direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.18D14, C24.58D14, (C2×D4).228D14, (C23×Dic7)⋊8C2, (C22×D4).10D7, (C2×C14).291C24, (C2×C28).642C23, Dic7⋊C4⋊72C22, C14.139(C22×D4), (C22×C4).269D14, (C22×C14).121D4, C23.67(C7⋊D4), C23.D7⋊57C22, (D4×C14).311C22, C14⋊5(C22.D4), (C23×C14).73C22, C22.305(C23×D7), C23.133(C22×D7), C22.77(D4⋊2D7), (C22×C14).227C23, (C22×C28).437C22, (C2×Dic7).281C23, (C22×Dic7)⋊48C22, (D4×C2×C14).21C2, (C2×C14).73(C2×D4), C7⋊6(C2×C22.D4), (C2×Dic7⋊C4)⋊47C2, C14.103(C2×C4○D4), C2.67(C2×D4⋊2D7), (C2×C23.D7)⋊24C2, C2.12(C22×C7⋊D4), (C2×C4).236(C22×D7), C22.108(C2×C7⋊D4), (C2×C14).175(C4○D4), SmallGroup(448,1249)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1172 in 342 conjugacy classes, 127 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×10], C22, C22 [×10], C22 [×22], C7, C2×C4 [×2], C2×C4 [×26], D4 [×8], C23, C23 [×8], C23 [×10], C14, C14 [×6], C14 [×6], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C24 [×2], Dic7 [×8], C28 [×2], C2×C14, C2×C14 [×10], C2×C14 [×22], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C2×Dic7 [×8], C2×Dic7 [×16], C2×C28 [×2], C2×C28 [×2], C7×D4 [×8], C22×C14, C22×C14 [×8], C22×C14 [×10], C2×C22.D4, Dic7⋊C4 [×8], C23.D7 [×12], C22×Dic7 [×8], C22×Dic7 [×4], C22×C28, D4×C14 [×4], D4×C14 [×4], C23×C14 [×2], C2×Dic7⋊C4 [×2], C23.18D14 [×8], C2×C23.D7, C2×C23.D7 [×2], C23×Dic7, D4×C2×C14, C2×C23.18D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×4], C24, D14 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C7⋊D4 [×4], C22×D7 [×7], C2×C22.D4, D4⋊2D7 [×4], C2×C7⋊D4 [×6], C23×D7, C23.18D14 [×4], C2×D4⋊2D7 [×2], C22×C7⋊D4, C2×C23.18D14
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >
►On 224 points
Generators in S224
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 55)(11 56)(12 43)(13 44)(14 45)(15 188)(16 189)(17 190)(18 191)(19 192)(20 193)(21 194)(22 195)(23 196)(24 183)(25 184)(26 185)(27 186)(28 187)(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 85)(57 216)(58 217)(59 218)(60 219)(61 220)(62 221)(63 222)(64 223)(65 224)(66 211)(67 212)(68 213)(69 214)(70 215)(71 123)(72 124)(73 125)(74 126)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(81 119)(82 120)(83 121)(84 122)(99 150)(100 151)(101 152)(102 153)(103 154)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(127 199)(128 200)(129 201)(130 202)(131 203)(132 204)(133 205)(134 206)(135 207)(136 208)(137 209)(138 210)(139 197)(140 198)(155 171)(156 172)(157 173)(158 174)(159 175)(160 176)(161 177)(162 178)(163 179)(164 180)(165 181)(166 182)(167 169)(168 170)
(1 85)(2 73)(3 87)(4 75)(5 89)(6 77)(7 91)(8 79)(9 93)(10 81)(11 95)(12 83)(13 97)(14 71)(15 157)(16 208)(17 159)(18 210)(19 161)(20 198)(21 163)(22 200)(23 165)(24 202)(25 167)(26 204)(27 155)(28 206)(29 63)(30 48)(31 65)(32 50)(33 67)(34 52)(35 69)(36 54)(37 57)(38 56)(39 59)(40 44)(41 61)(42 46)(43 121)(45 123)(47 125)(49 113)(51 115)(53 117)(55 119)(58 120)(60 122)(62 124)(64 126)(66 114)(68 116)(70 118)(72 221)(74 223)(76 211)(78 213)(80 215)(82 217)(84 219)(86 222)(88 224)(90 212)(92 214)(94 216)(96 218)(98 220)(99 164)(100 201)(101 166)(102 203)(103 168)(104 205)(105 156)(106 207)(107 158)(108 209)(109 160)(110 197)(111 162)(112 199)(127 149)(128 195)(129 151)(130 183)(131 153)(132 185)(133 141)(134 187)(135 143)(136 189)(137 145)(138 191)(139 147)(140 193)(142 172)(144 174)(146 176)(148 178)(150 180)(152 182)(154 170)(169 184)(171 186)(173 188)(175 190)(177 192)(179 194)(181 196)
