direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×D14⋊C4, C24.78D14, C23.64D28, (C23×C4)⋊2D7, (C23×C28)⋊3C2, (C23×D7)⋊7C4, (C2×C28)⋊12C23, D14⋊7(C22×C4), (C22×C4)⋊41D14, (D7×C24).3C2, C23.69(C4×D7), C2.3(C22×D28), C14.38(C23×C4), (C2×Dic7)⋊8C23, (C23×Dic7)⋊6C2, C22.76(C2×D28), (C2×C14).285C24, (C22×C28)⋊55C22, C14.131(C22×D4), (C22×C14).204D4, C22.42(C23×D7), C23.103(C7⋊D4), C23.335(C22×D7), (C23×C14).107C22, (C22×C14).414C23, (C22×Dic7)⋊46C22, (C23×D7).111C22, (C22×D7).236C23, C14⋊2(C2×C22⋊C4), C7⋊2(C22×C22⋊C4), C22.79(C2×C4×D7), C2.38(D7×C22×C4), (C2×C4)⋊10(C22×D7), (C2×C14)⋊6(C22⋊C4), C2.2(C22×C7⋊D4), (C2×C14).572(C2×D4), (C22×D7)⋊15(C2×C4), C22.101(C2×C7⋊D4), (C2×C14).158(C22×C4), (C22×C14).105(C2×C4), SmallGroup(448,1240)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3332 in 674 conjugacy classes, 247 normal (17 characteristic)
C1, C2 [×3], C2 [×12], C2 [×8], C4 [×8], C22, C22 [×34], C22 [×64], C7, C2×C4 [×4], C2×C4 [×28], C23 [×15], C23 [×84], D7 [×8], C14 [×3], C14 [×12], C22⋊C4 [×16], C22×C4 [×6], C22×C4 [×14], C24, C24 [×22], Dic7 [×4], C28 [×4], D14 [×8], D14 [×56], C2×C14, C2×C14 [×34], C2×C22⋊C4 [×12], C23×C4, C23×C4, C25, C2×Dic7 [×4], C2×Dic7 [×12], C2×C28 [×4], C2×C28 [×12], C22×D7 [×28], C22×D7 [×56], C22×C14 [×15], C22×C22⋊C4, D14⋊C4 [×16], C22×Dic7 [×6], C22×Dic7 [×4], C22×C28 [×6], C22×C28 [×4], C23×D7 [×14], C23×D7 [×8], C23×C14, C2×D14⋊C4 [×12], C23×Dic7, C23×C28, D7×C24, C22×D14⋊C4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], D7, C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, D14 [×7], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C4×D7 [×4], D28 [×4], C7⋊D4 [×4], C22×D7 [×7], C22×C22⋊C4, D14⋊C4 [×16], C2×C4×D7 [×6], C2×D28 [×6], C2×C7⋊D4 [×6], C23×D7, C2×D14⋊C4 [×12], D7×C22×C4, C22×D28, C22×C7⋊D4, C22×D14⋊C4
Generators and relations
G = < a,b,c,d,e | a2=b2=c14=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c7d >
►On 224 points
Generators in S224
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 122)(16 123)(17 124)(18 125)(19 126)(20 113)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 121)(29 158)(30 159)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 166)(38 167)(39 168)(40 155)(41 156)(42 157)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 194)(58 195)(59 196)(60 183)(61 184)(62 185)(63 186)(64 187)(65 188)(66 189)(67 190)(68 191)(69 192)(70 193)(71 176)(72 177)(73 178)(74 179)(75 180)(76 181)(77 182)(78 169)(79 170)(80 171)(81 172)(82 173)(83 174)(84 175)(85 214)(86 215)(87 216)(88 217)(89 218)(90 219)(91 220)(92 221)(93 222)(94 223)(95 224)(96 211)(97 212)(98 213)(99 205)(100 206)(101 207)(102 208)(103 209)(104 210)(105 197)(106 198)(107 199)(108 200)(109 201)(110 202)(111 203)(112 204)
(1 82)(2 83)(3 84)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 69)(16 70)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 111)(30 112)(31 99)(32 100)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(41 109)(42 110)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)(113 183)(114 184)(115 185)(116 186)(117 187)(118 188)(119 189)(120 190)(121 191)(122 192)(123 193)(124 194)(125 195)(126 196)(127 180)(128 181)(129 182)(130 169)(131 170)(132 171)(133 172)(134 173)(135 174)(136 175)(137 176)(138 177)(139 178)(140 179)(141 222)(142 223)(143 224)(144 211)(145 