metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.42D28, C24.45D14, C23.50(C4×D7), (C22×Dic7)⋊8C4, (C22×C14).61D4, (C22×C4).23D14, C22.41(C2×D28), C23.50(C7⋊D4), C7⋊2(C23.34D4), (C23×Dic7).3C2, C14.C42⋊10C2, C22.22(D14⋊C4), (C22×C28).22C22, (C23×C14).26C22, C23.276(C22×D7), C14.25(C42⋊C2), C22.42(D4⋊2D7), (C22×C14).318C23, C2.3(C22.D28), C2.1(C23.18D14), C14.70(C22.D4), C2.12(C23.11D14), (C22×Dic7).183C22, C2.7(C2×D14⋊C4), (C2×C22⋊C4).5D7, C22.122(C2×C4×D7), (C2×C14).149(C2×D4), (C14×C22⋊C4).6C2, C14.34(C2×C22⋊C4), C22.46(C2×C7⋊D4), (C2×C23.D7).5C2, (C22×C14).46(C2×C4), (C2×Dic7).93(C2×C4), (C2×C14).139(C4○D4), (C2×C14).13(C22⋊C4), (C2×C14).104(C22×C4), SmallGroup(448,477)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.42D28
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=b, f2=bcd, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce13 >
Subgroups: 852 in 218 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C2×Dic7, C2×Dic7, C2×C28, C22×C14, C22×C14, C22×C14, C23.34D4, C23.D7, C7×C22⋊C4, C22×Dic7, C22×Dic7, C22×C28, C23×C14, C14.C42, C2×C23.D7, C14×C22⋊C4, C23×Dic7, C23.42D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, C4○D4, D14, C2×C22⋊C4, C42⋊C2, C22.D4, C4×D7, D28, C7⋊D4, C22×D7, C23.34D4, D14⋊C4, C2×C4×D7, C2×D28, D4⋊2D7, C2×C7⋊D4, C23.11D14, C22.D28, C2×D14⋊C4, C23.18D14, C23.42D28
►On 224 points
Generators in S224
(1 101)(2 183)(3 103)(4 185)(5 105)(6 187)(7 107)(8 189)(9 109)(10 191)(11 111)(12 193)(13 85)(14 195)(15 87)(16 169)(17 89)(18 171)(19 91)(20 173)(21 93)(22 175)(23 95)(24 177)(25 97)(26 179)(27 99)(28 181)(29 192)(30 112)(31 194)(32 86)(33 196)(34 88)(35 170)(36 90)(37 172)(38 92)(39 174)(40 94)(41 176)(42 96)(43 178)(44 98)(45 180)(46 100)(47 182)(48 102)(49 184)(50 104)(51 186)(52 106)(53 188)(54 108)(55 190)(56 110)(57 139)(58 144)(59 113)(60 146)(61 115)(62 148)(63 117)(64 150)(65 119)(66 152)(67 121)(68 154)(69 123)(70 156)(71 125)(72 158)(73 127)(74 160)(75 129)(76 162)(77 131)(78 164)(79 133)(80 166)(81 135)(82 168)(83 137)(84 142)(114 214)(116 216)(118 218)(120 220)(122 222)(124 224)(126 198)(128 200)(130 202)(132 204)(134 206)(136 208)(138 210)(140 212)(141 209)(143 211)(145 213)(147 215)(149 217)(151 219)(153 221)(155 223)(157 197)(159 199)(161 201)(163 203)(165 205)(167 207)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 85)(28 86)(29 178)(30 179)(31 180)(32 181)(33 182)(34 183)(35 184)(36 185)(37 186)(38 187)(39 188)(40 189)(41 190)(42 191)(43 192)(44 193)(45 194)(46 195)(47 196)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 125)(58 126)(59 127)(60 128)(61 129)(62 130)(63 131)(64 132)(65 133)(66 134)(67 135)(68 136)(69 137)(70 138)(71 139)(72 140)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 121)(82 122)(83 123)(84 124)(141 223)(142 224)(143 197)(144 198)(145 199)(146 200)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(153 207)(154 208)(155 209)(156 210)(157 211)(158 212)(159 213)(160 214)(161 215)(162 216)(163 217)(164 218)(165 219)(166 220)(167 221)(168 222)