(1 42)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 173)(16 174)(17 175)(18 176)(19 177)(20 178)(21 179)(22 180)(23 181)(24 182)(25 169)(26 170)(27 171)(28 172)(43 96)(44 97)(45 98)(46 85)(47 86)(48 87)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 81)(58 82)(59 83)(60 84)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(99 128)(100 129)(101 130)(102 131)(103 132)(104 133)(105 134)(106 135)(107 136)(108 137)(109 138)(110 139)(111 140)(112 127)(113 224)(114 211)(115 212)(116 213)(117 214)(118 215)(119 216)(120 217)(121 218)(122 219)(123 220)(124 221)(125 222)(126 223)(141 205)(142 206)(143 207)(144 208)(145 209)(146 210)(147 197)(148 198)(149 199)(150 200)(151 201)(152 202)(153 203)(154 204)(155 186)(156 187)(157 188)(158 189)(159 190)(160 191)(161 192)(162 193)(163 194)(164 195)(165 196)(166 183)(167 184)(168 185)
(1 221)(2 222)(3 223)(4 224)(5 211)(6 212)(7 213)(8 214)(9 215)(10 216)(11 217)(12 218)(13 219)(14 220)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 125)(30 126)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 121)(40 122)(41 123)(42 124)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 57)(56 58)(71 98)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(127 179)(128 180)(129 181)(130 182)(131 169)(132 170)(133 171)(134 172)(135 173)(136 174)(137 175)(138 176)(139 177)(140 178)(141 186)(142 187)(143 188)(144 189)(145 190)(146 191)(147 192)(148 193)(149 194)(150 195)(151 196)(152 183)(153 184)(154 185)(155 205)(156 206)(157 207)(158 208)(159 209)(160 210)(161 197)(162 198)(163 199)(164 200)(165 201)(166 202)(167 203)(168 204)
(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)
(1 185 124 204)(2 167 125 153)(3 183 126 202)(4 165 113 151)(5 195 114 200)(6 163 115 149)(7 193 116 198)(8 161 117 147)(9 191 118 210)(10 159 119 145)(11 189 120 208)(12 157 121 143)(13 187 122 206)(14 155 123 141)(15 59 135 96)(16 82 136 56)(17 57 137 94)(18 80 138 54)(19 69 139 92)(20 78 140 52)(21 67 127 90)(22 76 128 50)(23 65 129 88)(24 74 130 48)(25 63 131 86)(26 72 132 46)(27 61 133 98)(28 84 134 44)(29 184 222 203)(30 166 223 152)(31 196 224 201)(32 164 211 150)(33 194 212 199)(34 162 213 148)(35 192 214 197)(36 160 215 146)(37 190 216 209)(38 158 217 144)(39 188 218 207)(40 156 219 142)(41 186 220 205)(42 168 221 154)(43 173 83 106)(45 171 71 104)(47 169 73 102)(49 181 75 100)(51 179 77 112)(53 177 79 110)(55 175 81 108)(58 107 95 174)(60 105 97 172)(62 103 85 170)(64 101 87 182)(66 99 89 180)(68 111 91 178)(70 109 93 176)
G:=sub<Sym(224)| (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,43)(13,44)(14,45)(15,188)(16,189)(17,190)(18,191)(19,192)(20,193)(21,194)(22,195)(23,196)(24,183)(25,184)(26,185)(27,186)(28,187)(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)(42,85)(57,216)(58,217)(59,218)(60,219)(61,220)(62,221)(63,222)(64,223)(65,224)(66,211)(67,212)(68,213)(69,214)(70,215)(71,123)(72,124)(73,125)(74,126)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122)(99,150)(100,151)(101,152)(102,153)(103,154)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149)(127,199)(128,200)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(136,208