212)(146 213)(147 214)(148 215)(149 216)(150 217)(151 218)(152 219)(153 220)(154 221)(155 200)(156 201)(157 202)(158 203)(159 204)(160 205)(161 206)(162 207)(163 208)(164 209)(165 210)(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 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 60)(8 59)(9 58)(10 57)(11 70)(12 69)(13 68)(14 67)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 72)(23 71)(24 84)(25 83)(26 82)(27 81)(28 80)(29 90)(30 89)(31 88)(32 87)(33 86)(34 85)(35 98)(36 97)(37 96)(38 95)(39 94)(40 93)(41 92)(42 91)(43 102)(44 101)(45 100)(46 99)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 104)(56 103)(113 179)(114 178)(115 177)(116 176)(117 175)(118 174)(119 173)(120 172)(121 171)(122 170)(123 169)(124 182)(125 181)(126 180)(127 196)(128 195)(129 194)(130 193)(131 192)(132 191)(133 190)(134 189)(135 188)(136 187)(137 186)(138 185)(139 184)(140 183)(141 200)(142 199)(143 198)(144 197)(145 210)(146 209)(147 208)(148 207)(149 206)(150 205)(151 204)(152 203)(153 202)(154 201)(155 222)(156 221)(157 220)(158 219)(159 218)(160 217)(161 216)(162 215)(163 214)(164 213)(165 212)(166 211)(167 224)(168 223)
(1 221 27 209)(2 222 28 210)(3 223 15 197)(4 224 16 198)(5 211 17 199)(6 212 18 200)(7 213 19 201)(8 214 20 202)(9 215 21 203)(10 216 22 204)(11 217 23 205)(12 218 24 206)(13 219 25 207)(14 220 26 208)(29 181 44 184)(30 182 45 185)(31 169 46 186)(32 170 47 187)(33 171 48 188)(34 172 49 189)(35 173 50 190)(36 174 51 191)(37 175 52 192)(38 176 53 193)(39 177 54 194)(40 178 55 195)(41 179 56 196)(42 180 43 183)(57 168 72 144)(58 155 73 145)(59 156 74 146)(60 157 75 147)(61 158 76 148)(62 159 77 149)(63 160 78 150)(64 161 79 151)(65 162 80 152)(66 163 81 153)(67 164 82 154)(68 165 83 141)(69 166 84 142)(70 167 71 143)(85 113 110 127)(86 114 111 128)(87 115 112 129)(88 116 99 130)(89 117 100 131)(90 118 101 132)(91 119 102 133)(92 120 103 134)(93 121 104 135)(94 122 105 136)(95 123 106 137)(96 124 107 138)(97 125 108 139)(98 126 109 140)
G:=sub<Sym(224)| (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,122)(16,123)(17,124)(18,125)(19,126)(20,113)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,121)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,155)(41,156)(42,157)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,194)(58,195)(59,196)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,214)(86,215)(87,216)(88,217)(89,218)(90,219)(91,220)(92,221)(93,222)(94,223)(95,224)(96,211)(97,212)(98,213)(99,205)(100,206)(101,207)(102,208)(103,209)(104,210)(105,197)(106,198)(107,199)(108,200)(109,201)(110,202)(111,203)(112,204), (1,82)(2,83)(3,84)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,69)(16,70)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,111)(30,112)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(127,180)(128,181)(129,182)(130,169)(131,170)(132,171)(133,172)(134,173)(135,174)(136,175)(137,176)(138,177)(139,178)(140,179)(141,222)(142,223)(143,224)(144,211)(145,212)(146,213)(147,214)(148,215)(149,216)(150,217)(151,218)(152,219)(153,220)(154,221)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(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), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,57)(11,70)(12,69)(13,68)(14,67)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,72)(23,71