(1 47)(2 48)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(57 211)(58 212)(59 213)(60 214)(61 215)(62 216)(63 217)(64 218)(65 219)(66 220)(67 221)(68 222)(69 223)(70 224)(71 197)(72 198)(73 199)(74 200)(75 201)(76 202)(77 203)(78 204)(79 205)(80 206)(81 207)(82 208)(83 209)(84 210)(85 194)(86 195)(87 196)(88 169)(89 170)(90 171)(91 172)(92 173)(93 174)(94 175)(95 176)(96 177)(97 178)(98 179)(99 180)(100 181)(101 182)(102 183)(103 184)(104 185)(105 186)(106 187)(107 188)(108 189)(109 190)(110 191)(111 192)(112 193)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(121 153)(122 154)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 161)(130 162)(131 163)(132 164)(133 165)(134 166)(135 167)(136 168)(137 141)(138 142)(139 143)(140 144)
(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)
(1 160 182 74)(2 199 183 127)(3 158 184 72)(4 197 185 125)(5 156 186 70)(6 223 187 123)(7 154 188 68)(8 221 189 121)(9 152 190 66)(10 219 191 119)(11 150 192 64)(12 217 193 117)(13 148 194 62)(14 215 195 115)(15 146 196 60)(16 213 169 113)(17 144 170 58)(18 211 171 139)(19 142 172 84)(20 209 173 137)(21 168 174 82)(22 207 175 135)(23 166 176 80)(24 205 177 133)(25 164 178 78)(26 203 179 131)(27 162 180 76)(28 201 181 129)(29 118 111 218)(30 63 112 149)(31 116 85 216)(32 61 86 147)(33 114 87 214)(34 59 88 145)(35 140 89 212)(36 57 90 143)(37 138 91 210)(38 83 92 141)(39 136 93 208)(40 81 94 167)(41 134 95 206)(42 79 96 165)(43 132 97 204)(44 77 98 163)(45 130 99 202)(46 75 100 161)(47 128 101 200)(48 73 102 159)(49 126 103 198)(50 71 104 157)(51 124 105 224)(52 69 106 155)(53 122 107 222)(54 67 108 153)(55 120 109 220)(56 65 110 151)
G:=sub<Sym(224)| (1,101)(2,183)(3,103)(4,185)(5,105)(6,187)(7,107)(8,189)(9,109)(10,191)(11,111)(12,193)(13,85)(14,195)(15,87)(16,169)(17,89)(18,171)(19,91)(20,173)(21,93)(22,175)(23,95)(24,177)(25,97)(26,179)(27,99)(28,181)(29,192)(30,112)(31,194)(32,86)(33,196)(34,88)(35,170)(36,90)(37,172)(38,92)(39,174)(40,94)(41,176)(42,96)(43,178)(44,98)(45,180)(46,100)(47,182)(48,102)(49,184)(50,104)(51,186)(52,106)(53,188)(54,108)(55,190)(56,110)(57,139)(58,144)(59,113)(60,146)(61,115)(62,148)(63,117)(64,150)(65,119)(66,152)(67,121)(68,154)(69,123)(70,156)(71,125)(72,158)(73,127)(74,160)(75,129)(76,162)(77,131)(78,164)(79,133)(80,166)(81,135)(82,168)(83,137)(84,142)(114,214)(116,216)(118,218)(120,220)(122,222)(124,224)(126,198)(128,200)(130,202)(132,204)(134,206)(136,208)(138,210)(140,212)(141,209)(143,211)(145,213)(147,215)(149,217)(151,219)(153,221)(155,223)(157,197)(159,199)(161,201)(163,203)(165,205)(167,207), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,85)(28,86)(29,178)(30,179)(31,180)(32,181)(33,182)(34,183)(35,184)(36,185)(37,186)(38,187)(39,188)(40,189)(41,190)(42,191)(43,192)(44,193)(45,194)(46,195)(47,196)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,125)(58,126)(59,127)(60,128)(61,129)(62,130)(63,131)(64,132)(65,133)(66,134)(67,135)(68,136)(69,137)(70,138)(71,139)(72,140)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(141,223)(142,224)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216)(163,