direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D28.C4, C56.46C23, C28.69C24, M4(2)⋊25D14, C14⋊2(C8○D4), C4○D28.4C4, C7⋊C8.35C23, D28.28(C2×C4), (C2×D28).16C4, (C2×C8).279D14, (C8×D7)⋊22C22, C8.43(C22×D7), C23.30(C4×D7), C4.68(C23×D7), C8⋊D7⋊18C22, (C14×M4(2))⋊9C2, (C2×M4(2))⋊17D7, C14.32(C23×C4), C28.92(C22×C4), (C4×D7).35C23, (C2×C28).882C23, (C2×C56).238C22, Dic14.29(C2×C4), (C2×Dic14).16C4, C4○D28.49C22, D14.13(C22×C4), (C22×C4).374D14, (C7×M4(2))⋊25C22, Dic7.13(C22×C4), (C22×C28).264C22, C7⋊2(C2×C8○D4), (D7×C2×C8)⋊29C2, C4.123(C2×C4×D7), C22.8(C2×C4×D7), (C2×C7⋊C8)⋊33C22, (C22×C7⋊C8)⋊10C2, (C2×C4).87(C4×D7), C7⋊D4.3(C2×C4), (C2×C8⋊D7)⋊27C2, C2.33(D7×C22×C4), (C4×D7).24(C2×C4), (C2×C7⋊D4).14C4, (C2×C28).131(C2×C4), (C2×C4○D28).21C2, (C2×C4×D7).303C22, (C2×C14).25(C22×C4), (C22×C14).78(C2×C4), (C2×Dic7).72(C2×C4), (C22×D7).46(C2×C4), (C2×C4).825(C22×D7), SmallGroup(448,1197)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D28.C4
G = < a,b,c,d | a2=b28=c2=1, d4=b14, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b15, dcd-1=b14c >
Subgroups: 932 in 266 conjugacy classes, 151 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C14, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C22×C8, C2×M4(2), C2×M4(2), C8○D4, C2×C4○D4, C7⋊C8, C56, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C2×C8○D4, C8×D7, C8⋊D7, C2×C7⋊C8, C2×C7⋊C8, C2×C56, C7×M4(2), C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C2×C7⋊D4, C22×C28, D7×C2×C8, C2×C8⋊D7, D28.C4, C22×C7⋊C8, C14×M4(2), C2×C4○D28, C2×D28.C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C24, D14, C8○D4, C23×C4, C4×D7, C22×D7, C2×C8○D4, C2×C4×D7, C23×D7, D28.C4, D7×C22×C4, C2×D28.C4
►On 224 points
Generators in S224
(1 172)(2 173)(3 174)(4 175)(5 176)(6 177)(7 178)(8 179)(9 180)(10 181)(11 182)(12 183)(13 184)(14 185)(15 186)(16 187)(17 188)(18 189)(19 190)(20 191)(21 192)(22 193)(23 194)(24 195)(25 196)(26 169)(27 170)(28 171)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 97)(37 98)(38 99)(39 100)(40 101)(41 102)(42 103)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 85)(53 86)(54 87)(55 88)(56 89)(57 128)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 136)(66 137)(67 138)(68 139)(69 140)(70 113)(71 114)(72 115)(73 116)(74 117)(75 118)(76 119)(77 120)(78 121)(79 122)(80 123)(81 124)(82 125)(83 126)(84 127)(141 216)(142 217)(143 218)(144 219)(145 220)(146 221)(147 222)(148 223)(149 224)(150 197)(151 198)(152 199)(153 200)(154 201)(155 202)(156 203)(157 204)(158 205)(159 206)(160 207)(161 208)(162 209)(163 210)(164 211)(165 212)(166 213)(167 214)(168 215)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(50 56)(51 55)(52 54)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(80 84)(81 83)(85 87)(88 112)(89 111)(90 110)(91 109)(92 108)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(113 137)(114 136)(115 135)(116 134)(117 133)(118 132)(119 131)(120 130)(121 129)(122 128)(123 127)(124 126)(138 140)(141 147)(142 146)(143 145)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)(169 181)(170 180)(171 179)(172 178)(173 177)(174 176)(182 196)(183 195)(184 194)(185 193)(186 192)(187 191)(188 190)(197 213)(198 212)(199 211)(200 210)(201 209)(202 208)(203 207)(204 206)(214 224)(215 223)(216 222)(217 221)(218 220)
