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), D4.Dic7⋊13C2, Q8.21(C7⋊D4), D4.28(C22×D7), (C7×D4).28C23, D4.D7.3C22, D4.9D14⋊12C2, D4.8D14⋊11C2, Q8.28(C22×D7), (C7×Q8).28C23, 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
Subgroups: 916 in 248 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C7, C8 [×4], C2×C4 [×3], C2×C4 [×12], D4, D4 [×3], D4 [×7], Q8, Q8 [×3], Q8 [×9], D7, C14, C14 [×4], C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×3], C2×Q8 [×5], C4○D4, C4○D4 [×3], C4○D4 [×9], Dic7 [×3], C28, C28 [×3], C28 [×3], D14, C2×C14 [×3], C2×C14, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- (1+4), 2- (1+4), C7⋊C8, C7⋊C8 [×3], Dic14 [×3], Dic14 [×3], C4×D7 [×3], D28, C2×Dic7 [×3], C7⋊D4 [×3], C2×C28 [×3], C2×C28 [×6], C7×D4, C7×D4 [×3], C7×D4 [×3], C7×Q8, C7×Q8 [×3], C7×Q8 [×3], Q8○D8, C2×C7⋊C8 [×3], C4.Dic7 [×3], D4⋊D7, D4.D7 [×3], Q8⋊D7 [×3], C7⋊Q16 [×9], C2×Dic14 [×3], C4○D28 [×3], D4⋊2D7 [×3], Q8×D7, Q8×C14 [×3], Q8×C14, C7×C4○D4, C7×C4○D4 [×3], C7×C4○D4 [×3], C28.C23 [×3], C2×C7⋊Q16 [×3], D4.Dic7, D4.8D14 [×3], D4.9D14 [×3], D4.10D14, C7×2- (1+4), D28.35C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, C7⋊D4 [×4], C22×D7 [×7], Q8○D8, C2×C7⋊D4 [×6], C23×D7, C22×C7⋊D4, D28.35C23
Generators and relations
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 >
►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)
(1 114)(2 113)(3 140)(4 139)(5 138)(6 137)(7 136)(8 135)(9 134)(10 133)(11 132)(12 131)(13 130)(14 129)(15 128)(16 127)(17 126)(18 125)(19 124)(20 123)(21 122)(22 121)(23 120)(24 119)(25 118)(26 117)(27 116)(28 115)(29 150)(30 149)(31 148)(32 147)(33 146)(34 145)(35 144)(36 143)(37 142)(38 141)(39 168)(40 167)(41 166)(42 165)(43 164)(44 163)(45 162)(46 161)(47 160)(48 159)(49 158)(50 157)(51 156)(52 155)(53 154)(54 153)(55 152)(56 151)(57 184)(58 183)(59 182)(60 181)(61 180)(62 179)(63 178)(64 177)(65 176)(66 175)(67 174)(68 173)(69 172)(70 171)(71 170)(72 169)(73 196)(74 195)(75 194)(76 193)(77 192)(78 191)(79 190)(80 189)(81 188)(82 187)(83 186)(84 185)(85 212)(86 211)(87 210)(88 209)(89 208)(90 207)(91 206)(92 205)(93 204)(94 203)(95 202)(96 201)(97 200)(98 199)(99 198)(100 197)(101 224)(102 223)(103 222)(104 221)(105 220)(106 219)(107 218)(108 217)(109 216)(110 215)(111 214)(112 213)
(1 59 15 73)(2 60 16 74)(3 61 17 75)(4 62 18 76)(5 63 19 77)(6 64 20 78)(7 65 21 79)(8 66 22 80)(9 67 23 81)(10 68 24 82)(11 69 25 83)(12 70 26 84)(13 71 27 57)(14 72 28 58)(29 108 43 94)(30 109 44 95)(31 110 45 96)(32 111 46 97)(33 112 47 98)(34 85 48 99)(35 86 49 100)(36 87 50 101)(37 88 51 102)(38 89 52 103)(39 90 53 104)(40 91 54 105)(41 92 55 106)(42 93 56 107)(113 181 127 195)(114 182 128 196)(115 183 129 169)(116 184 130 170)(117 185 131 171)(118 186 132 172)(119 187 133 173)(120 188 134 174)(121 189 135 175)(122 190 136 176)(123 191 137 177)(124 192 138 178)(125 193 139 179)(126 194 140 180)(141 208 155 222)(142 209 156 223)(143 210 157 224)(144 211 158 197)(145 212 159 198)(146 213 160 199)(147 214 161 200)(148 215 162 201)(149 216 163 202)(150 217 164 203)(151 218 165 204)(152 219 166 205)(153 220 167 206)(154 221 168 207)