217)(164,218)(165,219)(166,220)(167,221)(168,222), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(57,211)(58,212)(59,213)(60,214)(61,215)(62,216)(63,217)(64,218)(65,219)(66,220)(67,221)(68,222)(69,223)(70,224)(71,197)(72,198)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,194)(86,195)(87,196)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(109,190)(110,191)(111,192)(112,193)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,141)(138,142)(139,143)(140,144), (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), (1,160,182,74)(2,199,183,127)(3,158,184,72)(4,197,185,125)(5,156,186,70)(6,223,187,123)(7,154,188,68)(8,221,189,121)(9,152,190,66)(10,219,191,119)(11,150,192,64)(12,217,193,117)(13,148,194,62)(14,215,195,115)(15,146,196,60)(16,213,169,113)(17,144,170,58)(18,211,171,139)(19,142,172,84)(20,209,173,137)(21,168,174,82)(22,207,175,135)(23,166,176,80)(24,205,177,133)(25,164,178,78)(26,203,179,131)(27,162,180,76)(28,201,181,129)(29,118,111,218)(30,63,112,149)(31,116,85,216)(32,61,86,147)(33,114,87,214)(34,59,88,145)(35,140,89,212)(36,57,90,143)(37,138,91,210)(38,83,92,141)(39,136,93,208)(40,81,94,167)(41,134,95,206)(42,79,96,165)(43,132,97,204)(44,77,98,163)(45,130,99,202)(46,75,100,161)(47,128,101,200)(48,73,102,159)(49,126,103,198)(50,71,104,157)(51,124,105,224)(52,69,106,155)(53,122,107,222)(54,67,108,153)(55,120,109,220)(56,65,110,151)>;
G:=Group( (1,101)(2,183)(3,103)(4,185)(5,105)(6,187)(7,107)(8,189)(9,109)(10,191)(11,111)(12,193)(13,85)(14,195)(15,87)(16,169)(17,89)(18,171)(19,91)(20,173)(21,93)(22,175)(23,95)(24,177)(25,97)(26,179)(27,99)(28,181)(29,192)(30,112)(31,194)(32,86)(33,196)(34,88)(35,170)(36,90)(37,172)(38,92)(39,174)(40,94)(41,176)(42,96)(43,178)(44,98)(45,180)(46,100)(47,182)(48,102)(49,184)(50,104)(51,186)(52,106)(53,188)(54,108)(55,190)(56,110)(57,139)(58,144)(59,113)(60,146)(61,115)(62,148)(63,117)(64,150)(65,119)(66,152)(67,121)(68,154)(69,123)(70,156)(71,125)(72,158)(73,127)(74,160)(75,129)(76,162)(77,131)(78,164)(79,133)(80,166)(81,135)(82,168)(83,137)(84,142)(114,214)(116,216)(118,218)(120,220)(122,222)(124,224)(126,198)(128,200)(130,202)(132,204)(134,206)(136,208)(138,210)(140,212)(141,209)(143,211)(145,213)(147,215)(149,217)(151,219)(153,221)(155,223)(157,197)(159,199)(161,201)(163,203)(165,205)(167,207), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,85)(28,86)(29,178)(30,179)(31,180)(32,181)(33,182)(34,183)(35,184)(36,185)(37,186)(38,187)(39,188)(40,189)(41,190)(42,191)(43,192)(44,193)(45,194)(46,195)(47,196)(48,169)(49,170)(50,171)(51,172)(52,173)(53,174)(54,175)(55,176)(56,177)(57,125)(58,126)(59,127)(60,128)(61,129)(62,130)(63,131)(64,132)(65,133)(66,134)(67,135)(68,136)(69,137)(70,138)(71,139)(72,140)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(141,223)(142,224)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216)(163,217)(164,218)(165,219)(166,220)(167,221)(168,222), (1,47)(2,48)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(57,211)(58,212)(59,213)(60,214)(61,215)(62,216)(63,217)(64,218)(65,219)(66,220)(67,221)(68