)(24,84)(25,83)(26,82)(27,81)(28,80)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,102)(44,101)(45,100)(46,99)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(113,179)(114,178)(115,177)(116,176)(117,175)(118,174)(119,173)(120,172)(121,171)(122,170)(123,169)(124,182)(125,181)(126,180)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,200)(142,199)(143,198)(144,197)(145,210)(146,209)(147,208)(148,207)(149,206)(150,205)(151,204)(152,203)(153,202)(154,201)(155,222)(156,221)(157,220)(158,219)(159,218)(160,217)(161,216)(162,215)(163,214)(164,213)(165,212)(166,211)(167,224)(168,223), (1,221,27,209)(2,222,28,210)(3,223,15,197)(4,224,16,198)(5,211,17,199)(6,212,18,200)(7,213,19,201)(8,214,20,202)(9,215,21,203)(10,216,22,204)(11,217,23,205)(12,218,24,206)(13,219,25,207)(14,220,26,208)(29,181,44,184)(30,182,45,185)(31,169,46,186)(32,170,47,187)(33,171,48,188)(34,172,49,189)(35,173,50,190)(36,174,51,191)(37,175,52,192)(38,176,53,193)(39,177,54,194)(40,178,55,195)(41,179,56,196)(42,180,43,183)(57,168,72,144)(58,155,73,145)(59,156,74,146)(60,157,75,147)(61,158,76,148)(62,159,77,149)(63,160,78,150)(64,161,79,151)(65,162,80,152)(66,163,81,153)(67,164,82,154)(68,165,83,141)(69,166,84,142)(70,167,71,143)(85,113,110,127)(86,114,111,128)(87,115,112,129)(88,116,99,130)(89,117,100,131)(90,118,101,132)(91,119,102,133)(92,120,103,134)(93,121,104,135)(94,122,105,136)(95,123,106,137)(96,124,107,138)(97,125,108,139)(98,126,109,140)>;
G:=Group( (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,122)(16,123)(17,124)(18,125)(19,126)(20,113)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,121)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,166)(38,167)(39,168)(40,155)(41,156)(42,157)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,194)(58,195)(59,196)(60,183)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,176)(72,177)(73,178)(74,179)(75,180)(76,181)(77,182)(78,169)(79,170)(80,171)(81,172)(82,173)(83,174)(84,175)(85,214)(86,215)(87,216)(88,217)(89,218)(90,219)(91,220)(92,221)(93,222)(94,223)(95,224)(96,211)(97,212)(98,213)(99,205)(100,206)(101,207)(102,208)(103,209)(104,210)(105,197)(106,198)(107,199)(108,200)(109,201)(110,202)(111,203)(112,204), (1,82)(2,83)(3,84)(4,71)(5,72)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,80)(14,81)(15,69)(16,70)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,111)(30,112)(31,99)(32,100)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(41,109)(42,110)(43,85)(44,86)(45,87)(46,88)(47,89)(48,90)(49,91)(50,92)(51,93)(52,94)(53,95)(54,96)(55,97)(56,98)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(127,180)(128,181)(129,182)(130,169)(131,170)(132,171)(133,172)(134,173)(135,174)(136,175)(137,176)(138,177)(139,178)(140,179)(141,222)(142,223)(143,224)(144,211)(145,212)(146,213)(147,214)(148,215)(149,216)(150,217)(151,218)(152,219)(153,220)(154,221)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)(162,207)(163,208)(164,209)(165,210)(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), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,60)(8,59)(9,58)(10,57)(11,70)(12,69)(13,68)(14,67)(15,79)(16,78)(17,77)(18,76)(19,75)(20,74)(21,73)(22,72)(23,71)(24,84)(25,83)(26,82)(27,81)(28,80)(29,90)(30,89)(31,88)(32,87)(33,86)(34,85)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,102)(44,