(1 209 79 104 15 223 65 90)(2 224 80 91 16 210 66 105)(3 211 81 106 17 197 67 92)(4 198 82 93 18 212 68 107)(5 213 83 108 19 199 69 94)(6 200 84 95 20 214 70 109)(7 215 57 110 21 201 71 96)(8 202 58 97 22 216 72 111)(9 217 59 112 23 203 73 98)(10 204 60 99 24 218 74 85)(11 219 61 86 25 205 75 100)(12 206 62 101 26 220 76 87)(13 221 63 88 27 207 77 102)(14 208 64 103 28 222 78 89)(29 172 162 122 43 186 148 136)(30 187 163 137 44 173 149 123)(31 174 164 124 45 188 150 138)(32 189 165 139 46 175 151 125)(33 176 166 126 47 190 152 140)(34 191 167 113 48 177 153 127)(35 178 168 128 49 192 154 114)(36 193 141 115 50 179 155 129)(37 180 142 130 51 194 156 116)(38 195 143 117 52 181 157 131)(39 182 144 132 53 196 158 118)(40 169 145 119 54 183 159 133)(41 184 146 134 55 170 160 120)(42 171 147 121 56 185 161 135)
G:=sub<Sym(224)| (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,181)(11,182)(12,183)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,169)(27,170)(28,171)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,85)(53,86)(54,87)(55,88)(56,89)(57,128)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,197)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209)(163,210)(164,211)(165,212)(166,213)(167,214)(168,215), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(50,56)(51,55)(52,54)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(80,84)(81,83)(85,87)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(113,137)(114,136)(115,135)(116,134)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)(138,140)(141,147)(142,146)(143,145)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(169,181)(170,180)(171,179)(172,178)(173,177)(174,176)(182,196)(183,195)(184,194)(185,193)(186,192)(187,191)(188,190)(197,213)(198,212)(199,211)(200,210)(201,209)(202,208)(203,207)(204,206)(214,224)(215,223)(216,222)(217,221)(218,220), (1,209,79,104,15,223,65,90)(2,224,80,91,16,210,66,105)(3,211,81,106,17,197,67,92)(4,198,82,93,18,212,68,107)(5,213,83,108,19,199,69,94)(6,200,84,95,20,214,70,109)(7,215,57,110,21,201,71,96)(8,202,58,97,22,216,72,111)(9,217,59,112,23,203,73,98)(10,204,60,99,24,218,74,85)(11,219,61,86,25,205,75,100)(12,206,62,101,26,220,76,87)(13,221,63,88,27,207,77,102)(14,208,64,103,28,222,78,89)(29,172,162,122,43,186,148,136)(30,187,163,137,44,173,149,123)(31,174,164,124,45,188,150,138)(32,189,165,139,46,175,151,125)(33,176,166,126,47,190,152,140)(34,191,167,113,48,177,153,127)(35,178,168,128,49,192,154,114)(36,193,141,115,50,179,155,129)(37,180,142,130,51,194,156,116)(38,195,143,117,52,181,157,131)(39,182,144,132,53,196,158,118)(40,169,145,119,54,183,159,133)(41,184,146,134,55,170,160,120)(42,171,147,121,56,185,161,135)>;
