metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C28.40C24, D28.35C23, 2- 1+4⋊3D7, Dic14.35C23, C7⋊5(Q8○D8), (C7×D4).39D4, C7⋊C8.19C23, (C7×Q8).39D4, C4○D4.17D14, C28.272(C2×D4), D4⋊D7.2C22, (C2×Q8).93D14, C4.40(C23×D7), Q8⋊D7.3C22, D4.21(C7⋊D4), Q8.Dic7⋊13C2, Q8.21(C7⋊D4), (C7×D4).28C23, D4.28(C22×D7), D4.D7.3C22, D4.8D14⋊11C2, D4.9D14⋊12C2, (C7×Q8).28C23, Q8.28(C22×D7), C28.C23⋊12C2, C7⋊Q16.4C22, (C2×C28).121C23, C4○D28.34C22, C14.174(C22×D4), (C7×2- 1+4)⋊3C2, D4.10D14⋊10C2, (Q8×C14).154C22, C4.Dic7.32C22, (C2×Dic14).205C22, C4.78(C2×C7⋊D4), (C2×C7⋊Q16)⋊32C2, (C2×C14).88(C2×D4), C22.9(C2×C7⋊D4), (C2×C7⋊C8).185C22, C2.47(C22×C7⋊D4), (C7×C4○D4).30C22, (C2×C4).105(C22×D7), SmallGroup(448,1291)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D28.35C23
G = < a,b,c,d,e | a28=b2=e2=1, c2=d2=a14, bab=a-1, ac=ca, ad=da, eae=a15, bc=cb, bd=db, ebe=a7b, dcd-1=a14c, ce=ec, de=ed >
Subgroups: 916 in 248 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, D7, C14, C14, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, Dic7, C28, C28, C28, D14, C2×C14, C2×C14, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, 2- 1+4, C7⋊C8, C7⋊C8, Dic14, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×D4, C7×Q8, C7×Q8, C7×Q8, Q8○D8, C2×C7⋊C8, C4.Dic7, D4⋊D7, D4.D7, Q8⋊D7, C7⋊Q16, C2×Dic14, C4○D28, D4⋊2D7, Q8×D7, Q8×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×C4○D4, C28.C23, C2×C7⋊Q16, Q8.Dic7, D4.8D14, D4.9D14, D4.10D14, C7×2- 1+4, D28.35C23
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, C7⋊D4, C22×D7, Q8○D8, C2×C7⋊D4, C23×D7, C22×C7⋊D4, D28.35C23
►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)
(1 130)(2 129)(3 128)(4 127)(5 126)(6 125)(7 124)(8 123)(9 122)(10 121)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 140)(20 139)(21 138)(22 137)(23 136)(24 135)(25 134)(26 133)(27 132)(28 131)(29 160)(30 159)(31 158)(32 157)(33 156)(34 155)(35 154)(36 153)(37 152)(38 151)(39 150)(40 149)(41 148)(42 147)(43 146)(44 145)(45 144)(46 143)(47 142)(48 141)(49 168)(50 167)(51 166)(52 165)(53 164)(54 163)(55 162)(56 161)(57 176)(58 175)(59 174)(60 173)(61 172)(62 171)(63 170)(64 169)(65 196)(66 195)(67 194)(68 193)(69 192)(70 191)(71 190)(72 189)(73 188)(74 187)(75 186)(76 185)(77 184)(78 183)(79 182)(80 181)(81 180)(82 179)(83 178)(84 177)(85 201)(86 200)(87 199)(88 198)(89 197)(90 224)(91 223)(92 222)(93 221)(94 220)(95 219)(96 218)(97 217)(98 216)(99 215)(100 214)(101 213)(102 212)(103 211)(104 210)(105 209)(106 208)(107 207)(108 206)(109 205)(110 204)(111 203)(112 202)
(1 81 15 67)(2 82 16 68)(3 83 17 69)(4 84 18 70)(5 57 19 71)(6 58 20 72)(7 59 21 73)(8 60 22 74)(9 61 23 75)(10 62 24 76)(11 63 25 77)(12 64 26 78)(13 65 27 79)(14 66 28 80)(29 105 43 91)(30 106 44 92)(31 107 45 93)(32 108 46 94)(33 109 47 95)(34 110 48 96)(35 111 49 97)(36 112 50 98)(37 85 51 99)(38 86 52 100)(39 87 53 101)(40 88 54 102)(41 89 55 103)(42 90 56 104)(113 191 127 177)(114 192 128 178)(115 193 129 179)(116 194 130 180)(117 195 131 181)(118 196 132 182)(119 169 133 183)(120 170 134 184)(121 171 135 185)(122 172 136 186)(123 173 137 187)(124 174 138 188)(125 175 139 189)(126 176 140 190)(141 218 155 204)(142 219 156 205)(143 220 157 206)(144 221 158 207)(145 222 159 208)(146 223 160 209)(147 224 161 210)(148 197 162 211)(149 198 163 212)(150 199 164 213)(151 200 165 214)(152 201 166 215)(153 202 167 216)(154 203 168 217)
