Copied to
clipboard

## G = C4×C4○D28order 448 = 26·7

### Direct product of C4 and C4○D28

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4×C4○D28
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — C2×C4○D28 — C4×C4○D28
 Lower central C7 — C14 — C4×C4○D28
 Upper central C1 — C42 — C2×C42

Generators and relations for C4×C4○D28
G = < a,b,c,d | a4=b4=d2=1, c14=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c13 >

Subgroups: 1156 in 310 conjugacy classes, 159 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C42, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C42, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C22×D7, C22×C14, C4×C4○D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C4×C28, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, C22×C28, C4×Dic14, D7×C42, C42⋊D7, C4×D28, C23.21D14, C4×C7⋊D4, C2×C4×C28, C2×C4○D28, C4×C4○D28
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, C24, D14, C23×C4, C2×C4○D4, C4×D7, C22×D7, C4×C4○D4, C2×C4×D7, C4○D28, C23×D7, D7×C22×C4, C2×C4○D28, C4×C4○D28

Smallest permutation representation of C4×C4○D28
On 224 points
Generators in S224
(1 96 137 214)(2 97 138 215)(3 98 139 216)(4 99 140 217)(5 100 113 218)(6 101 114 219)(7 102 115 220)(8 103 116 221)(9 104 117 222)(10 105 118 223)(11 106 119 224)(12 107 120 197)(13 108 121 198)(14 109 122 199)(15 110 123 200)(16 111 124 201)(17 112 125 202)(18 85 126 203)(19 86 127 204)(20 87 128 205)(21 88 129 206)(22 89 130 207)(23 90 131 208)(24 91 132 209)(25 92 133 210)(26 93 134 211)(27 94 135 212)(28 95 136 213)(29 196 146 59)(30 169 147 60)(31 170 148 61)(32 171 149 62)(33 172 150 63)(34 173 151 64)(35 174 152 65)(36 175 153 66)(37 176 154 67)(38 177 155 68)(39 178 156 69)(40 179 157 70)(41 180 158 71)(42 181 159 72)(43 182 160 73)(44 183 161 74)(45 184 162 75)(46 185 163 76)(47 186 164 77)(48 187 165 78)(49 188 166 79)(50 189 167 80)(51 190 168 81)(52 191 141 82)(53 192 142 83)(54 193 143 84)(55 194 144 57)(56 195 145 58)
(1 161 15 147)(2 162 16 148)(3 163 17 149)(4 164 18 150)(5 165 19 151)(6 166 20 152)(7 167 21 153)(8 168 22 154)(9 141 23 155)(10 142 24 156)(11 143 25 157)(12 144 26 158)(13 145 27 159)(14 146 28 160)(29 136 43 122)(30 137 44 123)(31 138 45 124)(32 139 46 125)(33 140 47 126)(34 113 48 127)(35 114 49 128)(36 115 50 129)(37 116 51 130)(38 117 52 131)(39 118 53 132)(40 119 54 133)(41 120 55 134)(42 121 56 135)(57 93 71 107)(58 94 72 108)(59 95 73 109)(60 96 74 110)(61 97 75 111)(62 98 76 112)(63 99 77 85)(64 100 78 86)(65 101 79 87)(66 102 80 88)(67 103 81 89)(68 104 82 90)(69 105 83 91)(70 106 84 92)(169 214 183 200)(170 215 184 201)(171 216 185 202)(172 217 186 203)(173 218 187 204)(174 219 188 205)(175 220 189 206)(176 221 190 207)(177 222 191 208)(178 223 192 209)(179 224 193 210)(180 197 194 211)(181 198 195 212)(182 199 196 213)
(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)
(1 129)(2 128)(3 127)(4 126)(5 125)(6 124)(7 123)(8 122)(9 121)(10 120)(11 119)(12 118)(13 117)(14 116)(15 115)(16 114)(17 113)(18 140)(19 139)(20 138)(21 137)(22 136)(23 135)(24 134)(25 133)(26 132)(27 131)(28 130)(29 168)(30 167)(31 166)(32 165)(33 164)(34 163)(35 162)(36 161)(37 160)(38 159)(39 158)(40 157)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 144)(54 143)(55 142)(56 141)(57 192)(58 191)(59 190)(60 189)(61 188)(62 187)(63 186)(64 185)(65 184)(66 183)(67 182)(68 181)(69 180)(70 179)(71 178)(72 177)(73 176)(74 175)(75 174)(76 173)(77 172)(78 171)(79 170)(80 169)(81 196)(82 195)(83 194)(84 193)(85 217)(86 216)(87 215)(88 214)(89 213)(90 212)(91 211)(92 210)(93 209)(94 208)(95 207)(96 206)(97 205)(98 204)(99 203)(100 202)(101 201)(102 200)(103 199)(104 198)(105 197)(106 224)(107 223)(108 222)(109 221)(110 220)(111 219)(112 218)

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4L 4M ··· 4R 4S ··· 4AD 7A 7B 7C 14A ··· 14U 28A ··· 28BT order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 14 14 14 14 1 ··· 1 2 ··· 2 14 ··· 14 2 2 2 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D7 C4○D4 D14 D14 C4×D7 C4○D28 kernel C4×C4○D28 C4×Dic14 D7×C42 C42⋊D7 C4×D28 C23.21D14 C4×C7⋊D4 C2×C4×C28 C2×C4○D28 C4○D28 C2×C42 C28 C42 C22×C4 C2×C4 C4 # reps 1 2 2 2 2 1 4 1 1 16 3 8 12 9 24 48

Matrix representation of C4×C4○D28 in GL3(𝔽29) generated by

 12 0 0 0 12 0 0 0 12
,
 1 0 0 0 17 0 0 0 17
,
 28 0 0 0 14 3 0 23 5
,
 1 0 0 0 25 1 0 14 4
G:=sub<GL(3,GF(29))| [12,0,0,0,12,0,0,0,12],[1,0,0,0,17,0,0,0,17],[28,0,0,0,14,23,0,3,5],[1,0,0,0,25,14,0,1,4] >;

C4×C4○D28 in GAP, Magma, Sage, TeX

C_4\times C_4\circ D_{28}
% in TeX

G:=Group("C4xC4oD28");
// GroupNames label

G:=SmallGroup(448,927);
// by ID

G=gap.SmallGroup(448,927);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,80,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^14=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^13>;
// generators/relations

׿
×
𝔽