direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C28.C23, C28.31C24, D28.28C23, Dic14.27C23, (C2×Q8)⋊27D14, C7⋊C8.13C23, (C22×Q8)⋊4D7, C28.255(C2×D4), (C2×C28).211D4, Q8⋊D7⋊16C22, C4.31(C23×D7), C14⋊4(C8.C22), (Q8×C14)⋊34C22, C7⋊Q16⋊15C22, Q8.20(C22×D7), (C7×Q8).20C23, (C2×C28).548C23, C4○D28.57C22, (C22×C4).274D14, (C22×C14).210D4, C14.150(C22×D4), C23.93(C7⋊D4), C4.Dic7⋊33C22, (C2×D28).277C22, (C22×C28).280C22, (C2×Dic14).305C22, (Q8×C2×C14)⋊3C2, C7⋊5(C2×C8.C22), (C2×Q8⋊D7)⋊30C2, C4.25(C2×C7⋊D4), (C2×C7⋊Q16)⋊30C2, (C2×C4○D28).24C2, (C2×C14).585(C2×D4), (C2×C4).93(C7⋊D4), (C2×C7⋊C8).183C22, (C2×C4.Dic7)⋊27C2, C2.23(C22×C7⋊D4), (C2×C4).240(C22×D7), C22.113(C2×C7⋊D4), SmallGroup(448,1261)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1012 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×7], Q8 [×4], Q8 [×9], C23, C23, D7 [×2], C14, C14 [×2], C14 [×2], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8 [×6], C2×Q8 [×4], C4○D4 [×6], Dic7 [×2], C28 [×2], C28 [×2], C28 [×4], D14 [×4], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C7⋊C8 [×4], Dic14 [×2], Dic14, C4×D7 [×4], D28 [×2], D28, C2×Dic7, C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×6], C7×Q8 [×4], C7×Q8 [×6], C22×D7, C22×C14, C2×C8.C22, C2×C7⋊C8 [×2], C4.Dic7 [×4], Q8⋊D7 [×8], C7⋊Q16 [×8], C2×Dic14, C2×C4×D7, C2×D28, C4○D28 [×4], C4○D28 [×2], C2×C7⋊D4, C22×C28, C22×C28, Q8×C14 [×6], Q8×C14 [×3], C2×C4.Dic7, C2×Q8⋊D7 [×2], C28.C23 [×8], C2×C7⋊Q16 [×2], C2×C4○D28, Q8×C2×C14, C2×C28.C23
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8.C22 [×2], C22×D4, C7⋊D4 [×4], C22×D7 [×7], C2×C8.C22, C2×C7⋊D4 [×6], C23×D7, C28.C23 [×2], C22×C7⋊D4, C2×C28.C23
Generators and relations
G = < a,b,c,d,e | a2=b28=c2=1, d2=e2=b14, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b15, cd=dc, ece-1=b21c, ede-1=b14d >
►On 224 points
Generators in S224
(1 80)(2 81)(3 82)(4 83)(5 84)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 94)(38 95)(39 96)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 85)(113 185)(114 186)(115 187)(116 188)(117 189)(118 190)(119 191)(120 192)(121 193)(122 194)(123 195)(124 196)(125 169)(126 170)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)(133 177)(134 178)(135 179)(136 180)(137 181)(138 182)(139 183)(140 184)(141 200)(142 201)(143 202)(144 203)(145 204)(146 205)(147 206)(148 207)(149 208)(150 209)(151 210)(152 211)(153 212)(154 213)(155 214)(156 215)(157 216)(158 217)(159 218)(160 219)(161 220)(162 221)(163 222)(164 223)(165 224)(166 197)(167 198)(168 199)
(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)
(1 139)(2 138)(3 137)(4 136)(5 135)(6 134)(7 133)(8 132)(9 131)(10 130)(11 129)(12 128)(13 127)(14 126)(15 125)(16 124)(17 123)(18 122)(19 121)(20 120)(21 119)(22 118)(23 117)(24 116)(25 115)(26 114)(27 113)(28 140)(29 168)(30 167)(31 166)(32 165)(33 164)(34 163)(35 162)(36 161)(37 160)(38 159)(39 158)(40 157)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 144)(54 143)(55 142)(56 141)(57 178)(58 177)(59 176)(60 175)(61 174)(62 173)(63 172)(64 171)(65 170)(66 169)(67 196)(68 195)(69 194)(70 193)(71 192)(72 191)(73 190)(74 189)(75 188)(76 187)(77 186)(78 185)(79 184)(80 183)(81 182)(82 181)(83 180)(84 179)(85 200)(86 199)(87 198)(88 197)(89 224)(90 223)(91 222)(92 221)(93 220)(94 219)(95 218)(96 217)(97 216)(98 215)(99 214)(100 213)(101 212)(102 211)(103 210)(104 209)(105 208)(106 207)(107 206)(108 205)(109 204)(110 203)(111 202)(112 201)