(1 8 15 22)(2 9 16 23)(3 10 17 24)(4 11 18 25)(5 12 19 26)(6 13 20 27)(7 14 21 28)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)(57 78 71 64)(58 79 72 65)(59 80 73 66)(60 81 74 67)(61 82 75 68)(62 83 76 69)(63 84 77 70)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)(113 134 127 120)(114 135 128 121)(115 136 129 122)(116 137 130 123)(117 138 131 124)(118 139 132 125)(119 140 133 126)(141 148 155 162)(142 149 156 163)(143 150 157 164)(144 151 158 165)(145 152 159 166)(146 153 160 167)(147 154 161 168)(169 176 183 190)(170 177 184 191)(171 178 185 192)(172 179 186 193)(173 180 187 194)(174 181 188 195)(175 182 189 196)(197 218 211 204)(198 219 212 205)(199 220 213 206)(200 221 214 207)(201 222 215 208)(202 223 216 209)(203 224 217 210)
(1 41)(2 56)(3 43)(4 30)(5 45)(6 32)(7 47)(8 34)(9 49)(10 36)(11 51)(12 38)(13 53)(14 40)(15 55)(16 42)(17 29)(18 44)(19 31)(20 46)(21 33)(22 48)(23 35)(24 50)(25 37)(26 52)(27 39)(28 54)(57 93)(58 108)(59 95)(60 110)(61 97)(62 112)(63 99)(64 86)(65 101)(66 88)(67 103)(68 90)(69 105)(70 92)(71 107)(72 94)(73 109)(74 96)(75 111)(76 98)(77 85)(78 100)(79 87)(80 102)(81 89)(82 104)(83 91)(84 106)(113 166)(114 153)(115 168)(116 155)(117 142)(118 157)(119 144)(120 159)(121 146)(122 161)(123 148)(124 163)(125 150)(126 165)(127 152)(128 167)(129 154)(130 141)(131 156)(132 143)(133 158)(134 145)(135 160)(136 147)(137 162)(138 149)(139 164)(140 151)(169 221)(170 208)(171 223)(172 210)(173 197)(174 212)(175 199)(176 214)(177 201)(178 216)(179 203)(180 218)(181 205)(182 220)(183 207)(184 222)(185 209)(186 224)(187 211)(188 198)(189 213)(190 200)(191 215)(192 202)(193 217)(194 204)(195 219)(196 206)
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), (1,130)(2,129)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,140)(20,139)(21,138)(22,137)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,147)(43,146)(44,145)(45,144)(46,143)(47,142)(48,141)(49,168)(50,167)(51,166)(52,165)(53,164)(54,163)(55,162)(56,161)(57,176)(58,175)(59,174)(60,173)(61,172)(62,171)(63,170)(64,169)(65,196)(66,195)(67,194)(68,193)(69,192)(70,191)(71,190)(72,189)(73,188)(74,187)(75,186)(76,185)(77,184)(78,183)(79,182)(80,181)(81,180)(82,179)(83,178)(84,177)(85,201)(86,200)(87,199)(88,198)(89,197)(90,224)(91,223)(92,222)(93,221)(94,220)(95,219)(96,218)(97,217)(98,216)(99,215)(100,214)(101,213)(102,212)(103,211)(104,210)(105,209)(106,208)(107,207)(108,206)(109,205)(110,204)(111,203)(112,202), (1,81,15,67)(2,82,16,68)(3,83,17,69)(4,84,18,70)(5,57,19,71)(6,58,20,72)(7,59,21,73)(8,60,22,74)(9,61,23,75)(10,62,24,76)(11,63,25,77)(12,64,26,78)(13,65,27,79)(14