(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 90)(58 105)(59 92)(60 107)(61 94)(62 109)(63 96)(64 111)(65 98)(66 85)(67 100)(68 87)(69 102)(70 89)(71 104)(72 91)(73 106)(74 93)(75 108)(76 95)(77 110)(78 97)(79 112)(80 99)(81 86)(82 101)(83 88)(84 103)(113 144)(114 159)(115 146)(116 161)(117 148)(118 163)(119 150)(120 165)(121 152)(122 167)(123 154)(124 141)(125 156)(126 143)(127 158)(128 145)(129 160)(130 147)(131 162)(132 149)(133 164)(134 151)(135 166)(136 153)(137 168)(138 155)(139 142)(140 157)(169 199)(170 214)(171 201)(172 216)(173 203)(174 218)(175 205)(176 220)(177 207)(178 222)(179 209)(180 224)(181 211)(182 198)(183 213)(184 200)(185 215)(186 202)(187 217)(188 204)(189 219)(190 206)(191 221)(192 208)(193 223)(194 210)(195 197)(196 212)
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), (1,114)(2,113)(3,140)(4,139)(5,138)(6,137)(7,136)(8,135)(9,134)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,168)(40,167)(41,166)(42,165)(43,164)(44,163)(45,162)(46,161)(47,160)(48,159)(49,158)(50,157)(51,156)(52,155)(53,154)(54,153)(55,152)(56,151)(57,184)(58,183)(59,182)(60,181)(61,180)(62,179)(63,178)(64,177)(65,176)(66,175)(67,174)(68,173)(69,172)(70,171)(71,170)(72,169)(73,196)(74,195)(75,194)(76,193)(77,192)(78,191)(79,190)(80,189)(81,188)(82,187)(83,186)(84,185)(85,212)(86,211)(87,210)(88,209)(89,208)(90,207)(91,206)(92,205)(93,204)(94,203)(95,202)(96,201)(97,200)(98,199)(99,198)(100,197)(101,224)(102,223)(103,222)(104,221)(105,220)(106,219)(107,218)(108,217)(109,216)(110,215)(111,214)(112,213), (1,59,15,73)(2,60,16,74)(3,61,17,75)(4,62,18,76)(5,63,19,77)(6,64,20,78)(7,65,21,79)(8,66,22,80)(9,67,23,81)(10,68,24,82)(11,69,25,83)(12,70,26,84)(13,71,27,57)(14,72,28,58)(29,108,43,94)(30,109,44,95)(31,110,45,96)(32,111,46,97)(33,112,47,98)(34,85,48,99)(35,86,49,100)(36,87,50,101)(37,88,51,102)(38,89,52,103)(39,90,53,104)(40,91,54,105)(41,92,55,106)(42,93,56,107)(113,181,127,195)(114,182,128,196)(115,183,129,169)(116,184,130,170)(117,185,131,171)(118,186,132,172)(119,187,133,173)(120,188,134,174)(121,189,135,175)(122,190,136,176)(123,191,137,177)(124,192,138,178)(125,193,139,179)(126,194,140,180)(141,208,155,222)(142,209,156,223)(143,210,157,224)(144,211,158,197)(145,212,159,198)(146,213,160,199)(147,214,161,200)(148,215,162,201)(149,216,163,202)(150,217,164,203)(151,218,165,204)(152,219,166,205)(153,220,167,206)(154,221,168,207), (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)(57,90)(58,105)(59,92)(60,107)(61,94)(62,109)(63,96)(64,111)(65,98)(66,85)(67,100)(68,87)(69,102)(70,89)(71,104)(72,91)(73,106)(74,93)(75,108)(76,95)(77,110)(78,97)(79,112)(80,99)(81,86)(82,101)(83,88)(84,103)(113,144)(114,159)(115,146)(116,161)(117,148)(118,163)(119,150)(120,165)(121,152)(122,167)(123,154)(124,141)(125,156)(126,143)(127,158)(128,145)(129,160)(130,147)(131,162)(132,149)(133,164)(134,151)(135,166)(136,153)(137,168)(138,155)(139,142)(140,157)(169,199)(170,214)(171,201)(172,216)(173,203)(174,218)(175,205)(176,220)(177,207)(178,222)(179,209)(180,224)(181,211)(182,198)(183,213)(184,200)(185,215)(186,202)(187,217)(188,204)(189,219)(190,206)(191,221)(192,208)(193,223)(194,210)(195,197)(196,212)>;