)(137,209)(138,210)(139,197)(140,198)(155,171)(156,172)(157,173)(158,174)(159,175)(160,176)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,169)(168,170), (1,85)(2,73)(3,87)(4,75)(5,89)(6,77)(7,91)(8,79)(9,93)(10,81)(11,95)(12,83)(13,97)(14,71)(15,157)(16,208)(17,159)(18,210)(19,161)(20,198)(21,163)(22,200)(23,165)(24,202)(25,167)(26,204)(27,155)(28,206)(29,63)(30,48)(31,65)(32,50)(33,67)(34,52)(35,69)(36,54)(37,57)(38,56)(39,59)(40,44)(41,61)(42,46)(43,121)(45,123)(47,125)(49,113)(51,115)(53,117)(55,119)(58,120)(60,122)(62,124)(64,126)(66,114)(68,116)(70,118)(72,221)(74,223)(76,211)(78,213)(80,215)(82,217)(84,219)(86,222)(88,224)(90,212)(92,214)(94,216)(96,218)(98,220)(99,164)(100,201)(101,166)(102,203)(103,168)(104,205)(105,156)(106,207)(107,158)(108,209)(109,160)(110,197)(111,162)(112,199)(127,149)(128,195)(129,151)(130,183)(131,153)(132,185)(133,141)(134,187)(135,143)(136,189)(137,145)(138,191)(139,147)(140,193)(142,172)(144,174)(146,176)(148,178)(150,180)(152,182)(154,170)(169,184)(171,186)(173,188)(175,190)(177,192)(179,194)(181,196), (1,42)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,173)(16,174)(17,175)(18,176)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,169)(26,170)(27,171)(28,172)(43,96)(44,97)(45,98)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,81)(58,82)(59,83)(60,84)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,127)(113,224)(114,211)(115,212)(116,213)(117,214)(118,215)(119,216)(120,217)(121,218)(122,219)(123,220)(124,221)(125,222)(126,223)(141,205)(142,206)(143,207)(144,208)(145,209)(146,210)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,183)(167,184)(168,185), (1,221)(2,222)(3,223)(4,224)(5,211)(6,212)(7,213)(8,214)(9,215)(10,216)(11,217)(12,218)(13,219)(14,220)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,125)(30,126)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,121)(40,122)(41,123)(42,124)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,57)(56,58)(71,98)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(127,179)(128,180)(129,181)(130,182)(131,169)(132,170)(133,171)(134,172)(135,173)(136,174)(137,175)(138,176)(139,177)(140,178)(141,186)(142,187)(143,188)(144,189)(145,190)(146,191)(147,192)(148,193)(149,194)(150,195)(151,196)(152,183)(153,184)(154,185)(155,205)(156,206)(157,207)(158,208)(159,209)(160,210)(161,197)(162,198)(163,199)(164,200)(165,201)(166,202)(167,203)(168,204), (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), (1,185,124,204)(2,167,125,153)(3,183,126,202)(4,165,113,151)(5,195,114,200)(6,163,115,149)(7,193,116,198)(8,161,117,147)(9,191,118,210)(10,159,119,145)(11,189,120,208)(12,157,121,143)(13,187,122,206)(14,155,123,141)(15,59,135,96)(16,82,136,56)(17,57,137,94)(18,80,138,54)(19,69,139,92)(20,78,140,52)(21,67,127,90)(22,76,128,50)(23,65,129,88)(24,74,130,48)(25,63,131,86)(26,72,132,46)(27,61,133,98)(28,84,134,44)(29,184,222,203)(30,166,223,152)(31,196,224,201)(32,164,211,150)(33,194,212,199)(34,162,213,148)(35,192,214,197)(36,160,215,146)(37,190,216,209)(38,158,217,144)(39,188,218,207)(40,156,219,142)(41,186,220,205)(42,168,221,154)(43,173,83,106)(45,171,71,104)(47,169,73,102)(49,181,75,100)(51,179,77,112)(53,177,79,110)(55,175,81,108)(58,107,95,174)(60,105,97,172)(62,103,85,170)(64,101,87,182)(66,99,89,180)(68,111,91,178)(70,109,93,176)>;