(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 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 56)(57 64 71 78)(58 65 72 79)(59 66 73 80)(60 67 74 81)(61 68 75 82)(62 69 76 83)(63 70 77 84)(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)(141 162 155 148)(142 163 156 149)(143 164 157 150)(144 165 158 151)(145 166 159 152)(146 167 160 153)(147 168 161 154)(169 190 183 176)(170 191 184 177)(171 192 185 178)(172 193 186 179)(173 194 187 180)(174 195 188 181)(175 196 189 182)(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 52 15 38)(2 39 16 53)(3 54 17 40)(4 41 18 55)(5 56 19 42)(6 43 20 29)(7 30 21 44)(8 45 22 31)(9 32 23 46)(10 47 24 33)(11 34 25 48)(12 49 26 35)(13 36 27 50)(14 51 28 37)(57 100 71 86)(58 87 72 101)(59 102 73 88)(60 89 74 103)(61 104 75 90)(62 91 76 105)(63 106 77 92)(64 93 78 107)(65 108 79 94)(66 95 80 109)(67 110 81 96)(68 97 82 111)(69 112 83 98)(70 99 84 85)(113 154 127 168)(114 141 128 155)(115 156 129 142)(116 143 130 157)(117 158 131 144)(118 145 132 159)(119 160 133 146)(120 147 134 161)(121 162 135 148)(122 149 136 163)(123 164 137 150)(124 151 138 165)(125 166 139 152)(126 153 140 167)(169 197 183 211)(170 212 184 198)(171 199 185 213)(172 214 186 200)(173 201 187 215)(174 216 188 202)(175 203 189 217)(176 218 190 204)(177 205 191 219)(178 220 192 206)(179 207 193 221)(180 222 194 208)(181 209 195 223)(182 224 196 210)
G:=sub<Sym(224)| (1,80)(2,81)(3,82)(4,83)(5,84)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,85)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)(140,184)(141,200)(142,201)(143,202)(144,203)(145,204)(146,205)(147,206)(148,207)(149,208)(150,209)(151,210)(152,211)(153,212)(154,213)(155,214)(156,215)(157,216)(158,217)(159,218)(160,219)(161,220)(162,221)(163,222)(164,223)(165,224)(166,197)(167,198)(168,199), (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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,120)(21,119)(22,118)(23,117)(24,116)(25,115)(26,114)(27,113)(28,140)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,200)(86,199)(87,198)(88,197)(89,224)(90,223)(91,222)(92,221)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)(112,201), (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)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(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)(141,162,155,148)(142,163,156,149)(143,164,157,150)(144,165,158,151)(145,166,159,152)(146,167,160,153)(147,168,161,154)(169,190,183,176)(170,191,184,177)(171,192,185,178)(172,193,186,179)(173,194,187,180)(174,195,188,181)(175,196,189,182)(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), (1,52,15,38)(2,39,16,53)(3,54,17,40)(4,41,18,55)(5,56,19,42)(6,43,20,29)(7,30,21,44)(8,45,22,31)(9,32,23,46)(10,47,24,33)(11,34,25,48)(12,49,26,35)(13,36,27,50)(14,51,28,37)(57,100,71,86)(58,87,72,101)(59,102,73,88)(60,89,74,103)(61,104,75,90)(62,91,76,105)(63,106,77,92)(64,93,78,107)(65,108,79,94)(66,95,80,109)(67,110,81,96)(68,97,82,111)(69,112,83,98)(70,99,84,85)(113,154,127,168)(114,141,128,155)(115,156,129,142)(116,143,130,157)(117,158,131,144)(118,145,132,159)(119,160,133,146)(120,147,134,161)(121,162,135,148)(122,149,136,163)(123,164,137,150)(124,151,138,165)(125,166,139,152)(126,153,140,167)(169,197,183,211)(170,212,184,198)(171,199,185,213)(172,214,186,200)(173,201,187,215)(174,216,188,202)(175,203,189,217)(176,218,190,204)(177,205,191,219)(178,220,192,206)(179,207,193,221)(180,222,194,208)(181,209,195,223)(182,224,196,210)>;
G:=Group( (1,80)(2,81)(3,82)(4,83)(5,84)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,86)(30,87)(31,88)(32,89)(33,90)(34,91)(35,92)(36,93)(37,94)(38,95)(39,96)(40,97)(41,98)(42,99)(43,100)(44,101)(45,102)(46,103)(47,104)(48,105)(49,106)(50,107)(51,108)(52,109)(53,110)(54,111)(55,112)(56,85)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)(140,184)(141,200)(142,201)(143,202)(144,203)(145,204)(146,205)(147,206)(148,207)(149,208)(150,209)(151,210)(152,211)(153,212)(154,213)(155,214)(156,215)(157,216)(158,217)(159,218)(160,219)(161,220)(162,221)(163,222)(164,223