,222)(69,223)(70,224)(71,197)(72,198)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,194)(86,195)(87,196)(88,169)(89,170)(90,171)(91,172)(92,173)(93,174)(94,175)(95,176)(96,177)(97,178)(98,179)(99,180)(100,181)(101,182)(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(109,190)(110,191)(111,192)(112,193)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,141)(138,142)(139,143)(140,144), (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), (1,160,182,74)(2,199,183,127)(3,158,184,72)(4,197,185,125)(5,156,186,70)(6,223,187,123)(7,154,188,68)(8,221,189,121)(9,152,190,66)(10,219,191,119)(11,150,192,64)(12,217,193,117)(13,148,194,62)(14,215,195,115)(15,146,196,60)(16,213,169,113)(17,144,170,58)(18,211,171,139)(19,142,172,84)(20,209,173,137)(21,168,174,82)(22,207,175,135)(23,166,176,80)(24,205,177,133)(25,164,178,78)(26,203,179,131)(27,162,180,76)(28,201,181,129)(29,118,111,218)(30,63,112,149)(31,116,85,216)(32,61,86,147)(33,114,87,214)(34,59,88,145)(35,140,89,212)(36,57,90,143)(37,138,91,210)(38,83,92,141)(39,136,93,208)(40,81,94,167)(41,134,95,206)(42,79,96,165)(43,132,97,204)(44,77,98,163)(45,130,99,202)(46,75,100,161)(47,128,101,200)(48,73,102,159)(49,126,103,198)(50,71,104,157)(51,124,105,224)(52,69,106,155)(53,122,107,222)(54,67,108,153)(55,120,109,220)(56,65,110,151) );
G=PermutationGroup([[(1,101),(2,183),(3,103),(4,185),(5,105),(6,187),(7,107),(8,189),(9,109),(10,191),(11,111),(12,193),(13,85),(14,195),(15,87),(16,169),(17,89),(18,171),(19,91),(20,173),(21,93),(22,175),(23,95),(24,177),(25,97),(26,179),(27,99),(28,181),(29,192),(30,112),(31,194),(32,86),(33,196),(34,88),(35,170),(36,90),(37,172),(38,92),(39,174),(40,94),(41,176),(42,96),(43,178),(44,98),(45,180),(46,100),(47,182),(48,102),(49,184),(50,104),(51,186),(52,106),(53,188),(54,108),(55,190),(56,110),(57,139),(58,144),(59,113),(60,146),(61,115),(62,148),(63,117),(64,150),(65,119),(66,152),(67,121),(68,154),(69,123),(70,156),(71,125),(72,158),(73,127),(74,160),(75,129),(76,162),(77,131),(78,164),(79,133),(80,166),(81,135),(82,168),(83,137),(84,142),(114,214),(116,216),(118,218),(120,220),(122,222),(124,224),(126,198),(128,200),(130,202),(132,204),(134,206),(136,208),(138,210),(140,212),(141,209),(143,211),(145,213),(147,215),(149,217),(151,219),(153,221),(155,223),(157,197),(159,199),(161,201),(163,203),(165,205),(167,207)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,85),(28,86),(29,178),(30,179),(31,180),(32,181),(33,182),(34,183),(35,184),(36,185),(37,186),(38,187),(39,188),(40,189),(41,190),(42,191),(43,192),(44,193),(45,194),(46,195),(47,196),(48,169),(49,170),(50,171),(51,172),(52,173),(53,174),(54,175),(55,176),(56,177),(57,125),(58,126),(59,127),(60,128),(61,129),(62,130),(63,131),(64,132),(65,133),(66,134),(67,135),(68,136),(69,137),(70,138),(71,139),(72,140),(73,113),(74,114),(75,115),(76,116),(77,117),(78,118),(79,119),(80,120),(81,121),(82,122),(83,123),(84,124),(141,223),(142,224),(143,197),(144,198),(145,199),(146,200),(147,201),(148,202),(149,203),(150,204),(151,205),(152,206),(153,207),(154,208),(155