,66,28,80)(29,105,43,91)(30,106,44,92)(31,107,45,93)(32,108,46,94)(33,109,47,95)(34,110,48,96)(35,111,49,97)(36,112,50,98)(37,85,51,99)(38,86,52,100)(39,87,53,101)(40,88,54,102)(41,89,55,103)(42,90,56,104)(113,191,127,177)(114,192,128,178)(115,193,129,179)(116,194,130,180)(117,195,131,181)(118,196,132,182)(119,169,133,183)(120,170,134,184)(121,171,135,185)(122,172,136,186)(123,173,137,187)(124,174,138,188)(125,175,139,189)(126,176,140,190)(141,218,155,204)(142,219,156,205)(143,220,157,206)(144,221,158,207)(145,222,159,208)(146,223,160,209)(147,224,161,210)(148,197,162,211)(149,198,163,212)(150,199,164,213)(151,200,165,214)(152,201,166,215)(153,202,167,216)(154,203,168,217), (1,8,15,22)(2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,78,71,64)(58,79,72,65)(59,80,73,66)(60,81,74,67)(61,82,75,68)(62,83,76,69)(63,84,77,70)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112)(113,134,127,120)(114,135,128,121)(115,136,129,122)(116,137,130,123)(117,138,131,124)(118,139,132,125)(119,140,133,126)(141,148,155,162)(142,149,156,163)(143,150,157,164)(144,151,158,165)(145,152,159,166)(146,153,160,167)(147,154,161,168)(169,176,183,190)(170,177,184,191)(171,178,185,192)(172,179,186,193)(173,180,187,194)(174,181,188,195)(175,182,189,196)(197,218,211,204)(198,219,212,205)(199,220,213,206)(200,221,214,207)(201,222,215,208)(202,223,216,209)(203,224,217,210), (1,41)(2,56)(3,43)(4,30)(5,45)(6,32)(7,47)(8,34)(9,49)(10,36)(11,51)(12,38)(13,53)(14,40)(15,55)(16,42)(17,29)(18,44)(19,31)(20,46)(21,33)(22,48)(23,35)(24,50)(25,37)(26,52)(27,39)(28,54)(57,93)(58,108)(59,95)(60,110)(61,97)(62,112)(63,99)(64,86)(65,101)(66,88)(67,103)(68,90)(69,105)(70,92)(71,107)(72,94)(73,109)(74,96)(75,111)(76,98)(77,85)(78,100)(79,87)(80,102)(81,89)(82,104)(83,91)(84,106)(113,166)(114,153)(115,168)(116,155)(117,142)(118,157)(119,144)(120,159)(121,146)(122,161)(123,148)(124,163)(125,150)(126,165)(127,152)(128,167)(129,154)(130,141)(131,156)(132,143)(133,158)(134,145)(135,160)(136,147)(137,162)(138,149)(139,164)(140,151)(169,221)(170,208)(171,223)(172,210)(173,197)(174,212)(175,199)(176,214)(177,201)(178,216)(179,203)(180,218)(181,205)(182,220)(183,207)(184,222)(185,209)(186,224)(187,211)(188,198)(189,213)(190,200)(191,215)(192,202)(193,217)(194,204)(195,219)(196,206)>;
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), (1,130)(2,129)(3,128)(4,127)(5,126)(6,125)(7,124)(8,123)(9,122)(10,121)(11,120)(12,119)(13,118)(14,117)(15,116)(16,115)(17,114)(18,113)(19,140)(20,139)(21,138)(22,137)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,147)(43,146)(44,145)(45,144)(46,143)(47,142)(48,141)(49,168)(50,167)(51,166)(52,165)(53,164)(54,163)(55,162)(56,161)(57,176)(58,175)(59,174)(60,173)(61,172)(62,171)(63,170)(64,169)(65,196)(66,195)(67,194)(68,193)(69,192)(70,191)(71,190)(72,189)(73,188)(74,187)(75,186)(76,185)(77,184)(78,183)(79,182)(80,181)(81,180)(82,179)(83,178)(84,177)(85,201)(86,200)(87,199)(88,198)(89,197)(90,224)(91,223)(92,222)(93,221)(94,220)(95,219)(96,218)(97