64 181)(65 182)(66 183)(67 184)(68 185)(69 186)(70 187)(71 188)(72 189)(73 190)(74 191)(75 192)(76 193)(77 194)(78 195)(79 196)(80 197)(81 198)(82 199)(83 200)(84 201)(85 202)(86 203)(87 204)(88 205)(89 206)(90 207)(91 208)(92 209)(93 210)(94 211)(95 212)(96 213)(97 214)(98 215)(99 216)(100 217)(101 218)(102 219)(103 220)(104 221)(105 222)(106 223)(107 224)(108 169)(109 170)(110 171)(111 172)(112 173)
(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 69 29 97)(2 68 30 96)(3 67 31 95)(4 66 32 94)(5 65 33 93)(6 64 34 92)(7 63 35 91)(8 62 36 90)(9 61 37 89)(10 60 38 88)(11 59 39 87)(12 58 40 86)(13 57 41 85)(14 112 42 84)(15 111 43 83)(16 110 44 82)(17 109 45 81)(18 108 46 80)(19 107 47 79)(20 106 48 78)(21 105 49 77)(22 104 50 76)(23 103 51 75)(24 102 52 74)(25 101 53 73)(26 100 54 72)(27 99 55 71)(28 98 56 70)(113 173 141 201)(114 172 142 200)(115 171 143 199)(116 170 144 198)(117 169 145 197)(118 224 146 196)(119 223 147 195)(120 222 148 194)(121 221 149 193)(122 220 150 192)(123 219 151 191)(124 218 152 190)(125 217 153 189)(126 216 154 188)(127 215 155 187)(128 214 156 186)(129 213 157 185)(130 212 158 184)(131 211 159 183)(132 210 160 182)(133 209 161 181)(134 208 162 180)(135 207 163 179)(136 206 164 178)(137 205 165 177)(138 204 166 176)(139 203 167 175)(140 202 168 174)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 28A ··· 28L 56A ··· 56X order 1 2 2 2 4 4 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 28 28 28 28 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + + + - image C1 C2 C2 C2 D4 D4 D7 Q16 D14 D14 D28 D28 Dic28 kernel C2×Dic28 Dic28 C2×C56 C2×Dic14 C28 C2×C14 C2×C8 C14 C8 C2×C4 C4 C22 C2 # reps 1 4 1 2 1 1 3 4 6 3 6 6 24

Matrix representation of C2×Dic28 in GL3(𝔽113) generated by

 112 0 0 0 112 0 0 0 112
,
 112 0 0 0 70 91 0 22 27
,
 1 0 0 0 99 13 0 37 14
G:=sub<GL(3,GF(113))| [112,0,0,0,112,0,0,0,112],[112,0,0,0,70,22,0,91,27],[1,0,0,0,99,37,0,13,14] >;

C2×Dic28 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{28}
% in TeX

G:=Group("C2xDic28");
// GroupNames label

G:=SmallGroup(224,100);
// by ID

G=gap.SmallGroup(224,100);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,218,122,579,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^2=b^56=1,c^2=b^28,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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