)(165,224)(166,197)(167,198)(168,199), (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), (1,139)(2,138)(3,137)(4,136)(5,135)(6,134)(7,133)(8,132)(9,131)(10,130)(11,129)(12,128)(13,127)(14,126)(15,125)(16,124)(17,123)(18,122)(19,121)(20,120)(21,119)(22,118)(23,117)(24,116)(25,115)(26,114)(27,113)(28,140)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,196)(68,195)(69,194)(70,193)(71,192)(72,191)(73,190)(74,189)(75,188)(76,187)(77,186)(78,185)(79,184)(80,183)(81,182)(82,181)(83,180)(84,179)(85,200)(86,199)(87,198)(88,197)(89,224)(90,223)(91,222)(92,221)(93,220)(94,219)(95,218)(96,217)(97,216)(98,215)(99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)(107,206)(108,205)(109,204)(110,203)(111,202)(112,201), (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)(29,36,43,50)(30,37,44,51)(31,38,45,52)(32,39,46,53)(33,40,47,54)(34,41,48,55)(35,42,49,56)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(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)(141,162,155,148)(142,163,156,149)(143,164,157,150)(144,165,158,151)(145,166,159,152)(146,167,160,153)(147,168,161,154)(169,190,183,176)(170,191,184,177)(171,192,185,178)(172,193,186,179)(173,194,187,180)(174,195,188,181)(175,196,189,182)(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), (1,52,15,38)(2,39,16,53)(3,54,17,40)(4,41,18,55)(5,56,19,42)(6,43,20,29)(7,30,21,44)(8,45,22,31)(9,32,23,46)(10,47,24,33)(11,34,25,48)(12,49,26,35)(13,36,27,50)(14,51,28,37)(57,100,71,86)(58,87,72,101)(59,102,73,88)(60,89,74,103)(61,104,75,90)(62,91,76,105)(63,106,77,92)(64,93,78,107)(65,108,79,94)(66,95,80,109)(67,110,81,96)(68,97,82,111)(69,112,83,98)(70,99,84,85)(113,154,127,168)(114,141,128,155)(115,156,129,142)(116,143,130,157)(117,158,131,144)(118,145,132,159)(119,160,133,146)(120,147,134,161)(121,162,135,148)(122,149,136,163)(123,164,137,150)(124,151,138,165)(125,166,139,152)(126,153,140,167)(169,197,183,211)(170,212,184,198)(171,199,185,213)(172,214,186,200)(173,201,187,215)(174,216,188,202)(175,203,189,217)(176,218,190,204)(177,205,191,219)(178,220,192,206)(179,207,193,221)(180,222,194,208)(181,209,195,223)(182,224,196,210) );
G=PermutationGroup([(1,80),(2,81),(3,82),(4,83),(5,84),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,86),(30,87),(31,88),(32,89),(33,90),(34,91),(35,92),(36,93),(37,94),(38,95),(39,96),(40,97),(41,98),(42,99),(43,100),(44,101),(45,102),(46,103),(47,104),(48,105),(49,106),(50,107),(51,108),(52,109),(53,110),(54,111),(55,112),(56,85),(113,185),(114,186),(115,187),(116,188),(117,189),(118,190),(119,191),(120,192),(121,193),(122,194),(123,195),(124,196),(125,169),(126,170),(127,171),(128,172),(129,173),(130,174),(131,175),(132,176),(133,177),(134,178),(135,179),(136,180),(137,181),(138,182),(139,183),(140,184),(141,200),(142,201),(143,202),(144,203),(145,204),(146,205),(147,206),(148,207),(149,208),(150,209),(151,210),(152,211),(153,212),(154,213),(155,214),(156,215),(157,216),(158,217),(159,218),(160,219),(161,220),(162,221),(163,222),(164,223),(165,224),(166,197),(167,198),(168,199)], [(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)], [(1,139),(2,138),(3,137),(4,136),(5,135),(6,134),(7,133),(8,132),(9,131),(10,130),(11,129),(12,128),(13,127),(14,126),(15,125),(16,124),(17,123),(18,122),(19,121),(20,120),(21,119),(22,118),(23,117),(24,116),(25,115),(26,114),(27,113),(28,140),(29,168),(30,167),(31,166),(32,165),(33,164),(34,163),(35,162),(36,161),(37,160),(38,159),(39,158),(40,157),(41,156),(42,155),(43,154),(44,153),(45,152),(46,151),(47,150),(48,149),(49,148),(50,147),(51,146),(52,145),(53,144),(54,143),(55,142),(56,141),(57,178),(58,177),(59,176),(60,175),(61,174),(62,173),(63,172),(64,171),(65,170),(66,169),(67,196),(68,195),(69,194),(70,193),(71,192),(72,191),(73,190),(74