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), (1,114)(2,113)(3,140)(4,139)(5,138)(6,137)(7,136)(8,135)(9,134)(10,133)(11,132)(12,131)(13,130)(14,129)(15,128)(16,127)(17,126)(18,125)(19,124)(20,123)(21,122)(22,121)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,150)(30,149)(31,148)(32,147)(33,146)(34,145)(35,144)(36,143)(37,142)(38,141)(39,168)(40,167)(41,166)(42,165)(43,164)(44,163)(45,162)(46,161)(47,160)(48,159)(49,158)(50,157)(51,156)(52,155)(53,154)(54,153)(55,152)(56,151)(57,184)(58,183)(59,182)(60,181)(61,180)(62,179)(63,178)(64,177)(65,176)(66,175)(67,174)(68,173)(69,172)(70,171)(71,170)(72,169)(73,196)(74,195)(75,194)(76,193)(77,192)(78,191)(79,190)(80,189)(81,188)(82,187)(83,186)(84,185)(85,212)(86,211)(87,210)(88,209)(89,208)(90,207)(91,206)(92,205)(93,204)(94,203)(95,202)(96,201)(97,200)(98,199)(99,198)(100,197)(101,224)(102,223)(103,222)(104,221)(105,220)(106,219)(107,218)(108,217)(109,216)(110,215)(111,214)(112,213), (1,59,15,73)(2,60,16,74)(3,61,17,75)(4,62,18,76)(5,63,19,77)(6,64,20,78)(7,65,21,79)(8,66,22,80)(9,67,23,81)(10,68,24,82)(11,69,25,83)(12,70,26,84)(13,71,27,57)(14,72,28,58)(29,108,43,94)(30,109,44,95)(31,110,45,96)(32,111,46,97)(33,112,47,98)(34,85,48,99)(35,86,49,100)(36,87,50,101)(37,88,51,102)(38,89,52,103)(39,90,53,104)(40,91,54,105)(41,92,55,106)(42,93,56,107)(113,181,127,195)(114,182,128,196)(115,183,129,169)(116,184,130,170)(117,185,131,171)(118,186,132,172)(119,187,133,173)(120,188,134,174)(121,189,135,175)(122,190,136,176)(123,191,137,177)(124,192,138,178)(125,193,139,179)(126,194,140,180)(141,208,155,222)(142,209,156,223)(143,210,157,224)(144,211,158,197)(145,212,159,198)(146,213,160,199)(147,214,161,200)(148,215,162,201)(149,216,163,202)(150,217,164,203)(151,218,165,204)(152,219,166,205)(153,220,167,206)(154,221,168,207), (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)(57,90)(58,105)(59,92)(60,107)(61,94)(62,109)(63,96)(64,111)(65,98)(66,85)(67,100)(68,87)(69,102)(70,89)(71,104)(72,91)(73,106)(74,93)(75,108)(76,95)(77,110)(78,97)(79,112)(80,99)(81,86)(82,101)(83,88)(84,103)(113,144)(114,159)(115,146)(116,161)(117,148)(118,163)(119,150)(120,165)(121,152)(122,167)(123,154)(124,141)(125,156)(126,143)(127,158)(128,145)(129,160)(130,147)(131,162)(132,149)(133,164)(134,151)(135,166)(136,153)(137,168)(138,155)(139,142)(140,157)(169,199)(170,214)(171,201)(172,216)(173,203)(174,218)(175,205)(176,220)(177,207)(178,222)(179,209)(180,224)(181,211)(182,198)(183,213)(184,200)(185,215)(186,202)(187,217)(188,204)(189,219)(190,206)(191,221)(192,208)(193,223)(194,210)(195,197)(196,212) );
