metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4⋊2Dic7, C28.213(C2×D4), (C2×C28).501D4, D4.6(C2×Dic7), Q8.6(C2×Dic7), (C2×D4).202D14, D4⋊Dic7⋊44C2, Q8⋊Dic7⋊44C2, C28.84(C22×C4), (C2×Q8).171D14, C14.111(C4○D8), C28.98(C22⋊C4), (C2×C28).480C23, (C22×C4).355D14, (C22×C14).112D4, C23.44(C7⋊D4), C7⋊5(C23.24D4), C4.32(C23.D7), C4.14(C22×Dic7), C2.7(D4.8D14), (D4×C14).243C22, C4⋊Dic7.355C22, (Q8×C14).206C22, C22.3(C23.D7), C23.21D14⋊19C2, (C22×C28).206C22, (C7×C4○D4)⋊2C4, (C22×C7⋊C8)⋊8C2, (C2×C4○D4).2D7, C4.95(C2×C7⋊D4), (C14×C4○D4).2C2, (C7×D4).23(C2×C4), (C7×Q8).24(C2×C4), (C2×C28).123(C2×C4), (C2×C14).566(C2×D4), (C2×C7⋊C8).282C22, C14.82(C2×C22⋊C4), (C2×C4).52(C2×Dic7), C22.97(C2×C7⋊D4), C2.18(C2×C23.D7), (C2×C4).281(C7⋊D4), (C2×C4).565(C22×D7), (C2×C14).25(C22⋊C4), SmallGroup(448,767)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C28.(C2×D4)
G = < a,b,c,d,e | a4=d14=1, b2=c2=a2, e2=d7, bab-1=eae-1=a-1, ac=ca, ad=da, bc=cb, bd=db, ebe-1=a-1b, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 468 in 158 conjugacy classes, 71 normal (27 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, C28, C2×C14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C23.24D4, C2×C7⋊C8, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C23.D7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, D4⋊Dic7, Q8⋊Dic7, C22×C7⋊C8, C23.21D14, C14×C4○D4, C28.(C2×D4)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C4○D8, C2×Dic7, C7⋊D4, C22×D7, C23.24D4, C23.D7, C22×Dic7, C2×C7⋊D4, D4.8D14, C2×C23.D7, C28.(C2×D4)
►On 224 points
Generators in S224
(1 206 17 134)(2 207 18 135)(3 208 19 136)(4 209 20 137)(5 210 21 138)(6 197 22 139)(7 198 23 140)(8 199 24 127)(9 200 25 128)(10 201 26 129)(11 202 27 130)(12 203 28 131)(13 204 15 132)(14 205 16 133)(29 88 113 43)(30 89 114 44)(31 90 115 45)(32 91 116 46)(33 92 117 47)(34 93 118 48)(35 94 119 49)(36 95 120 50)(37 96 121 51)(38 97 122 52)(39 98 123 53)(40 85 124 54)(41 86 125 55)(42 87 126 56)(57 158 222 104)(58 159 223 105)(59 160 224 106)(60 161 211 107)(61 162 212 108)(62 163 213 109)(63 164 214 110)(64 165 215 111)(65 166 216 112)(66 167 217 99)(67 168 218 100)(68 155 219 101)(69 156 220 102)(70 157 221 103)(71 182 149 190)(72 169 150 191)(73 170 151 192)(74 171 152 193)(75 172 153 194)(76 173 154 195)(77 174 141 196)(78 175 142 183)(79 176 143 184)(80 177 144 185)(81 178 145 186)(82 179 146 187)(83 180 147 188)(84 181 148 189)