G:=Group( (1,46)(2,47)(3,48)(4,49)(5,50)(6,51)(7,52)(8,53)(9,54)(10,55)(11,56)(12,43)(13,44)(14,45)(15,188)(16,189)(17,190)(18,191)(19,192)(20,193)(21,194)(22,195)(23,196)(24,183)(25,184)(26,185)(27,186)(28,187)(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)(42,85)(57,216)(58,217)(59,218)(60,219)(61,220)(62,221)(63,222)(64,223)(65,224)(66,211)(67,212)(68,213)(69,214)(70,215)(71,123)(72,124)(73,125)(74,126)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,119)(82,120)(83,121)(84,122)(99,150)(100,151)(101,152)(102,153)(103,154)(104,141)(105,142)(106,143)(107,144)(108,145)(109,146)(110,147)(111,148)(112,149)(127,199)(128,200)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(136,208)(137,209)(138,210)(139,197)(140,198)(155,171)(156,172)(157,173)(158,174)(159,175)(160,176)(161,177)(162,178)(163,179)(164,180)(165,181)(166,182)(167,169)(168,170), (1,85)(2,73)(3,87)(4,75)(5,89)(6,77)(7,91)(8,79)(9,93)(10,81)(11,95)(12,83)(13,97)(14,71)(15,157)(16,208)(17,159)(18,210)(19,161)(20,198)(21,163)(22,200)(23,165)(24,202)(25,167)(26,204)(27,155)(28,206)(29,63)(30,48)(31,65)(32,50)(33,67)(34,52)(35,69)(36,54)(37,57)(38,56)(39,59)(40,44)(41,61)(42,46)(43,121)(45,123)(47,125)(49,113)(51,115)(53,117)(55,119)(58,120)(60,122)(62,124)(64,126)(66,114)(68,116)(70,118)(72,221)(74,223)(76,211)(78,213)(80,215)(82,217)(84,219)(86,222)(88,224)(90,212)(92,214)(94,216)(96,218)(98,220)(99,164)(100,201)(101,166)(102,203)(103,168)(104,205)(105,156)(106,207)(107,158)(108,209)(109,160)(110,197)(111,162)(112,199)(127,149)(128,195)(129,151)(130,183)(131,153)(132,185)(133,141)(134,187)(135,143)(136,189)(137,145)(138,191)(139,147)(140,193)(142,172)(144,174)(146,176)(148,178)(150,180)(152,182)(154,170)(169,184)(171,186)(173,188)(175,190)(177,192)(179,194)(181,196), (1,42)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,173)(16,174)(17,175)(18,176)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,169)(26,170)(27,171)(28,172)(43,96)(44,97)(45,98)(46,85)(47,86)(48,87)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,81)(58,82)(59,83)(60,84)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80)(99,128)(100,129)(101,130)(102,131)(103,132)(104,133)(105,134)(106,135)(107,136)(108,137)(109,138)(110,139)(111,140)(112,127)(113,224)(114,211)(115,212)(116,213)(117,214)(118,215)(119,216)(120,217)(121,218)(122,219)(123,220)(124,221)(125,222)(126,223)(141,205)(142,206)(143,207)(144,208)(145,209)(146,210)(147,197)(148,198)(149,199)(150,200)(151,201)(152,202)(153,203)(154,204)(155,186)(156,187)(157,188)(158,189)(159,190)(160,191)(161,192)(162,193)(163,194)(164,195)(165,196)(166,183)(167,184)(168,185), (1,221)(2,222)(3,223)(4,224)(5,211)(6,212)(7,213)(8,214)(9,215)(10,216)(11,217)(12,218)(13,219)(14,220)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,125)(30,126)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,121)(40,122)(41,123)(42,124)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,57)(56,58)(71,98)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(127,179)(128,180)(129,181)(130,182)(131,169)(132,170)(133,171)(134,172)(135,173)(136,174)(137,175)(138,176)(139,177)(140,178)(141,186)(142,187)(143,188)(144,189)(145,190)(146,191)(147,192)(148,193)(149,194)(150,195)(151,196)(152,183)(153,184)(154,185)(155,205)(156,206)(157,207)(158,208)(159,209)(160,210)(161