,189),(75,188),(76,187),(77,186),(78,185),(79,184),(80,183),(81,182),(82,181),(83,180),(84,179),(85,200),(86,199),(87,198),(88,197),(89,224),(90,223),(91,222),(92,221),(93,220),(94,219),(95,218),(96,217),(97,216),(98,215),(99,214),(100,213),(101,212),(102,211),(103,210),(104,209),(105,208),(106,207),(107,206),(108,205),(109,204),(110,203),(111,202),(112,201)], [(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),(29,36,43,50),(30,37,44,51),(31,38,45,52),(32,39,46,53),(33,40,47,54),(34,41,48,55),(35,42,49,56),(57,64,71,78),(58,65,72,79),(59,66,73,80),(60,67,74,81),(61,68,75,82),(62,69,76,83),(63,70,77,84),(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),(141,162,155,148),(142,163,156,149),(143,164,157,150),(144,165,158,151),(145,166,159,152),(146,167,160,153),(147,168,161,154),(169,190,183,176),(170,191,184,177),(171,192,185,178),(172,193,186,179),(173,194,187,180),(174,195,188,181),(175,196,189,182),(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)], [(1,52,15,38),(2,39,16,53),(3,54,17,40),(4,41,18,55),(5,56,19,42),(6,43,20,29),(7,30,21,44),(8,45,22,31),(9,32,23,46),(10,47,24,33),(11,34,25,48),(12,49,26,35),(13,36,27,50),(14,51,28,37),(57,100,71,86),(58,87,72,101),(59,102,73,88),(60,89,74,103),(61,104,75,90),(62,91,76,105),(63,106,77,92),(64,93,78,107),(65,108,79,94),(66,95,80,109),(67,110,81,96),(68,97,82,111),(69,112,83,98),(70,99,84,85),(113,154,127,168),(114,141,128,155),(115,156,129,142),(116,143,130,157),(117,158,131,144),(118,145,132,159),(119,160,133,146),(120,147,134,161),(121,162,135,148),(122,149,136,163),(123,164,137,150),(124,151,138,165),(125,166,139,152),(126,153,140,167),(169,197,183,211),(170,212,184,198),(171,199,185,213),(172,214,186,200),(173,201,187,215),(174,216,188,202),(175,203,189,217),(176,218,190,204),(177,205,191,219),(178,220,192,206),(179,207,193,221),(180,222,194,208),(181,209,195,223),(182,224,196,210)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 57 | 0 | 0 |
| 0 | 0 | 28 | 85 | 0 | 0 |
| 0 | 0 | 56 | 20 | 0 | 4 |
| 0 | 0 | 37 | 20 | 109 | 0 |
| 93 | 22 | 0 | 0 | 0 | 0 |
| 100 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 80 | 0 | 111 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 92 | 0 | 33 | 0 |
| 0 | 0 | 92 | 1 | 33 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 2 | 0 | 0 |
| 0 | 0 | 112 | 1 | 0 | 0 |
| 0 | 0 | 0 | 80 | 0 | 112 |
| 0 | 0 | 33 | 80 | 1 | 0 |
| 83 | 106 | 0 | 0 | 0 | 0 |
| 80 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 99 | 63 | 0 | 0 |
| 0 | 0 | 74 | 14 | 0 | 0 |
| 0 | 0 | 44 | 34 | 25 | 39 |
| 0 | 0 | 10 | 34 | 39 | 88 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,28,28,56,37,0,0,57,85,20,20,0,0,0,0,0,109,0,0,0,0,4,0],[93,100,0,0,0,0,22,20,0,0,0,0,0,0,80,0,92,92,0,0,0,0,0,1,0,0,111,112,33,33,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,112,0,33,0,0,2,1,80,80,0,0,0,0,0,1,0,0,0,0,112,0],[83,80,0,0,0,0,106,30,0,0,0,0,0,0,99,74,44,10,0,0,63,14,34,34,0,0,0,0,25,39,0,0,0,0,39,88] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8.C22 | C28.C23 |
| kernel | C2×C28.C23 | C2×C4.Dic7 | C2×Q8⋊D7 | C28.C23 | C2×C7⋊Q16 | C2×C4○D28 | Q8×C2×C14 | C2×C28 | C22×C14 | C22×Q8 | C22×C4 | C2×Q8 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 3 | 1 | 3 | 3 | 18 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_2\times C_{28}.C_2^3 % in TeX
G:=Group("C2xC28.C2^3"); // GroupNames label
G:=SmallGroup(448,1261);
// by ID
G=gap.SmallGroup(448,1261);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,675,297,136,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^28=c^2=1,d^2=e^2=b^14,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^15,c*d=d*c,e*c*e^-1=b^21*c,e*d*e^-1=b^14*d>;
// generators/relations
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