(1 190 17 182)(2 191 18 169)(3 192 19 170)(4 193 20 171)(5 194 21 172)(6 195 22 173)(7 196 23 174)(8 183 24 175)(9 184 25 176)(10 185 26 177)(11 186 27 178)(12 187 28 179)(13 188 15 180)(14 189 16 181)(29 103 113 157)(30 104 114 158)(31 105 115 159)(32 106 116 160)(33 107 117 161)(34 108 118 162)(35 109 119 163)(36 110 120 164)(37 111 121 165)(38 112 122 166)(39 99 123 167)(40 100 124 168)(41 101 125 155)(42 102 126 156)(43 70 88 221)(44 57 89 222)(45 58 90 223)(46 59 91 224)(47 60 92 211)(48 61 93 212)(49 62 94 213)(50 63 95 214)(51 64 96 215)(52 65 97 216)(53 66 98 217)(54 67 85 218)(55 68 86 219)(56 69 87 220)(71 206 149 134)(72 207 150 135)(73 208 151 136)(74 209 152 137)(75 210 153 138)(76 197 154 139)(77 198 141 140)(78 199 142 127)(79 200 143 128)(80 201 144 129)(81 202 145 130)(82 203 146 131)(83 204 147 132)(84 205 148 133)
(1 127 17 199)(2 128 18 200)(3 129 19 201)(4 130 20 202)(5 131 21 203)(6 132 22 204)(7 133 23 205)(8 134 24 206)(9 135 25 207)(10 136 26 208)(11 137 27 209)(12 138 28 210)(13 139 15 197)(14 140 16 198)(29 95 113 50)(30 96 114 51)(31 97 115 52)(32 98 116 53)(33 85 117 54)(34 86 118 55)(35 87 119 56)(36 88 120 43)(37 89 121 44)(38 90 122 45)(39 91 123 46)(40 92 124 47)(41 93 125 48)(42 94 126 49)(57 111 222 165)(58 112 223 166)(59 99 224 167)(60 100 211 168)(61 101 212 155)(62 102 213 156)(63 103 214 157)(64 104 215 158)(65 105 216 159)(66 106 217 160)(67 107 218 161)(68 108 219 162)(69 109 220 163)(70 110 221 164)(71 175 149 183)(72 176 150 184)(73 177 151 185)(74 178 152 186)(75 179 153 187)(76 180 154 188)(77 181 141 189)(78 182 142 190)(79 169 143 191)(80 170 144 192)(81 171 145 193)(82 172 146 194)(83 173 147 195)(84 174 148 196)
(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 42 8 35)(2 41 9 34)(3 40 10 33)(4 39 11 32)(5 38 12 31)(6 37 13 30)(7 36 14 29)(15 114 22 121)(16 113 23 120)(17 126 24 119)(18 125 25 118)(19 124 26 117)(20 123 27 116)(21 122 28 115)(43 198 50 205)(44 197 51 204)(45 210 52 203)(46 209 53 202)(47 208 54 201)(48 207 55 200)(49 206 56 199)(57 195 64 188)(58 194 65 187)(59 193 66 186)(60 192 67 185)(61 191 68 184)(62 190 69 183)(63 189 70 196)(71 102 78 109)(72 101 79 108)(73 100 80 107)(74 99 81 106)(75 112 82 105)(76 111 83 104)(77 110 84 103)(85 129 92 136)(86 128 93 135)(87 127 94 134)(88 140 95 133)(89 139 96 132)(90 138 97 131)(91 137 98 130)(141 164 148 157)(142 163 149 156)(143 162 150 155)(144 161 151 168)(145 160 152 167)(146 159 153 166)(147 158 154 165)(169 219 176 212)(170 218 177 211)(171 217 178 224)(172 216 179 223)(173 215 180 222)(174 214 181 221)(175 213 182 220)
G:=sub<Sym(224)| (1,206,17,134)(2,207,18,135)(3,208,19,136)(4,209,20,137)(5,210,21,138)(6,197,22,139)(7,198,23,140)(8,199,24,127)(9,200,25,128)(10,201,26,129)(11,202,27,130)(12,203,28,131)(13,204,15,132)(14,205,16,133)(29,88,113,43)(30,89,114,44)(31,90,115,45)(32,91,116,46)(33,92,117,47)(34,93,118,48)(35,94,119,49)(36,95,120,50)(37,96,121,51)(38,97,122,52)(39,98,123,53)(40,85,124,54)(41,86,125,55)(42,87,126,56)(57,158,222,104)(58,159,223,105)(59,160,224,106)(60,161,211,107)(61,162,212,108)(62,163,213,109)(63,164,214,110)(64,165,215,111)(65,166,216,112)(66,167,217,99)(67,168,218,100)(68,155,219,101)(69,156,220,102)(70,157,221,103)(71,182,149,190)(72,169