,197)(162,198)(163,199)(164,200)(165,201)(166,202)(167,203)(168,204), (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), (1,185,124,204)(2,167,125,153)(3,183,126,202)(4,165,113,151)(5,195,114,200)(6,163,115,149)(7,193,116,198)(8,161,117,147)(9,191,118,210)(10,159,119,145)(11,189,120,208)(12,157,121,143)(13,187,122,206)(14,155,123,141)(15,59,135,96)(16,82,136,56)(17,57,137,94)(18,80,138,54)(19,69,139,92)(20,78,140,52)(21,67,127,90)(22,76,128,50)(23,65,129,88)(24,74,130,48)(25,63,131,86)(26,72,132,46)(27,61,133,98)(28,84,134,44)(29,184,222,203)(30,166,223,152)(31,196,224,201)(32,164,211,150)(33,194,212,199)(34,162,213,148)(35,192,214,197)(36,160,215,146)(37,190,216,209)(38,158,217,144)(39,188,218,207)(40,156,219,142)(41,186,220,205)(42,168,221,154)(43,173,83,106)(45,171,71,104)(47,169,73,102)(49,181,75,100)(51,179,77,112)(53,177,79,110)(55,175,81,108)(58,107,95,174)(60,105,97,172)(62,103,85,170)(64,101,87,182)(66,99,89,180)(68,111,91,178)(70,109,93,176) );
G=PermutationGroup([(1,46),(2,47),(3,48),(4,49),(5,50),(6,51),(7,52),(8,53),(9,54),(10,55),(11,56),(12,43),(13,44),(14,45),(15,188),(16,189),(17,190),(18,191),(19,192),(20,193),(21,194),(22,195),(23,196),(24,183),(25,184),(26,185),(27,186),(28,187),(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),(42,85),(57,216),(58,217),(59,218),(60,219),(61,220),(62,221),(63,222),(64,223),(65,224),(66,211),(67,212),(68,213),(69,214),(70,215),(71,123),(72,124),(73,125),(74,126),(75,113),(76,114),(77,115),(78,116),(79,117),(80,118),(81,119),(82,120),(83,121),(84,122),(99,150),(100,151),(101,152),(102,153),(103,154),(104,141),(105,142),(106,143),(107,144),(108,145),(109,146),(110,147),(111,148),(112,149),(127,199),(128,200),(129,201),(130,202),(131,203),(132,204),(133,205),(134,206),(135,207),(136,208),(137,209),(138,210),(139,197),(140,198),(155,171),(156,172),(157,173),(158,174),(159,175),(160,176),(161,177),(162,178),(163,179),(164,180),(165,181),(166,182),(167,169),(168,170)], [(1,85),(2,73),(3,87),(4,75),(5,89),(6,77),(7,91),(8,79),(9,93),(10,81),(11,95),(12,83),(13,97),(14,71),(15,157),(16,208),(17,159),(18,210),(19,161),(20,198),(21,163),(22,200),(23,165),(24,202),(25,167),(26,204),(27,155),(28,206),(29,63),(30,48),(31,65),(32,50),(33,67),(34,52),(35,69),(36,54),(37,57),(38,56),(39,59),(40,44),(41,61),(42,46),(43,121),(45,123),(47,125),(49,113),(51,115),(53,117),(55,119),(58,120),(60,122),(62,124),(64,126),(66,114),(68,116),(70,118),(72,221),(74,223),(76,211),(78,213),(80,215),(82,217),(84,219),(86,222),(88,224),(90,212),(92,214),(94,216),(96,218),(98,220),(99,164),(100,201),(101,166),(102,203),(103,168),(104,205),(105,156),(106,207),(107,158),(108,209),(109,160),(110,197),(111,162),(112,199),(127,149),(128,195),(129,151),(130,183),(131,153),(132,185),(133,141),(134,187),(135,143),(136,189),(137,145),(138,191),(139,147),(140,193),(142,172),(144,174),(146,176),(148,178),(150,180),(152,182),(154,170),(169,184),(171,186),(173,188),(175,190),(177,192),(179,194),(181,196)], [(1,42),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,173),(16,174),(17,175),(18,176),(19,177),(20,178),(21,179),(22,180),(23,181),(24,182),(25,169),(26,170),(27,171),(28,172),(43,96),(44,97),(45,98),(46,85),(47,86),(48,87),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95),(57,81),(58,82),(59,83),(60,84),