,209),(156,210),(157,211),(158,212),(159,213),(160,214),(161,215),(162,216),(163,217),(164,218),(165,219),(166,220),(167,221),(168,222)], [(1,47),(2,48),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(57,211),(58,212),(59,213),(60,214),(61,215),(62,216),(63,217),(64,218),(65,219),(66,220),(67,221),(68,222),(69,223),(70,224),(71,197),(72,198),(73,199),(74,200),(75,201),(76,202),(77,203),(78,204),(79,205),(80,206),(81,207),(82,208),(83,209),(84,210),(85,194),(86,195),(87,196),(88,169),(89,170),(90,171),(91,172),(92,173),(93,174),(94,175),(95,176),(96,177),(97,178),(98,179),(99,180),(100,181),(101,182),(102,183),(103,184),(104,185),(105,186),(106,187),(107,188),(108,189),(109,190),(110,191),(111,192),(112,193),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152),(121,153),(122,154),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,161),(130,162),(131,163),(132,164),(133,165),(134,166),(135,167),(136,168),(137,141),(138,142),(139,143),(140,144)], [(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)], [(1,160,182,74),(2,199,183,127),(3,158,184,72),(4,197,185,125),(5,156,186,70),(6,223,187,123),(7,154,188,68),(8,221,189,121),(9,152,190,66),(10,219,191,119),(11,150,192,64),(12,217,193,117),(13,148,194,62),(14,215,195,115),(15,146,196,60),(16,213,169,113),(17,144,170,58),(18,211,171,139),(19,142,172,84),(20,209,173,137),(21,168,174,82),(22,207,175,135),(23,166,176,80),(24,205,177,133),(25,164,178,78),(26,203,179,131),(27,162,180,76),(28,201,181,129),(29,118,111,218),(30,63,112,149),(31,116,85,216),(32,61,86,147),(33,114,87,214),(34,59,88,145),(35,140,89,212),(36,57,90,143),(37,138,91,210),(38,83,92,141),(39,136,93,208),(40,81,94,167),(41,134,95,206),(42,79,96,165),(43,132,97,204),(44,77,98,163),(45,130,99,202),(46,75,100,161),(47,128,101,200),(48,73,102,159),(49,126,103,198),(50,71,104,157),(51,124,105,224),(52,69,106,155),(53,122,107,222),(54,67,108,153),(55,120,109,220),(56,65,110,151)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | D4⋊2D7 |
| kernel | C23.42D28 | C14.C42 | C2×C23.D7 | C14×C22⋊C4 | C23×Dic7 | C22×Dic7 | C22×C14 | C2×C22⋊C4 | C2×C14 | C22×C4 | C24 | C23 | C23 | C23 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 3 | 8 | 6 | 3 | 12 | 12 | 12 | 12 |
Matrix representation of C23.42D28 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 28 | 28 | 0 | 0 |
| 0 | 0 | 0 | 22 | 7 |
| 0 | 0 | 0 | 22 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 26 | 7 |
| 0 | 0 | 0 | 11 | 3 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,1,0,0,0,0,2,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,1,28,0,0,0,0,28,0,0,0,0,0,22,22,0,0,0,7,3],[12,0,0,0,0,0,17,12,0,0,0,0,12,0,0,0,0,0,26,11,0,0,0,7,3] >;
C23.42D28 in GAP, Magma, Sage, TeX
C_2^3._{42}D_{28} % in TeX
G:=Group("C2^3.42D28"); // GroupNames label
G:=SmallGroup(448,477);
// by ID
G=gap.SmallGroup(448,477);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,422,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=b,f^2=b*c*d,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^13>;
// generators/relations