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## G = C2×C4×C28order 224 = 25·7

### Abelian group of type [2,4,28]

Aliases: C2×C4×C28, SmallGroup(224,149)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4×C28
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C4×C28 — C2×C4×C28
 Lower central C1 — C2×C4×C28
 Upper central C1 — C2×C4×C28

Generators and relations for C2×C4×C28
G = < a,b,c | a2=b4=c28=1, ab=ba, ac=ca, bc=cb >

Subgroups: 108, all normal (8 characteristic)
C1, C2, C4, C22, C22, C7, C2×C4, C23, C14, C42, C22×C4, C28, C2×C14, C2×C14, C2×C42, C2×C28, C22×C14, C4×C28, C22×C28, C2×C4×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C42, C22×C4, C28, C2×C14, C2×C42, C2×C28, C22×C14, C4×C28, C22×C28, C2×C4×C28

Smallest permutation representation of C2×C4×C28
Regular action on 224 points
Generators in S224
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(57 139)(58 140)(59 113)(60 114)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(73 127)(74 128)(75 129)(76 130)(77 131)(78 132)(79 133)(80 134)(81 135)(82 136)(83 137)(84 138)(85 190)(86 191)(87 192)(88 193)(89 194)(90 195)(91 196)(92 169)(93 170)(94 171)(95 172)(96 173)(97 174)(98 175)(99 176)(100 177)(101 178)(102 179)(103 180)(104 181)(105 182)(106 183)(107 184)(108 185)(109 186)(110 187)(111 188)(112 189)(141 217)(142 218)(143 219)(144 220)(145 221)(146 222)(147 223)(148 224)(149 197)(150 198)(151 199)(152 200)(153 201)(154 202)(155 203)(156 204)(157 205)(158 206)(159 207)(160 208)(161 209)(162 210)(163 211)(164 212)(165 213)(166 214)(167 215)(168 216)
(1 186 64 157)(2 187 65 158)(3 188 66 159)(4 189 67 160)(5 190 68 161)(6 191 69 162)(7 192 70 163)(8 193 71 164)(9 194 72 165)(10 195 73 166)(11 196 74 167)(12 169 75 168)(13 170 76 141)(14 171 77 142)(15 172 78 143)(16 173 79 144)(17 174 80 145)(18 175 81 146)(19 176 82 147)(20 177 83 148)(21 178 84 149)(22 179 57 150)(23 180 58 151)(24 181 59 152)(25 182 60 153)(26 183 61 154)(27 184 62 155)(28 185 63 156)(29 89 126 213)(30 90 127 214)(31 91 128 215)(32 92 129 216)(33 93 130 217)(34 94 131 218)(35 95 132 219)(36 96 133 220)(37 97 134 221)(38 98 135 222)(39 99 136 223)(40 100 137 224)(41 101 138 197)(42 102 139 198)(43 103 140 199)(44 104 113 200)(45 105 114 201)(46 106 115 202)(47 107 116 203)(48 108 117 204)(49 109 118 205)(50 110 119 206)(51 111 120 207)(52 112 121 208)(53 85 122 209)(54 86 123 210)(55 87 124 211)(56 88 125 212)
(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)

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

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

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

224 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4X 7A ··· 7F 14A ··· 14AP 28A ··· 28EN order 1 2 ··· 2 4 ··· 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

224 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C7 C14 C14 C28 kernel C2×C4×C28 C4×C28 C22×C28 C2×C28 C2×C42 C42 C22×C4 C2×C4 # reps 1 4 3 24 6 24 18 144

Matrix representation of C2×C4×C28 in GL3(𝔽29) generated by

 28 0 0 0 1 0 0 0 1
,
 1 0 0 0 28 0 0 0 17
,
 1 0 0 0 3 0 0 0 15
G:=sub<GL(3,GF(29))| [28,0,0,0,1,0,0,0,1],[1,0,0,0,28,0,0,0,17],[1,0,0,0,3,0,0,0,15] >;

C2×C4×C28 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_{28}
% in TeX

G:=Group("C2xC4xC28");
// GroupNames label

G:=SmallGroup(224,149);
// by ID

G=gap.SmallGroup(224,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,336,679]);
// Polycyclic

G:=Group<a,b,c|a^2=b^4=c^28=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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