,217)(98,216)(99,215)(100,214)(101,213)(102,212)(103,211)(104,210)(105,209)(106,208)(107,207)(108,206)(109,205)(110,204)(111,203)(112,202), (1,81,15,67)(2,82,16,68)(3,83,17,69)(4,84,18,70)(5,57,19,71)(6,58,20,72)(7,59,21,73)(8,60,22,74)(9,61,23,75)(10,62,24,76)(11,63,25,77)(12,64,26,78)(13,65,27,79)(14,66,28,80)(29,105,43,91)(30,106,44,92)(31,107,45,93)(32,108,46,94)(33,109,47,95)(34,110,48,96)(35,111,49,97)(36,112,50,98)(37,85,51,99)(38,86,52,100)(39,87,53,101)(40,88,54,102)(41,89,55,103)(42,90,56,104)(113,191,127,177)(114,192,128,178)(115,193,129,179)(116,194,130,180)(117,195,131,181)(118,196,132,182)(119,169,133,183)(120,170,134,184)(121,171,135,185)(122,172,136,186)(123,173,137,187)(124,174,138,188)(125,175,139,189)(126,176,140,190)(141,218,155,204)(142,219,156,205)(143,220,157,206)(144,221,158,207)(145,222,159,208)(146,223,160,209)(147,224,161,210)(148,197,162,211)(149,198,163,212)(150,199,164,213)(151,200,165,214)(152,201,166,215)(153,202,167,216)(154,203,168,217), (1,8,15,22)(2,9,16,23)(3,10,17,24)(4,11,18,25)(5,12,19,26)(6,13,20,27)(7,14,21,28)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,78,71,64)(58,79,72,65)(59,80,73,66)(60,81,74,67)(61,82,75,68)(62,83,76,69)(63,84,77,70)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112)(113,134,127,120)(114,135,128,121)(115,136,129,122)(116,137,130,123)(117,138,131,124)(118,139,132,125)(119,140,133,126)(141,148,155,162)(142,149,156,163)(143,150,157,164)(144,151,158,165)(145,152,159,166)(146,153,160,167)(147,154,161,168)(169,176,183,190)(170,177,184,191)(171,178,185,192)(172,179,186,193)(173,180,187,194)(174,181,188,195)(175,182,189,196)(197,218,211,204)(198,219,212,205)(199,220,213,206)(200,221,214,207)(201,222,215,208)(202,223,216,209)(203,224,217,210), (1,41)(2,56)(3,43)(4,30)(5,45)(6,32)(7,47)(8,34)(9,49)(10,36)(11,51)(12,38)(13,53)(14,40)(15,55)(16,42)(17,29)(18,44)(19,31)(20,46)(21,33)(22,48)(23,35)(24,50)(25,37)(26,52)(27,39)(28,54)(57,93)(58,108)(59,95)(60,110)(61,97)(62,112)(63,99)(64,86)(65,101)(66,88)(67,103)(68,90)(69,105)(70,92)(71,107)(72,94)(73,109)(74,96)(75,111)(76,98)(77,85)(78,100)(79,87)(80,102)(81,89)(82,104)(83,91)(84,106)(113,166)(114,153)(115,168)(116,155)(117,142)(118,157)(119,144)(120,159)(121,146)(122,161)(123,148)(124,163)(125,150)(126,165)(127,152)(128,167)(129,154)(130,141)(131,156)(132,143)(133,158)(134,145)(135,160)(136,147)(137,162)(138,149)(139,164)(140,151)(169,221)(170,208)(171,223)(172,210)(173,197)(174,212)(175,199)(176,214)(177,201)(178,216)(179,203)(180,218)(181,205)(182,220)(183,207)(184,222)(185,209)(186,224)(187,211)(188,198)(189,213)(190,200)(191,215)(192,202)(193,217)(194,204)(195,219)(196,206) );