,150,191)(73,170,151,192)(74,171,152,193)(75,172,153,194)(76,173,154,195)(77,174,141,196)(78,175,142,183)(79,176,143,184)(80,177,144,185)(81,178,145,186)(82,179,146,187)(83,180,147,188)(84,181,148,189), (1,190,17,182)(2,191,18,169)(3,192,19,170)(4,193,20,171)(5,194,21,172)(6,195,22,173)(7,196,23,174)(8,183,24,175)(9,184,25,176)(10,185,26,177)(11,186,27,178)(12,187,28,179)(13,188,15,180)(14,189,16,181)(29,103,113,157)(30,104,114,158)(31,105,115,159)(32,106,116,160)(33,107,117,161)(34,108,118,162)(35,109,119,163)(36,110,120,164)(37,111,121,165)(38,112,122,166)(39,99,123,167)(40,100,124,168)(41,101,125,155)(42,102,126,156)(43,70,88,221)(44,57,89,222)(45,58,90,223)(46,59,91,224)(47,60,92,211)(48,61,93,212)(49,62,94,213)(50,63,95,214)(51,64,96,215)(52,65,97,216)(53,66,98,217)(54,67,85,218)(55,68,86,219)(56,69,87,220)(71,206,149,134)(72,207,150,135)(73,208,151,136)(74,209,152,137)(75,210,153,138)(76,197,154,139)(77,198,141,140)(78,199,142,127)(79,200,143,128)(80,201,144,129)(81,202,145,130)(82,203,146,131)(83,204,147,132)(84,205,148,133), (1,127,17,199)(2,128,18,200)(3,129,19,201)(4,130,20,202)(5,131,21,203)(6,132,22,204)(7,133,23,205)(8,134,24,206)(9,135,25,207)(10,136,26,208)(11,137,27,209)(12,138,28,210)(13,139,15,197)(14,140,16,198)(29,95,113,50)(30,96,114,51)(31,97,115,52)(32,98,116,53)(33,85,117,54)(34,86,118,55)(35,87,119,56)(36,88,120,43)(37,89,121,44)(38,90,122,45)(39,91,123,46)(40,92,124,47)(41,93,125,48)(42,94,126,49)(57,111,222,165)(58,112,223,166)(59,99,224,167)(60,100,211,168)(61,101,212,155)(62,102,213,156)(63,103,214,157)(64,104,215,158)(65,105,216,159)(66,106,217,160)(67,107,218,161)(68,108,219,162)(69,109,220,163)(70,110,221,164)(71,175,149,183)(72,176,150,184)(73,177,151,185)(74,178,152,186)(75,179,153,187)(76,180,154,188)(77,181,141,189)(78,182,142,190)(79,169,143,191)(80,170,144,192)(81,171,145,193)(82,172,146,194)(83,173,147,195)(84,174,148,196), (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), (1,42,8,35)(2,41,9,34)(3,40,10,33)(4,39,11,32)(5,38,12,31)(6,37,13,30)(7,36,14,29)(15,114,22,121)(16,113,23,120)(17,126,24,119)(18,125,25,118)(19,124,26,117)(20,123,27,116)(21,122,28,115)(43,198,50,205)(44,197,51,204)(45,210,52,203)(46,209,53,202)(47,208,54,201)(48,207,55,200)(49,206,56,199)(57,195,64,188)(58,194,65,187)(59,193,66,186)(60,192,67,185)(61,191,68,184)(62,190,69,183)(63,189,70,196)(71,102,78,109)(72,101,79,108)(73,100,80,107)(74,99,81,106)(75,112,82,105)(76,111,83,104)(77,110,84,103)(85,129,92,136)(86,128,93,135)(87,127,94,134)(88,140,95,133)(89,139,96,132)(90,138,97,131)(91,137,98,130)(141,164,148,157)(142,163,149,156)(143,162,150,155)(144,161,151,168)(145,160,152,167)(146,159,153,166)(147,158,154,165)(169,219,176,212)(170,218,177,211)(171,217,178,224)(172,216,179,223)(173,215,180,222)(174,214,181,221)(175,213,182,220)>;