G:=Group( (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,181)(11,182)(12,183)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,169)(27,170)(28,171)(29,90)(30,91)(31,92)(32,93)(33,94)(34,95)(35,96)(36,97)(37,98)(38,99)(39,100)(40,101)(41,102)(42,103)(43,104)(44,105)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,85)(53,86)(54,87)(55,88)(56,89)(57,128)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,136)(66,137)(67,138)(68,139)(69,140)(70,113)(71,114)(72,115)(73,116)(74,117)(75,118)(76,119)(77,120)(78,121)(79,122)(80,123)(81,124)(82,125)(83,126)(84,127)(141,216)(142,217)(143,218)(144,219)(145,220)(146,221)(147,222)(148,223)(149,224)(150,197)(151,198)(152,199)(153,200)(154,201)(155,202)(156,203)(157,204)(158,205)(159,206)(160,207)(161,208)(162,209)(163,210)(164,211)(165,212)(166,213)(167,214)(168,215), (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), (1,7)(2,6)(3,5)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(15,21)(16,20)(17,19)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,40)(50,56)(51,55)(52,54)(57,79)(58,78)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(80,84)(81,83)(85,87)(88,112)(89,111)(90,110)(91,109)(92,108)(93,107)(94,106)(95,105)(96,104)(97,103)(98,102)(99,101)(113,137)(114,136)(115,135)(116,134)(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)(138,140)(141,147)(142,146)(143,145)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(169,181)(170,180)(171,179)(172,178)(173,177)(174,176)(182,196)(183,195)(184,194)(185,193)(186,192)(187,191)(188,190)(197,213)(198,212)(199,211)(200,210)(201,209)(202,208)(203,207)(204,206)(214,224)(215,223)(216,222)(217,221)(218,220), (1,209,79,104,15,223,65,90)(2,224,80,91,16,210,66,105)(3,211,81,106,17,197,67,92)(4,198,82,93,18,212,68,107)(5,213,83,108,19,199,69,94)(6,200,84,95,20,214,70,109)(7,215,57,110,21,201,71,96)(8,202,58,97,22,216,72,111)(9,217,59,112,23,203,73,98)(10,204,60,99,24,218,74,85)(11,219,61,86,25,205,75,100)(12,206,62,101,26,220,76,87)(13,221,63,88,27,207,77,102)(14,208,64,103,28,222,78,89)(29,172,162,122,43,186,148,136)(30,187,163,137,44,173,149,123)(31,174,164,124,45,188,150,138)(32,189,165,139,46,175,151,125)(33,176,166,126,47,190,152,140)(34,191,167,113,48,177,153,127)(35,178,168,128,49,192,154,114)(36,193,141,115,50,179,155,129)(37,180,142,130,51,194,156,116)(38,195,143,117,52,181,157,131)(39,182,144,132,53,196,158,118)(40,169,145,119,54,183,159,133)(41,184,146,134,55,170,160,120)(42,171,147,121,56,185,161,135) );
G=PermutationGroup([[(1,172),(2,173),(3,174),(4,175),(5,176),(6,177),(7,178),(8,179),(9,180),(10,181),(11,182),(12,183),(13,184),(14,185),(15,186),(16,187),(17,188),(18,189),(19,190),(20,191),(21,192),(22,193),(23,194),(24,195),(25,196),(26,169),(27,170),(28,171),(29,90),(30,91),(31,92),(32,93),(33,94),(34,95),(35,96),(36,97),(37,98),(38,99),(39,100),(40,101),(41,102),(42,103),(43,104),(44,105),(45,106),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,85),(53,86),(54,87),(55,88),(56,89),(57,128),(58,129),(59,130),(60,131),(61,132),(62,133),(63,134),(64,135),(65,136),(66,137),(67,138),(68,139),(69,140),(70,113),(71,114),(72,115),(73,116),(74,117),(75,118),(76,119),(77,120),(78,121),(79,122),(80,123),(81,124),(82,125),(83,126),(84,127),(141,216),(142,217),(143,218),(144,219),(145,220),(146,221),(147,222),(148,223),(149,224),(150,197),(151,198),(152,199),(153,200),(154,201),(155,202),(156,203),(157,204),(158,205),(159,206),(160,207),(161,208),(162,209),(163,210),(164,211),(165,212),(166,213),(167,214),(168,215)], [(