direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C8○D4, C56.80C23, C28.93C24, C4○D4.4C28, D4.8(C2×C28), Q8.9(C2×C28), (C2×C56)⋊54C22, (C22×C56)⋊27C2, (C22×C8)⋊13C14, (D4×C14).24C4, (C2×D4).12C28, (Q8×C14).20C4, (C2×Q8).10C28, C8.17(C22×C14), C4.22(C22×C28), C2.11(C23×C28), C4.17(C23×C14), C14.63(C23×C4), C23.20(C2×C28), (C2×M4(2))⋊17C14, M4(2)⋊11(C2×C14), (C14×M4(2))⋊35C2, C28.167(C22×C4), (C2×C28).969C23, C22.4(C22×C28), (C7×M4(2))⋊40C22, (C22×C28).600C22, (C2×C8)⋊16(C2×C14), (C7×C4○D4).8C4, (C2×C4).53(C2×C28), (C7×D4).30(C2×C4), (C7×Q8).33(C2×C4), (C2×C28).274(C2×C4), (C14×C4○D4).28C2, (C2×C4○D4).14C14, C4○D4.14(C2×C14), (C2×C14).35(C22×C4), (C22×C14).86(C2×C4), (C7×C4○D4).59C22, (C22×C4).127(C2×C14), (C2×C4).139(C22×C14), SmallGroup(448,1350)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 290 in 266 conjugacy classes, 242 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C7, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C14, C14 [×2], C14 [×6], C2×C8, C2×C8 [×15], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C28 [×2], C28 [×6], C2×C14, C2×C14 [×6], C2×C14 [×6], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C56 [×8], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×C14 [×3], C2×C8○D4, C2×C56, C2×C56 [×15], C7×M4(2) [×12], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C22×C56 [×3], C14×M4(2) [×3], C7×C8○D4 [×8], C14×C4○D4, C14×C8○D4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C7, C2×C4 [×28], C23 [×15], C14 [×15], C22×C4 [×14], C24, C28 [×8], C2×C14 [×35], C8○D4 [×2], C23×C4, C2×C28 [×28], C22×C14 [×15], C2×C8○D4, C22×C28 [×14], C23×C14, C7×C8○D4 [×2], C23×C28, C14×C8○D4
Generators and relations
G = < a,b,c,d | a14=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
►On 224 points
Generators in S224
(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 15 91 146 131 204 182 82)(2 16 92 147 132 205 169 83)(3 17 93 148 133 206 170 84)(4 18 94 149 134 207 171 71)(5 19 95 150 135 208 172 72)(6 20 96 151 136 209 173 73)(7 21 97 152 137 210 174 74)(8 22 98 153 138 197 175 75)(9 23 85 154 139 198 176 76)(10 24 86 141 140 199 177 77)(11 25 87 142 127 200 178 78)(12 26 88 143 128 201 179 79)(13 27 89 144 129 202 180 80)(14 28 90 145 130 203 181 81)(29 192 54 222 158 60 124 101)(30 193 55 223 159 61 125 102)(31 194 56 224 160 62 126 103)(32 195 43 211 161 63 113 104)(33 196 44 212 162 64 114 105)(34 183 45 213 163 65 115 106)(35 184 46 214 164 66 116 107)(36 185 47 215 165 67 117 108)(37 186 48 216 166 68 118 109)(38 187 49 217 167 69 119 110)(39 188 50 218 168 70 120 111)(40 189 51 219 155 57 121 112)(41 190 52 220 156 58 122 99)(42 191 53 221 157 59 123 100)