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)], [(1,130),(2,129),(3,128),(4,127),(5,126),(6,125),(7,124),(8,123),(9,122),(10,121),(11,120),(12,119),(13,118),(14,117),(15,116),(16,115),(17,114),(18,113),(19,140),(20,139),(21,138),(22,137),(23,136),(24,135),(25,134),(26,133),(27,132),(28,131),(29,160),(30,159),(31,158),(32,157),(33,156),(34,155),(35,154),(36,153),(37,152),(38,151),(39,150),(40,149),(41,148),(42,147),(43,146),(44,145),(45,144),(46,143),(47,142),(48,141),(49,168),(50,167),(51,166),(52,165),(53,164),(54,163),(55,162),(56,161),(57,176),(58,175),(59,174),(60,173),(61,172),(62,171),(63,170),(64,169),(65,196),(66,195),(67,194),(68,193),(69,192),(70,191),(71,190),(72,189),(73,188),(74,187),(75,186),(76,185),(77,184),(78,183),(79,182),(80,181),(81,180),(82,179),(83,178),(84,177),(85,201),(86,200),(87,199),(88,198),(89,197),(90,224),(91,223),(92,222),(93,221),(94,220),(95,219),(96,218),(97,217),(98,216),(99,215),(100,214),(101,213),(102,212),(103,211),(104,210),(105,209),(106,208),(107,207),(108,206),(109,205),(110,204),(111,203),(112,202)], [(1,81,15,67),(2,82,16,68),(3,83,17,69),(4,84,18,70),(5,57,19,71),(6,58,20,72),(7,59,21,73),(8,60,22,74),(9,61,23,75),(10,62,24,76),(11,63,25,77),(12,64,26,78),(13,65,27,79),(14,66,28,80),(29,105,43,91),(30,106,44,92),(31,107,45,93),(32,108,46,94),(33,109,47,95),(34,110,48,96),(35,111,49,97),(36,112,50,98),(37,85,51,99),(38,86,52,100),(39,87,53,101),(40,88,54,102),(41,89,55,103),(42,90,56,104),(113,191,127,177),(114,192,128,178),(115,193,129,179),(116,194,130,180),(117,195,131,181),(118,196,132,182),(119,169,133,183),(120,170,134,184),(121,171,135,185),(122,172,136,186),(123,173,137,187),(124,174,138,188),(125,175,139,189),(126,176,140,190),(141,218,155,204),(142,219,156,205),(143,220,157,206),(144,221,158,207),(145,222,159,208),(146,223,160,209),(147,224,161,210),(148,197,162,211),(149,198,163,212),(150,199,164,213),(151,200,165,214),(152,201,166,215),(153,202,167,216),(154,203,168,217)], [(1,8,15,22),(2,9,16,23),(3,10,17,24),(4,11,18,25),(5,12,19,26),(6,13,20,27),(7,14,21,28),(29,50,43,36),(30,51,44,37),(31,52,45,38),(32,53,46,39),(33,54,47,40),(34,55,48,41),(35,56,49,42),(57,78,71,64),(58,79,72,65),(59,80,73,66),(60,81,74,67),(61,82,75,68),(62,83,76,69),(63,84,77,70),(85,92,99,106),(86,93,100,107),(87,94,101,108),(88,95,102,109),(89,96,103,110),(90,97,104,111),(91,98,105,112),(113,134,127,120),(114,135,128,121),(115,136,129,122),(116,137,130,123),(117,138,131,124),(118,139,132,125),(119,140,133,126),(141,148,155,162),(142,149,156,163),(143,150,157,164),(144,151,158,165),(145,152,159,166),(146,153,160,167),(147,154,161,168),(169,176,183,190),(170,177,184,191),(171,178,185,192),(172,179,186,193),(173,180,187,194),(174,181,188,195),(175,182,189,196),(197,218,211,204),(198,219,212,205),(199,220,213,206),(200,221,214,207),(201,222,215,208),(202,223,216,209),(203,224,217,210)], [(1,41),(2,56),(3,43),(4,30),(5,45),(6,32),(7,47),(8,34),(9,49),(10,36),(11,51),(12,38),(13,53),(14,40),(15,55),(16,42),(17,29),(18,44),(19,31),(20,46),(21,33),(22,48),(23,35),(24,50),(25,37),(26,52),(27,39),(28,54),(57,93),(58,108),(59,95),(60,110),(61,97),(62,112),(63,99),(64,86),(65,101),(66,88),(67,103),(68,90),(69,105),(70,92),(71,107),(72,94),(73,109),(74,96),(75,111),(76,98),(77,85),(78,100),(79,87),(80,102),(81,89),(82,104),(83,91),(84,106),(113