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80),(99,128),(100,129),(101,130),(102,131),(103,132),(104,133),(105,134),(106,135),(107,136),(108,137),(109,138),(110,139),(111,140),(112,127),(113,224),(114,211),(115,212),(116,213),(117,214),(118,215),(119,216),(120,217),(121,218),(122,219),(123,220),(124,221),(125,222),(126,223),(141,205),(142,206),(143,207),(144,208),(145,209),(146,210),(147,197),(148,198),(149,199),(150,200),(151,201),(152,202),(153,203),(154,204),(155,186),(156,187),(157,188),(158,189),(159,190),(160,191),(161,192),(162,193),(163,194),(164,195),(165,196),(166,183),(167,184),(168,185)], [(1,221),(2,222),(3,223),(4,224),(5,211),(6,212),(7,213),(8,214),(9,215),(10,216),(11,217),(12,218),(13,219),(14,220),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,125),(30,126),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,121),(40,122),(41,123),(42,124),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,57),(56,58),(71,98),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(127,179),(128,180),(129,181),(130,182),(131,169),(132,170),(133,171),(134,172),(135,173),(136,174),(137,175),(138,176),(139,177),(140,178),(141,186),(142,187),(143,188),(144,189),(145,190),(146,191),(147,192),(148,193),(149,194),(150,195),(151,196),(152,183),(153,184),(154,185),(155,205),(156,206),(157,207),(158,208),(159,209),(160,210),(161,197),(162,198),(163,199),(164,200),(165,201),(166,202),(167,203),(168,204)], [(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)], [(1,185,124,204),(2,167,125,153),(3,183,126,202),(4,165,113,151),(5,195,114,200),(6,163,115,149),(7,193,116,198),(8,161,117,147),(9,191,118,210),(10,159,119,145),(11,189,120,208),(12,157,121,143),(13,187,122,206),(14,155,123,141),(15,59,135,96),(16,82,136,56),(17,57,137,94),(18,80,138,54),(19,69,139,92),(20,78,140,52),(21,67,127,90),(22,76,128,50),(23,65,129,88),(24,74,130,48),(25,63,131,86),(26,72,132,46),(27,61,133,98),(28,84,134,44),(29,184,222,203),(30,166,223,152),(31,196,224,201),(32,164,211,150),(33,194,212,199),(34,162,213,148),(35,192,214,197),(36,160,215,146),(37,190,216,209),(38,158,217,144),(39,188,218,207),(40,156,219,142),(41,186,220,205),(42,168,221,154),(43,173,83,106),(45,171,71,104),(47,169,73,102),(49,181,75,100),(51,179,77,112),(53,177,79,110),(55,175,81,108),(58,107,95,174),(60,105,97,172),(62,103,85,170),(64,101,87,182),(66,99,89,180),(68,111,91,178),(70,109,93,176)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 25 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 15 | 9 | 0 | 0 | 0 | 0 |
| 20 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,22,22,0,0,0,0,25,0,0,0,0,0,0,0,0,28,0,0,0,0,28,0],[15,20,0,0,0,0,9,14,0,0,0,0,0,0,28,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,17,0] >;
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | ··· | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | D4⋊2D7 |
| kernel | C2×C23.18D14 | C2×Dic7⋊C4 | C23.18D14 | C2×C23.D7 | C23×Dic7 | D4×C2×C14 | C22×C14 | C22×D4 | C2×C14 | C22×C4 | C2×D4 | C24 | C23 | C22 |
| # reps | 1 | 2 | 8 | 3 | 1 | 1 | 4 | 3 | 8 | 3 | 12 | 6 | 24 | 12 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{18}D_{14} % in TeX
G:=Group("C2xC2^3.18D14"); // GroupNames label
G:=SmallGroup(448,1249);
// by ID
G=gap.SmallGroup(448,1249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