101)(45,100)(46,99)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(113,179)(114,178)(115,177)(116,176)(117,175)(118,174)(119,173)(120,172)(121,171)(122,170)(123,169)(124,182)(125,181)(126,180)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,200)(142,199)(143,198)(144,197)(145,210)(146,209)(147,208)(148,207)(149,206)(150,205)(151,204)(152,203)(153,202)(154,201)(155,222)(156,221)(157,220)(158,219)(159,218)(160,217)(161,216)(162,215)(163,214)(164,213)(165,212)(166,211)(167,224)(168,223), (1,221,27,209)(2,222,28,210)(3,223,15,197)(4,224,16,198)(5,211,17,199)(6,212,18,200)(7,213,19,201)(8,214,20,202)(9,215,21,203)(10,216,22,204)(11,217,23,205)(12,218,24,206)(13,219,25,207)(14,220,26,208)(29,181,44,184)(30,182,45,185)(31,169,46,186)(32,170,47,187)(33,171,48,188)(34,172,49,189)(35,173,50,190)(36,174,51,191)(37,175,52,192)(38,176,53,193)(39,177,54,194)(40,178,55,195)(41,179,56,196)(42,180,43,183)(57,168,72,144)(58,155,73,145)(59,156,74,146)(60,157,75,147)(61,158,76,148)(62,159,77,149)(63,160,78,150)(64,161,79,151)(65,162,80,152)(66,163,81,153)(67,164,82,154)(68,165,83,141)(69,166,84,142)(70,167,71,143)(85,113,110,127)(86,114,111,128)(87,115,112,129)(88,116,99,130)(89,117,100,131)(90,118,101,132)(91,119,102,133)(92,120,103,134)(93,121,104,135)(94,122,105,136)(95,123,106,137)(96,124,107,138)(97,125,108,139)(98,126,109,140) );
G=PermutationGroup([(1,134),(2,135),(3,136),(4,137),(5,138),(6,139),(7,140),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,122),(16,123),(17,124),(18,125),(19,126),(20,113),(21,114),(22,115),(23,116),(24,117),(25,118),(26,119),(27,120),(28,121),(29,158),(30,159),(31,160),(32,161),(33,162),(34,163),(35,164),(36,165),(37,166),(38,167),(39,168),(40,155),(41,156),(42,157),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,141),(52,142),(53,143),(54,144),(55,145),(56,146),(57,194),(58,195),(59,196),(60,183),(61,184),(62,185),(63,186),(64,187),(65,188),(66,189),(67,190),(68,191),(69,192),(70,193),(71,176),(72,177),(73,178),(74,179),(75,180),(76,181),(77,182),(78,169),(79,170),(80,171),(81,172),(82,173),(83,174),(84,175),(85,214),(86,215),(87,216),(88,217),(89,218),(90,219),(91,220),(92,221),(93,222),(94,223),(95,224),(96,211),(97,212),(98,213),(99,205),(100,206),(101,207),(102,208),(103,209),(104,210),(105,197),(106,198),(107,199),(108,200),(109,201),(110,202),(111,203),(112,204)], [(1,82),(2,83),(3,84),(4,71),(5,72),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,80),(14,81),(15,69),(16,70),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,111),(30,112),(31,99),(32,100),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(41,109),(42,110),(43,85),(44,86),(45,87),(46,88),(47,89),(48,90),(49,91),(50,92),(51,93),(52,94),(53,95),(54,96),(55,97),(56,98),(113,183),(114,184),(115,185),(116,186),(117,187),(118,188),(119,189),(120,190),(121,191),(122,192),(123,193),(124,194),(125,195),(126,196),(127,180),(128,181),(129,182),(130,169),(131,170),(132,171),(133,172),(134,173),(135,174),(136,175),(137,176),(138,177),(139,178),(140,179),(141,222),(142,223),(143,224),(144,211),(145,212),(146,213),(147,214),(148,215),(149,216),(150,217),(151,218),(152,219),(153,220),(154,221),(155,200),(156,201),(157,202),(158,203),(159,204),(160,205),(161,206),(162,207),(163,208),(164,209),(165,210),(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