G:=Group( (1,206,17,134)(2,207,18,135)(3,208,19,136)(4,209,20,137)(5,210,21,138)(6,197,22,139)(7,198,23,140)(8,199,24,127)(9,200,25,128)(10,201,26,129)(11,202,27,130)(12,203,28,131)(13,204,15,132)(14,205,16,133)(29,88,113,43)(30,89,114,44)(31,90,115,45)(32,91,116,46)(33,92,117,47)(34,93,118,48)(35,94,119,49)(36,95,120,50)(37,96,121,51)(38,97,122,52)(39,98,123,53)(40,85,124,54)(41,86,125,55)(42,87,126,56)(57,158,222,104)(58,159,223,105)(59,160,224,106)(60,161,211,107)(61,162,212,108)(62,163,213,109)(63,164,214,110)(64,165,215,111)(65,166,216,112)(66,167,217,99)(67,168,218,100)(68,155,219,101)(69,156,220,102)(70,157,221,103)(71,182,149,190)(72,169,150,191)(73,170,151,192)(74,171,152,193)(75,172,153,194)(76,173,154,195)(77,174,141,196)(78,175,142,183)(79,176,143,184)(80,177,144,185)(81,178,145,186)(82,179,146,187)(83,180,147,188)(84,181,148,189), (1,190,17,182)(2,191,18,169)(3,192,19,170)(4,193,20,171)(5,194,21,172)(6,195,22,173)(7,196,23,174)(8,183,24,175)(9,184,25,176)(10,185,26,177)(11,186,27,178)(12,187,28,179)(13,188,15,180)(14,189,16,181)(29,103,113,157)(30,104,114,158)(31,105,115,159)(32,106,116,160)(33,107,117,161)(34,108,118,162)(35,109,119,163)(36,110,120,164)(37,111,121,165)(38,112,122,166)(39,99,123,167)(40,100,124,168)(41,101,125,155)(42,102,126,156)(43,70,88,221)(44,57,89,222)(45,58,90,223)(46,59,91,224)(47,60,92,211)(48,61,93,212)(49,62,94,213)(50,63,95,214)(51,64,96,215)(52,65,97,216)(53,66,98,217)(54,67,85,218)(55,68,86,219)(56,69,87,220)(71,206,149,134)(72,207,150,135)(73,208,151,136)(74,209,152,137)(75,210,153,138)(76,197,154,139)(77,198,141,140)(78,199,142,127)(79,200,143,128)(80,201,144,129)(81,202,145,130)(82,203,146,131)(83,204,147,132)(84,205,148,133), (1,127,17,199)(2,128,18,200)(3,129,19,201)(4,130,20,202)(5,131,21,203)(6,132,22,204)(7,133,23,205)(8,134,24,206)(9,135,25,207)(10,136,26,208)(11,137,27,209)(12,138,28,210)(13,139,15,197)(14,140,16,198)(29,95,113,50)(30,96,114,51)(31,97,115,52)(32,98,116,53)(33,85,117,54)(34,86,118,55)(35,87,119,56)(36,88,120,43)(37,89,121,44)(38,90,122,45)(39,91,123,46)(40,92,124,47)(41,93,125,48)(42,94,126,49)(57,111,222,165)(58,112,223,166)(59,99,224,167)(60,100,211,168)(61,101,212,155)(62,102,213,156)(63,103,214,157)(64,104,215,158)(65,105,216,159)(66,106,217,160)(67,107,218,161)(68,108,219,162)(69,109,220,163)(70,110,221,164)(71,175,149,183)(72,176,150,184)(73,177,151,185)(74,178,152,186)(75,179,153,187)(76,180,154,188)(77,181,141,189)(78,182,142,190)(79,169,143,191)(80,170,144,192)(81,171,145,193)(82,172,146,194)(83,173,147,195)(84,174,148,196), (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), (1,42,8,35)(2,41,9,34)(3,40,10,33)(4,39,11,32)(5,38,12,31)(6,37,13,30)(7,36,14,29)(15,114,22,121)(16,113,23,120)(17,126,24,119)(18,125,25,118)(19,124,26,117)(20,123,27,116)(21,122,28,115)(43,198,50,205)(44,197,51,204)(45,210,52,203)(46,209,53,202)(47,208,