metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.19D14, (D4×C14)⋊7C4, (C2×D4)⋊7Dic7, C28⋊3(C22⋊C4), (C2×Dic7)⋊12D4, (C2×C28).191D4, C14.102(C4×D4), C4⋊1(C23.D7), C2.19(D4×Dic7), (C22×D4).7D7, C2.5(C28⋊2D4), C2.4(C28⋊D4), C22.121(D4×D7), C14.36(C4⋊1D4), (C22×C4).353D14, C14.129(C4⋊D4), C23.10(C2×Dic7), C2.4(C28.17D4), C14.47(C4.4D4), (C23×C14).47C22, C7⋊4(C24.3C22), C23.306(C22×D7), C22.62(D4⋊2D7), (C22×C14).366C23, (C22×C28).199C22, C22.52(C22×Dic7), (C22×Dic7).197C22, (C2×C4×Dic7)⋊3C2, (D4×C2×C14).5C2, (C2×C4⋊Dic7)⋊35C2, (C2×C28).118(C2×C4), (C2×C14).555(C2×D4), C14.76(C2×C22⋊C4), (C2×C23.D7)⋊10C2, (C2×C4).50(C2×Dic7), C22.92(C2×C7⋊D4), C2.12(C2×C23.D7), (C2×C4).148(C7⋊D4), (C22×C14).73(C2×C4), (C2×C14).162(C4○D4), (C2×C14).197(C22×C4), SmallGroup(448,755)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.19D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=d, f2=db=bd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 916 in 258 conjugacy classes, 91 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C2×Dic7, C2×Dic7, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C24.3C22, C4×Dic7, C4⋊Dic7, C23.D7, C22×Dic7, C22×C28, D4×C14, D4×C14, C23×C14, C2×C4×Dic7, C2×C4⋊Dic7, C2×C23.D7, D4×C2×C14, C24.19D14
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, C4○D4, Dic7, D14, C2×C22⋊C4, C4×D4, C4⋊D4, C4.4D4, C4⋊1D4, C2×Dic7, C7⋊D4, C22×D7, C24.3C22, C23.D7, D4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, D4×Dic7, C28.17D4, C28⋊2D4, C28⋊D4, C2×C23.D7, C24.19D14
►On 224 points
Generators in S224
(1 100)(2 87)(3 102)(4 89)(5 104)(6 91)(7 106)(8 93)(9 108)(10 95)(11 110)(12 97)(13 112)(14 99)(15 86)(16 101)(17 88)(18 103)(19 90)(20 105)(21 92)(22 107)(23 94)(24 109)(25 96)(26 111)(27 98)(28 85)(29 167)(30 154)(31 141)(32 156)(33 143)(34 158)(35 145)(36 160)(37 147)(38 162)(39 149)(40 164)(41 151)(42 166)(43 153)(44 168)(45 155)(46 142)(47 157)(48 144)(49 159)(50 146)(51 161)(52 148)(53 163)(54 150)(55 165)(56 152)(57 123)(58 138)(59 125)(60 140)(61 127)(62 114)(63 129)(64 116)(65 131)(66 118)(67 133)(68 120)(69 135)(70 122)(71 137)(72 124)(73 139)(74 126)(75 113)(76 128)(77 115)(78 130)(79 117)(80 132)(81 119)(82 134)(83 121)(84 136)(169 216)(170 203)(171 218)(172 205)(173 220)(174 207)(175 222)(176 209)(177 224)(178 211)(179 198)(180 213)(181 200)(182 215)(183 202)(184 217)(185 204)(186 219)(187 206)(188 221)(189 208)(190 223)(191 210)(192 197)(193 212)(194 199)(195 214)(196 201)
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 83)(38 84)(39 57)(40 58)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 69)(52 70)(53 71)(54 72)(55 73)(56 74)(113 167)(114 168)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 147)(122 148)(123 149)(124 150)(125 151)(126 152)(127 153)(128 154)(129 155)(130 156)(131 157)(132 158)(133 159)(134 160)(135 161)(136 162)(137 163)(138 164)(139 165)(140 166)(169 202)(170 203)(171 204)(172 205)(173 206)(174 207)(175 208)(176 209)(177 210)(178 211)(179 212)(180 213)(181 214)(182 215)(183 216)(184 217)(185 218)(186 219)(187 220)(188 221)(189 222)(190 223)(191 224)(192 197)(193 198)(194 199)(195 200)(196 201)