)], [(1,66),(2,65),(3,64),(4,63),(5,62),(6,61),(7,60),(8,59),(9,58),(10,57),(11,70),(12,69),(13,68),(14,67),(15,79),(16,78),(17,77),(18,76),(19,75),(20,74),(21,73),(22,72),(23,71),(24,84),(25,83),(26,82),(27,81),(28,80),(29,90),(30,89),(31,88),(32,87),(33,86),(34,85),(35,98),(36,97),(37,96),(38,95),(39,94),(40,93),(41,92),(42,91),(43,102),(44,101),(45,100),(46,99),(47,112),(48,111),(49,110),(50,109),(51,108),(52,107),(53,106),(54,105),(55,104),(56,103),(113,179),(114,178),(115,177),(116,176),(117,175),(118,174),(119,173),(120,172),(121,171),(122,170),(123,169),(124,182),(125,181),(126,180),(127,196),(128,195),(129,194),(130,193),(131,192),(132,191),(133,190),(134,189),(135,188),(136,187),(137,186),(138,185),(139,184),(140,183),(141,200),(142,199),(143,198),(144,197),(145,210),(146,209),(147,208),(148,207),(149,206),(150,205),(151,204),(152,203),(153,202),(154,201),(155,222),(156,221),(157,220),(158,219),(159,218),(160,217),(161,216),(162,215),(163,214),(164,213),(165,212),(166,211),(167,224),(168,223)], [(1,221,27,209),(2,222,28,210),(3,223,15,197),(4,224,16,198),(5,211,17,199),(6,212,18,200),(7,213,19,201),(8,214,20,202),(9,215,21,203),(10,216,22,204),(11,217,23,205),(12,218,24,206),(13,219,25,207),(14,220,26,208),(29,181,44,184),(30,182,45,185),(31,169,46,186),(32,170,47,187),(33,171,48,188),(34,172,49,189),(35,173,50,190),(36,174,51,191),(37,175,52,192),(38,176,53,193),(39,177,54,194),(40,178,55,195),(41,179,56,196),(42,180,43,183),(57,168,72,144),(58,155,73,145),(59,156,74,146),(60,157,75,147),(61,158,76,148),(62,159,77,149),(63,160,78,150),(64,161,79,151),(65,162,80,152),(66,163,81,153),(67,164,82,154),(68,165,83,141),(69,166,84,142),(70,167,71,143),(85,113,110,127),(86,114,111,128),(87,115,112,129),(88,116,99,130),(89,117,100,131),(90,118,101,132),(91,119,102,133),(92,120,103,134),(93,121,104,135),(94,122,105,136),(95,123,106,137),(96,124,107,138),(97,125,108,139),(98,126,109,140)])
Matrix representation ►G ⊆ GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 18 | 25 |
| 0 | 0 | 0 | 4 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 18 | 25 |
| 0 | 0 | 0 | 1 | 11 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 11 | 2 |
| 0 | 0 | 0 | 27 | 18 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,18,4,0,0,0,25,4],[1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,18,1,0,0,0,25,11],[1,0,0,0,0,0,17,0,0,0,0,0,1,0,0,0,0,0,11,27,0,0,0,2,18] >;
136 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 4A | ··· | 4H | 4I | ··· | 4P | 7A | 7B | 7C | 14A | ··· | 14AS | 28A | ··· | 28AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 14 | ··· | 14 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | D14 | D14 | C4×D7 | D28 | C7⋊D4 |
| kernel | C22×D14⋊C4 | C2×D14⋊C4 | C23×Dic7 | C23×C28 | D7×C24 | C23×D7 | C22×C14 | C23×C4 | C22×C4 | C24 | C23 | C23 | C23 |
| # reps | 1 | 12 | 1 | 1 | 1 | 16 | 8 | 3 | 18 | 3 | 24 | 24 | 24 |
In GAP, Magma, Sage, TeX
C_2^2\times D_{14}\rtimes C_4 % in TeX
G:=Group("C2^2xD14:C4"); // GroupNames label
G:=SmallGroup(448,1240);
// by ID
G=gap.SmallGroup(448,1240);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,1123,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^14=d^2=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^7*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