54,201)(48,207,55,200)(49,206,56,199)(57,195,64,188)(58,194,65,187)(59,193,66,186)(60,192,67,185)(61,191,68,184)(62,190,69,183)(63,189,70,196)(71,102,78,109)(72,101,79,108)(73,100,80,107)(74,99,81,106)(75,112,82,105)(76,111,83,104)(77,110,84,103)(85,129,92,136)(86,128,93,135)(87,127,94,134)(88,140,95,133)(89,139,96,132)(90,138,97,131)(91,137,98,130)(141,164,148,157)(142,163,149,156)(143,162,150,155)(144,161,151,168)(145,160,152,167)(146,159,153,166)(147,158,154,165)(169,219,176,212)(170,218,177,211)(171,217,178,224)(172,216,179,223)(173,215,180,222)(174,214,181,221)(175,213,182,220) );
G=PermutationGroup([[(1,206,17,134),(2,207,18,135),(3,208,19,136),(4,209,20,137),(5,210,21,138),(6,197,22,139),(7,198,23,140),(8,199,24,127),(9,200,25,128),(10,201,26,129),(11,202,27,130),(12,203,28,131),(13,204,15,132),(14,205,16,133),(29,88,113,43),(30,89,114,44),(31,90,115,45),(32,91,116,46),(33,92,117,47),(34,93,118,48),(35,94,119,49),(36,95,120,50),(37,96,121,51),(38,97,122,52),(39,98,123,53),(40,85,124,54),(41,86,125,55),(42,87,126,56),(57,158,222,104),(58,159,223,105),(59,160,224,106),(60,161,211,107),(61,162,212,108),(62,163,213,109),(63,164,214,110),(64,165,215,111),(65,166,216,112),(66,167,217,99),(67,168,218,100),(68,155,219,101),(69,156,220,102),(70,157,221,103),(71,182,149,190),(72,169,150,191),(73,170,151,192),(74,171,152,193),(75,172,153,194),(76,173,154,195),(77,174,141,196),(78,175,142,183),(79,176,143,184),(80,177,144,185),(81,178,145,186),(82,179,146,187),(83,180,147,188),(84,181,148,189)], [(1,190,17,182),(2,191,18,169),(3,192,19,170),(4,193,20,171),(5,194,21,172),(6,195,22,173),(7,196,23,174),(8,183,24,175),(9,184,25,176),(10,185,26,177),(11,186,27,178),(12,187,28,179),(13,188,15,180),(14,189,16,181),(29,103,113,157),(30,104,114,158),(31,105,115,159),(32,106,116,160),(33,107,117,161),(34,108,118,162),(35,109,119,163),(36,110,120,164),(37,111,121,165),(38,112,122,166),(39,99,123,167),(40,100,124,168),(41,101,125,155),(42,102,126,156),(43,70,88,221),(44,57,89,222),(45,58,90,223),(46,59,91,224),(47,60,92,211),(48,61,93,212),(49,62,94,213),(50,63,95,214),(51,64,96,215),(52,65,97,216),(53,66,98,217),(54,67,85,218),(55,68,86,219),(56,69,87,220),(71,206,149,134),(72,207,150,135),(73,208,151,136),(74,209,152,137),(75,210,153,138),(76,197,154,139),(77,198,141,140),(78,199,142,127),(79,200,143,128),(80,201,144,129),(81,202,145,130),(82,203,146,131),(83,204,147,132),(84,205,148,133)], [(1,127,17,199),(2,128,18,200),(3,129,19,201),(4,130,20,202),(5,131,21,203),(6,132,22,204),(7,133,23,205),(8,134,24,206),(9,135,25,207),(10,136,26,208),(11,137,27,209),(12,138,28,210),(13,139,15,197),(14,140,16,198),(29,95,113,50),(30,96,114,51),(31,97,115,52),(32,98,116,53),(33,85,117,54),(34,86,118,55),(35,87,119,56),(36,88,120,43),(37,89,121,44),(38,90,122,45),(39,91,123,46),(40,92,124,47),(41,93,125,48),