(1 156 131 41)(2 157 132 42)(3 158 133 29)(4 159 134 30)(5 160 135 31)(6 161 136 32)(7 162 137 33)(8 163 138 34)(9 164 139 35)(10 165 140 36)(11 166 127 37)(12 167 128 38)(13 168 129 39)(14 155 130 40)(15 58 204 190)(16 59 205 191)(17 60 206 192)(18 61 207 193)(19 62 208 194)(20 63 209 195)(21 64 210 196)(22 65 197 183)(23 66 198 184)(24 67 199 185)(25 68 200 186)(26 69 201 187)(27 70 202 188)(28 57 203 189)(43 96 113 173)(44 97 114 174)(45 98 115 175)(46 85 116 176)(47 86 117 177)(48 87 118 178)(49 88 119 179)(50 89 120 180)(51 90 121 181)(52 91 122 182)(53 92 123 169)(54 93 124 170)(55 94 125 171)(56 95 126 172)(71 223 149 102)(72 224 150 103)(73 211 151 104)(74 212 152 105)(75 213 153 106)(76 214 154 107)(77 215 141 108)(78 216 142 109)(79 217 143 110)(80 218 144 111)(81 219 145 112)(82 220 146 99)(83 221 147 100)(84 222 148 101)
(1 138)(2 139)(3 140)(4 127)(5 128)(6 129)(7 130)(8 131)(9 132)(10 133)(11 134)(12 135)(13 136)(14 137)(15 197)(16 198)(17 199)(18 200)(19 201)(20 202)(21 203)(22 204)(23 205)(24 206)(25 207)(26 208)(27 209)(28 210)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(83 154)(84 141)(85 169)(86 170)(87 171)(88 172)(89 173)(90 174)(91 175)(92 176)(93 177)(94 178)(95 179)(96 180)(97 181)(98 182)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)(161 168)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
G:=sub<Sym(224)| (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), (1,15,91,146,131,204,182,82)(2,16,92,147,132,205,169,83)(3,17,93,148,133,206,170,84)(4,18,94,149,134,207,171,71)(5,19,95,150,135,208,172,72)(6,20,96,151,136,209,173,73)(7,21,97,152,137,210,174,74)(8,22,98,153,138,197,175,75)(9,23,85,154,139,198,176,76)(10,24,86,141,140,199,177,77)(11,25,87,142,127,200,178,78)(12,26,88,143,128,201,179,79)(13,27,89,144,129,202,180,80)(14,28,90,145,130,203,181,81)(29,192,54,222,158,60,124,101)(30,193,55,223,159,61,125,102)(31,194,56,224,160,62,126,103)(32,195,43,211,161,63,113,104)(33,196,44,212,162,64,114,105)(34,183,45,213,163,65,115,106)(35,184,46,214,164,66,116,107)(36,185,47,215,165,67,117,108)(37,186,48,216,166,68,118,109)(38,187,49,217,167,69,119,110)(39,188,50,218,168,70,120,111)(40,189,51,219,155,57,121,112)(41,190,52,220,156,58,122,99)(42,191,53,221,157,59,123,100), (1,156,131,41)(2,157,132,42)(3,158,133,29)(4,159,134,30)(5,160,135,31)(6,161,136,32)(7,162,137,33)(8,163,138,34)(9,164,139,35)(10,165,140,36)(11,166,127,37)(12,167,128,38)(13,168,129,39)(14,155,130,40)(15,58,204,190)(16,59,205,191)(17,60,206,192)(18,61,207,193)(19,62,208,194)(20,63,209,195)(21,64,210,196)(22,65,197,183)(23,66,198,184)(24,67,199,185)(25,68,200,186)(26,69,201,187)(27,70,202,188)(28,57,203,189)(43,96,113,173)(44,97,114,174)(45,98,115,175)(46,85,116,176)(47,86,117,177)(48,87,118,178)(49,88,119,179)(50,89,120,180)(51,90,121,181)(52,91,122,182)(53,92,123,169)(54,93,124,170)(55,94,125,171)(56,95,126,172)(71,223,149,102)(72,224,150,103)(73,211,151,104)(74,212,152,105)(75,213,153,106)(76,214,154,107)(77,215,141,108)(78,216,142,109)(79,217,143,110)(80,218,144,111)(81,219,145,112)(82,220,146,99)(83,221,147,100)(84,222,148,101), (1,138)(2,139)(3,140)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,197)(16,198)(17,199)(18,200)(19,201)(20,202)(21,203)(22,204)(23,205)(24,206)(25,207)(26,208)(27,209)(28,210)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,141)(85,169)(86,170)(87,171)(88,172)(89,173)(90,174)(91,175)(92,176)(93,177)(94,178)(95,179)(96,180)(97,181)(98,182)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224)>;