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)], [(1,7),(2,6),(3,5),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(15,21),(16,20),(17,19),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,40),(50,56),(51,55),(52,54),(57,79),(58,78),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(80,84),(81,83),(85,87),(88,112),(89,111),(90,110),(91,109),(92,108),(93,107),(94,106),(95,105),(96,104),(97,103),(98,102),(99,101),(113,137),(114,136),(115,135),(116,134),(117,133),(118,132),(119,131),(120,130),(121,129),(122,128),(123,127),(124,126),(138,140),(141,147),(142,146),(143,145),(148,168),(149,167),(150,166),(151,165),(152,164),(153,163),(154,162),(155,161),(156,160),(157,159),(169,181),(170,180),(171,179),(172,178),(173,177),(174,176),(182,196),(183,195),(184,194),(185,193),(186,192),(187,191),(188,190),(197,213),(198,212),(199,211),(200,210),(201,209),(202,208),(203,207),(204,206),(214,224),(215,223),(216,222),(217,221),(218,220)], [(1,209,79,104,15,223,65,90),(2,224,80,91,16,210,66,105),(3,211,81,106,17,197,67,92),(4,198,82,93,18,212,68,107),(5,213,83,108,19,199,69,94),(6,200,84,95,20,214,70,109),(7,215,57,110,21,201,71,96),(8,202,58,97,22,216,72,111),(9,217,59,112,23,203,73,98),(10,204,60,99,24,218,74,85),(11,219,61,86,25,205,75,100),(12,206,62,101,26,220,76,87),(13,221,63,88,27,207,77,102),(14,208,64,103,28,222,78,89),(29,172,162,122,43,186,148,136),(30,187,163,137,44,173,149,123),(31,174,164,124,45,188,150,138),(32,189,165,139,46,175,151,125),(33,176,166,126,47,190,152,140),(34,191,167,113,48,177,153,127),(35,178,168,128,49,192,154,114),(36,193,141,115,50,179,155,129),(37,180,142,130,51,194,156,116),(38,195,143,117,52,181,157,131),(39,182,144,132,53,196,158,118),(40,169,145,119,54,183,159,133),(41,184,146,134,55,170,160,120),(42,171,147,121,56,185,161,135)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 8Q | 8R | 8S | 8T | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D7 | D14 | D14 | D14 | C8○D4 | C4×D7 | C4×D7 | D28.C4 |
| kernel | C2×D28.C4 | D7×C2×C8 | C2×C8⋊D7 | D28.C4 | C22×C7⋊C8 | C14×M4(2) | C2×C4○D28 | C2×Dic14 | C2×D28 | C4○D28 | C2×C7⋊D4 | C2×M4(2) | C2×C8 | M4(2) | C22×C4 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 3 | 6 | 12 | 3 | 8 | 18 | 6 | 12 |
Matrix representation of C2×D28.C4 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 15 | 89 | 0 | 0 |
| 48 | 89 | 0 | 0 |
| 0 | 0 | 85 | 18 |
| 0 | 0 | 38 | 28 |
| 79 | 1 | 0 | 0 |
| 88 | 34 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 22 | 1 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 102 | 112 |
| 0 | 0 | 106 | 11 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[15,48,0,0,89,89,0,0,0,0,85,38,0,0,18,28],[79,88,0,0,1,34,0,0,0,0,112,22,0,0,0,1],[15,0,0,0,0,15,0,0,0,0,102,106,0,0,112,11] >;
C2×D28.C4 in GAP, Magma, Sage, TeX
C_2\times D_{28}.C_4 % in TeX
G:=Group("C2xD28.C4"); // GroupNames label
G:=SmallGroup(448,1197);
// by ID
G=gap.SmallGroup(448,1197);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,297,80,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^28=c^2=1,d^4=b^14,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^15,d*c*d^-1=b^14*c>;
// generators/relations