,166),(114,153),(115,168),(116,155),(117,142),(118,157),(119,144),(120,159),(121,146),(122,161),(123,148),(124,163),(125,150),(126,165),(127,152),(128,167),(129,154),(130,141),(131,156),(132,143),(133,158),(134,145),(135,160),(136,147),(137,162),(138,149),(139,164),(140,151),(169,221),(170,208),(171,223),(172,210),(173,197),(174,212),(175,199),(176,214),(177,201),(178,216),(179,203),(180,218),(181,205),(182,220),(183,207),(184,222),(185,209),(186,224),(187,211),(188,198),(189,213),(190,200),(191,215),(192,202),(193,217),(194,204),(195,219),(196,206)]])
73 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 14A | 14B | 14C | 14D | ··· | 14R | 28A | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 28 | 28 | 28 | 2 | 2 | 2 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
73 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C7⋊D4 | C7⋊D4 | Q8○D8 | D28.35C23 |
| kernel | D28.35C23 | C28.C23 | C2×C7⋊Q16 | Q8.Dic7 | D4.8D14 | D4.9D14 | D4.10D14 | C7×2- 1+4 | C7×D4 | C7×Q8 | 2- 1+4 | C2×Q8 | C4○D4 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 3 | 1 | 3 | 9 | 12 | 18 | 6 | 2 | 3 |
Matrix representation of D28.35C23 ►in GL6(𝔽113)
| 33 | 9 | 0 | 0 | 0 | 0 |
| 104 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 69 | 0 | 0 |
| 0 | 0 | 36 | 112 | 0 | 0 |
| 0 | 0 | 36 | 37 | 112 | 91 |
| 0 | 0 | 40 | 89 | 72 | 1 |
| 107 | 31 | 0 | 0 | 0 | 0 |
| 90 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 33 | 0 | 62 |
| 0 | 0 | 32 | 110 | 9 | 85 |
| 0 | 0 | 12 | 101 | 61 | 92 |
| 0 | 0 | 65 | 18 | 59 | 55 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 76 | 100 | 21 |
| 0 | 0 | 104 | 17 | 1 | 50 |
| 0 | 0 | 97 | 74 | 53 | 101 |
| 0 | 0 | 38 | 29 | 107 | 39 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 44 | 0 | 0 |
| 0 | 0 | 77 | 1 | 0 | 0 |
| 0 | 0 | 95 | 39 | 112 | 91 |
| 0 | 0 | 20 | 77 | 72 | 1 |
| 34 | 5 | 0 | 0 | 0 | 0 |
| 108 | 79 | 0 | 0 | 0 | 0 |
| 0 | 0 | 83 | 36 | 41 | 0 |
| 0 | 0 | 0 | 103 | 0 | 36 |
| 0 | 0 | 58 | 93 | 30 | 18 |
| 0 | 0 | 0 | 82 | 0 | 10 |
G:=sub<GL(6,GF(113))| [33,104,0,0,0,0,9,1,0,0,0,0,0,0,1,36,36,40,0,0,69,112,37,89,0,0,0,0,112,72,0,0,0,0,91,1],[107,90,0,0,0,0,31,6,0,0,0,0,0,0,0,32,12,65,0,0,33,110,101,18,0,0,0,9,61,59,0,0,62,85,92,55],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,104,97,38,0,0,76,17,74,29,0,0,100,1,53,107,0,0,21,50,101,39],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,77,95,20,0,0,44,1,39,77,0,0,0,0,112,72,0,0,0,0,91,1],[34,108,0,0,0,0,5,79,0,0,0,0,0,0,83,0,58,0,0,0,36,103,93,82,0,0,41,0,30,0,0,0,0,36,18,10] >;
D28.35C23 in GAP, Magma, Sage, TeX
D_{28}._{35}C_2^3 % in TeX
G:=Group("D28.35C2^3"); // GroupNames label
G:=SmallGroup(448,1291);
// by ID
G=gap.SmallGroup(448,1291);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,675,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^28=b^2=e^2=1,c^2=d^2=a^14,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^15,b*c=c*b,b*d=d*b,e*b*e=a^7*b,d*c*d^-1=a^14*c,c*e=e*c,d*e=e*d>;
// generators/relations