G:=Group( (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), (1,15,91,146,131,204,182,82)(2,16,92,147,132,205,169,83)(3,17,93,148,133,206,170,84)(4,18,94,149,134,207,171,71)(5,19,95,150,135,208,172,72)(6,20,96,151,136,209,173,73)(7,21,97,152,137,210,174,74)(8,22,98,153,138,197,175,75)(9,23,85,154,139,198,176,76)(10,24,86,141,140,199,177,77)(11,25,87,142,127,200,178,78)(12,26,88,143,128,201,179,79)(13,27,89,144,129,202,180,80)(14,28,90,145,130,203,181,81)(29,192,54,222,158,60,124,101)(30,193,55,223,159,61,125,102)(31,194,56,224,160,62,126,103)(32,195,43,211,161,63,113,104)(33,196,44,212,162,64,114,105)(34,183,45,213,163,65,115,106)(35,184,46,214,164,66,116,107)(36,185,47,215,165,67,117,108)(37,186,48,216,166,68,118,109)(38,187,49,217,167,69,119,110)(39,188,50,218,168,70,120,111)(40,189,51,219,155,57,121,112)(41,190,52,220,156,58,122,99)(42,191,53,221,157,59,123,100), (1,156,131,41)(2,157,132,42)(3,158,133,29)(4,159,134,30)(5,160,135,31)(6,161,136,32)(7,162,137,33)(8,163,138,34)(9,164,139,35)(10,165,140,36)(11,166,127,37)(12,167,128,38)(13,168,129,39)(14,155,130,40)(15,58,204,190)(16,59,205,191)(17,60,206,192)(18,61,207,193)(19,62,208,194)(20,63,209,195)(21,64,210,196)(22,65,197,183)(23,66,198,184)(24,67,199,185)(25,68,200,186)(26,69,201,187)(27,70,202,188)(28,57,203,189)(43,96,113,173)(44,97,114,174)(45,98,115,175)(46,85,116,176)(47,86,117,177)(48,87,118,178)(49,88,119,179)(50,89,120,180)(51,90,121,181)(52,91,122,182)(53,92,123,169)(54,93,124,170)(55,94,125,171)(56,95,126,172)(71,223,149,102)(72,224,150,103)(73,211,151,104)(74,212,152,105)(75,213,153,106)(76,214,154,107)(77,215,141,108)(78,216,142,109)(79,217,143,110)(80,218,144,111)(81,219,145,112)(82,220,146,99)(83,221,147,100)(84,222,148,101), (1,138)(2,139)(3,140)(4,127)(5,128)(6,129)(7,130)(8,131)(9,132)(10,133)(11,134)(12,135)(13,136)(14,137)(15,197)(16,198)(17,199)(18,200)(19,201)(20,202)(21,203)(22,204)(23,205)(24,206)(25,207)(26,208)(27,209)(28,210)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,141)(85,169)(86,170)(87,171)(88,172)(89,173)(90,174)(91,175)(92,176)(93,177)(94,178)(95,179)(96,180)(97,181)(98,182)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167)(161,168)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224) );
G=PermutationGroup([(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)], [(1,15,91,146,131,204,182,82),(2,16,92,147,132,205,169,83),(3,17,93,148,133,206,170,84),(4,18,94,149,134,207,171,71),(5,19,95,150,135,208,172,72),(6,20,96,151,136,209,173,73),(7,21,97,152,137,210,174,74),(8,22,98,153,138,197,175,75),(9,23,85,154,139,198,176,76),(10,24,86,141,140,199,177,77),(11,25,87,142,127,200,178,78),(12,26,88,143,128,201,179,79),(13,27,89,144,129,202,180,80),(14,28,90,145,130,203,181,81),(29,192,54,222,158,60,124,101),(30,193,55,223,159,61,125,102),(31,194,56,224,160,62,126,103),(32,195,43,211,161,63,113,104),(33,196,44,212,162,64,114,105),