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)], [(1,114),(2,113),(3,140),(4,139),(5,138),(6,137),(7,136),(8,135),(9,134),(10,133),(11,132),(12,131),(13,130),(14,129),(15,128),(16,127),(17,126),(18,125),(19,124),(20,123),(21,122),(22,121),(23,120),(24,119),(25,118),(26,117),(27,116),(28,115),(29,150),(30,149),(31,148),(32,147),(33,146),(34,145),(35,144),(36,143),(37,142),(38,141),(39,168),(40,167),(41,166),(42,165),(43,164),(44,163),(45,162),(46,161),(47,160),(48,159),(49,158),(50,157),(51,156),(52,155),(53,154),(54,153),(55,152),(56,151),(57,184),(58,183),(59,182),(60,181),(61,180),(62,179),(63,178),(64,177),(65,176),(66,175),(67,174),(68,173),(69,172),(70,171),(71,170),(72,169),(73,196),(74,195),(75,194),(76,193),(77,192),(78,191),(79,190),(80,189),(81,188),(82,187),(83,186),(84,185),(85,212),(86,211),(87,210),(88,209),(89,208),(90,207),(91,206),(92,205),(93,204),(94,203),(95,202),(96,201),(97,200),(98,199),(99,198),(100,197),(101,224),(102,223),(103,222),(104,221),(105,220),(106,219),(107,218),(108,217),(109,216),(110,215),(111,214),(112,213)], [(1,59,15,73),(2,60,16,74),(3,61,17,75),(4,62,18,76),(5,63,19,77),(6,64,20,78),(7,65,21,79),(8,66,22,80),(9,67,23,81),(10,68,24,82),(11,69,25,83),(12,70,26,84),(13,71,27,57),(14,72,28,58),(29,108,43,94),(30,109,44,95),(31,110,45,96),(32,111,46,97),(33,112,47,98),(34,85,48,99),(35,86,49,100),(36,87,50,101),(37,88,51,102),(38,89,52,103),(39,90,53,104),(40,91,54,105),(41,92,55,106),(42,93,56,107),(113,181,127,195),(114,182,128,196),(115,183,129,169),(116,184,130,170),(117,185,131,171),(118,186,132,172),(119,187,133,173),(120,188,134,174),(121,189,135,175),(122,190,136,176),(123,191,137,177),(124,192,138,178),(125,193,139,179),(126,194,140,180),(141,208,155,222),(142,209,156,223),(143,210,157,224),(144,211,158,197),(145,212,159,198),(146,213,160,199),(147,214,161,200),(148,215,162,201),(149,216,163,202),(150,217,164,203),(151,218,165,204),(152,219,166,205),(153,220,167,206),(154,221,168,207)], [(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),(57,90),(58,105),(59,92),(60,107),(61,94),(62,109),(63,96),(64,111),(65,98),(66,85),(67,100),(68,87),(69,102),(70,89),(71,104),(72,91),(73,106),(74,93),(75,108),(76,95),(77,110),(78,97),(79,112),(80,99),(81,86),(82,101),(83,88),(84,103),(113,144),(114,159),(115,146),(116,161),(117,148),(118,163),(119,150),(120,165),(121,152),(122,167),(123,154),(124,141),(125,156),(126,143),(127,158),(128,145),(129,160),(130,147),(131,162),(132,149),(133,164),(134,151),(135,166),(136,153),(137,168),(138,155),(139,142),(140,157),(169,199),(170,214),(171,201),(172,216),(173,203),(174,218),(175,205),(176,220),(177,207),(178,222),(179,209),(180,224),(181,211),(182,198),(183,213),(184,200),(185,215),(186,202),(187,217),(188,204),(189,219),(190,206),(191,221),(192,208),(193,223),(194,210),(195,197),(196,212)])
Matrix representation ►G ⊆ 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] >;
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 | D4.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 |
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
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