(42,94,126,49),(57,111,222,165),(58,112,223,166),(59,99,224,167),(60,100,211,168),(61,101,212,155),(62,102,213,156),(63,103,214,157),(64,104,215,158),(65,105,216,159),(66,106,217,160),(67,107,218,161),(68,108,219,162),(69,109,220,163),(70,110,221,164),(71,175,149,183),(72,176,150,184),(73,177,151,185),(74,178,152,186),(75,179,153,187),(76,180,154,188),(77,181,141,189),(78,182,142,190),(79,169,143,191),(80,170,144,192),(81,171,145,193),(82,172,146,194),(83,173,147,195),(84,174,148,196)], [(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)], [(1,42,8,35),(2,41,9,34),(3,40,10,33),(4,39,11,32),(5,38,12,31),(6,37,13,30),(7,36,14,29),(15,114,22,121),(16,113,23,120),(17,126,24,119),(18,125,25,118),(19,124,26,117),(20,123,27,116),(21,122,28,115),(43,198,50,205),(44,197,51,204),(45,210,52,203),(46,209,53,202),(47,208,54,201),(48,207,55,200),(49,206,56,199),(57,195,64,188),(58,194,65,187),(59,193,66,186),(60,192,67,185),(61,191,68,184),(62,190,69,183),(63,189,70,196),(71,102,78,109),(72,101,79,108),(73,100,80,107),(74,99,81,106),(75,112,82,105),(76,111,83,104),(77,110,84,103),(85,129,92,136),(86,128,93,135),(87,127,94,134),(88,140,95,133),(89,139,96,132),(90,138,97,131),(91,137,98,130),(141,164,148,157),(142,163,149,156),(143,162,150,155),(144,161,151,168),(145,160,152,167),(146,159,153,166),(147,158,154,165),(169,219,176,212),(170,218,177,211),(171,217,178,224),(172,216,179,223),(173,215,180,222),(174,214,181,221),(175,213,182,220)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | D14 | D14 | Dic7 | C4○D8 | C7⋊D4 | C7⋊D4 | D4.8D14 |
| kernel | C28.(C2×D4) | D4⋊Dic7 | Q8⋊Dic7 | C22×C7⋊C8 | C23.21D14 | C14×C4○D4 | C7×C4○D4 | C2×C28 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 3 | 3 | 3 | 12 | 8 | 18 | 6 | 12 |
Matrix representation of C28.(C2×D4) ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 65 | 15 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 65 | 30 |
| 0 | 0 | 55 | 48 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 1 | 0 | 0 |
| 112 | 89 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 105 | 104 | 0 | 0 |
| 70 | 8 | 0 | 0 |
| 0 | 0 | 73 | 25 |
| 0 | 0 | 58 | 40 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,98,65,0,0,0,15],[112,0,0,0,0,112,0,0,0,0,65,55,0,0,30,48],[112,0,0,0,0,112,0,0,0,0,98,0,0,0,0,98],[0,112,0,0,1,89,0,0,0,0,112,0,0,0,0,112],[105,70,0,0,104,8,0,0,0,0,73,58,0,0,25,40] >;
C28.(C2×D4) in GAP, Magma, Sage, TeX
C_{28}.(C_2\times D_4) % in TeX
G:=Group("C28.(C2xD4)"); // GroupNames label
G:=SmallGroup(448,767);
// by ID
G=gap.SmallGroup(448,767);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,254,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=d^14=1,b^2=c^2=a^2,e^2=d^7,b*a*b^-1=e*a*e^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,e*b*e^-1=a^-1*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations