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G = C2×C4×C28order 224 = 25·7

Abelian group of type [2,4,28]

direct product, abelian, monomial, 2-elementary

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

Series: Derived Chief Lower central Upper central

C1 — C2×C4×C28
C1C2C22C2×C14C2×C28C4×C28 — C2×C4×C28
C1 — C2×C4×C28
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 [×7], C4 [×12], C22, C22 [×6], C7, C2×C4 [×18], C23, C14 [×7], C42 [×4], C22×C4 [×3], C28 [×12], C2×C14, C2×C14 [×6], C2×C42, C2×C28 [×18], C22×C14, C4×C28 [×4], C22×C28 [×3], C2×C4×C28
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C7, C2×C4 [×18], C23, C14 [×7], C42 [×4], C22×C4 [×3], C28 [×12], C2×C14 [×7], C2×C42, C2×C28 [×18], C22×C14, C4×C28 [×4], C22×C28 [×3], C2×C4×C28

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

C2×C4×C28 is a maximal subgroup of
C28.8C42  (C2×C28)⋊3C8  C287M4(2)  C42.6Dic7  C42.7Dic7  C284(C4⋊C4)  (C2×C28)⋊10Q8  C424Dic7  (C2×C42).D7  C428Dic7  C429Dic7  C425Dic7  (C2×C4)⋊6D28  (C2×C42)⋊D7  C42.274D14  C42.276D14  C42.277D14

224 conjugacy classes

class 1 2A···2G4A···4X7A···7F14A···14AP28A···28EN
order12···24···47···714···1428···28
size11···11···11···11···11···1

224 irreducible representations

dim11111111
type+++
imageC1C2C2C4C7C14C14C28
kernelC2×C4×C28C4×C28C22×C28C2×C28C2×C42C42C22×C4C2×C4
# reps1432462418144

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

2800
010
001
,
100
0280
0017
,
100
030
0015
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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