(34,183,45,213,163,65,115,106),(35,184,46,214,164,66,116,107),(36,185,47,215,165,67,117,108),(37,186,48,216,166,68,118,109),(38,187,49,217,167,69,119,110),(39,188,50,218,168,70,120,111),(40,189,51,219,155,57,121,112),(41,190,52,220,156,58,122,99),(42,191,53,221,157,59,123,100)], [(1,156,131,41),(2,157,132,42),(3,158,133,29),(4,159,134,30),(5,160,135,31),(6,161,136,32),(7,162,137,33),(8,163,138,34),(9,164,139,35),(10,165,140,36),(11,166,127,37),(12,167,128,38),(13,168,129,39),(14,155,130,40),(15,58,204,190),(16,59,205,191),(17,60,206,192),(18,61,207,193),(19,62,208,194),(20,63,209,195),(21,64,210,196),(22,65,197,183),(23,66,198,184),(24,67,199,185),(25,68,200,186),(26,69,201,187),(27,70,202,188),(28,57,203,189),(43,96,113,173),(44,97,114,174),(45,98,115,175),(46,85,116,176),(47,86,117,177),(48,87,118,178),(49,88,119,179),(50,89,120,180),(51,90,121,181),(52,91,122,182),(53,92,123,169),(54,93,124,170),(55,94,125,171),(56,95,126,172),(71,223,149,102),(72,224,150,103),(73,211,151,104),(74,212,152,105),(75,213,153,106),(76,214,154,107),(77,215,141,108),(78,216,142,109),(79,217,143,110),(80,218,144,111),(81,219,145,112),(82,220,146,99),(83,221,147,100),(84,222,148,101)], [(1,138),(2,139),(3,140),(4,127),(5,128),(6,129),(7,130),(8,131),(9,132),(10,133),(11,134),(12,135),(13,136),(14,137),(15,197),(16,198),(17,199),(18,200),(19,201),(20,202),(21,203),(22,204),(23,205),(24,206),(25,207),(26,208),(27,209),(28,210),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151),(81,152),(82,153),(83,154),(84,141),(85,169),(86,170),(87,171),(88,172),(89,173),(90,174),(91,175),(92,176),(93,177),(94,178),(95,179),(96,180),(97,181),(98,182),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167),(161,168),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)])
Matrix representation ►G ⊆ GL3(𝔽113) generated by
| 112 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 98 | 0 | 0 |
| 0 | 69 | 0 |
| 0 | 0 | 69 |
| 112 | 0 | 0 |
| 0 | 74 | 2 |
| 0 | 30 | 39 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 39 | 112 |
G:=sub<GL(3,GF(113))| [112,0,0,0,7,0,0,0,7],[98,0,0,0,69,0,0,0,69],[112,0,0,0,74,30,0,2,39],[1,0,0,0,1,39,0,0,112] >;
280 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | ··· | 7F | 8A | ··· | 8H | 8I | ··· | 8T | 14A | ··· | 14R | 14S | ··· | 14BB | 28A | ··· | 28X | 28Y | ··· | 28BH | 56A | ··· | 56AV | 56AW | ··· | 56DP |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | C28 | C28 | C8○D4 | C7×C8○D4 |
| kernel | C14×C8○D4 | C22×C56 | C14×M4(2) | C7×C8○D4 | C14×C4○D4 | D4×C14 | Q8×C14 | C7×C4○D4 | C2×C8○D4 | C22×C8 | C2×M4(2) | C8○D4 | C2×C4○D4 | C2×D4 | C2×Q8 | C4○D4 | C14 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 2 | 8 | 6 | 18 | 18 | 48 | 6 | 36 | 12 | 48 | 8 | 48 |
In GAP, Magma, Sage, TeX
C_{14}\times C_8\circ D_4 % in TeX
G:=Group("C14xC8oD4"); // GroupNames label
G:=SmallGroup(448,1350);
// by ID
G=gap.SmallGroup(448,1350);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,2403,124]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