(1 188)(2 189)(3 190)(4 191)(5 192)(6 193)(7 194)(8 195)(9 196)(10 169)(11 170)(12 171)(13 172)(14 173)(15 174)(16 175)(17 176)(18 177)(19 178)(20 179)(21 180)(22 181)(23 182)(24 183)(25 184)(26 185)(27 186)(28 187)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 141)(78 142)(79 143)(80 144)(81 145)(82 146)(83 147)(84 148)(85 206)(86 207)(87 208)(88 209)(89 210)(90 211)(91 212)(92 213)(93 214)(94 215)(95 216)(96 217)(97 218)(98 219)(99 220)(100 221)(101 222)(102 223)(103 224)(104 197)(105 198)(106 199)(107 200)(108 201)(109 202)(110 203)(111 204)(112 205)
(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 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 84 86 52)(2 69 87 37)(3 82 88 50)(4 67 89 35)(5 80 90 48)(6 65 91 33)(7 78 92 46)(8 63 93 31)(9 76 94 44)(10 61 95 29)(11 74 96 42)(12 59 97 55)(13 72 98 40)(14 57 99 53)(15 70 100 38)(16 83 101 51)(17 68 102 36)(18 81 103 49)(19 66 104 34)(20 79 105 47)(21 64 106 32)(22 77 107 45)(23 62 108 30)(24 75 109 43)(25 60 110 56)(26 73 111 41)(27 58 112 54)(28 71 85 39)(113 169 153 216)(114 182 154 201)(115 195 155 214)(116 180 156 199)(117 193 157 212)(118 178 158 197)(119 191 159 210)(120 176 160 223)(121 189 161 208)(122 174 162 221)(123 187 163 206)(124 172 164 219)(125 185 165 204)(126 170 166 217)(127 183 167 202)(128 196 168 215)(129 181 141 200)(130 194 142 213)(131 179 143 198)(132 192 144 211)(133 177 145 224)(134 190 146 209)(135 175 147 222)(136 188 148 207)(137 173 149 220)(138 186 150 205)(139 171 151 218)(140 184 152 203)
G:=sub<Sym(224)| (1,100)(2,87)(3,102)(4,89)(5,104)(6,91)(7,106)(8,93)(9,108)(10,95)(11,110)(12,97)(13,112)(14,99)(15,86)(16,101)(17,88)(18,103)(19,90)(20,105)(21,92)(22,107)(23,94)(24,109)(25,96)(26,111)(27,98)(28,85)(29,167)(30,154)(31,141)(32,156)(33,143)(34,158)(35,145)(36,160)(37,147)(38,162)(39,149)(40,164)(41,151)(42,166)(43,153)(44,168)(45,155)(46,142)(47,157)(48,144)(49,159)(50,146)(51,161)(52,148)(53,163)(54,150)(55,165)(56,152)(57,123)(58,138)(59,125)(60,140)(61,127)(62,114)(63,129)(64,116)(65,131)(66,118)(67,133)(68,120)(69,135)(70,122)(71,137)(72,124)(73,139)(74,126)(75,113)(76,128)(77,115)(78,130)(79,117)(80,132)(81,119)(82,134)(83,121)(84,136)(169,216)(170,203)(171,218)(172,205)(173,220)(174,207)(175,222)(176,209)(177,224)(178,211)(179,198)(180,213)(181,200)(182,215)(183,202)(184,217)(185,204)(186,219)(187,206)(188,221)(189,208)(190,223)(191,210)(192,197)(193,212)(194,199)(195,214)(196,201), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(113,167)(114,168)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,211)(179,212)(180,213)(181,214)(182,215)(183,216)(184,217)(185,218)(186,219)(187,220)(188,221)(189,222)(190,223)(191,224)(192,197)(193,198)(194,199)(195,200)(196,201), (1,188)(2,189)(3,190)(4,191)(5,192)(6,193)(7,194)(8,195)(9,196)(10,169)(11,170)(12,171)(13,172)(14,173)(15,174)(16,175)(17,176)(18,177)(19,178)(20,179)(21,180)(22,181)(23,182)(24,183)(25,184)(26,185)(27,186)(28,187)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,141)(78,142)(79,143)(80,144)(81,145)(82,146)(83,147)(84,148)(85,206)(86,207)(87,208)(88,209)(89,210)(90,211)(91,212)(92,213)(93,214)(94,215)(95,216)(96,217)(97,218)(98,219)(99,220)(100,221)(101,222)(102,223)(103,224)(104,197)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(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), (1,84,86,52)(2,69,87,37)(3,82,88,50)(4,67,89,35)(5,80,90,48)(6,65,91,33)(7,78,92,46)(8,63,93,31)(9,76,94,44)(10,61,95,29)(11,74,96,42)(12,59,97,55)(13,72,98,40)(14,57,99,53)(15,70,100,38)(16,83,101,51)(17,68,102,36)(18,81,103,49)(19,66,104,34)(20,79,105,47)(21,64,106,32)(22,77,107,45)(23,62,108,30)(24,75,109,43)(25,60,110,56)(26,73,111,41)(27,58,112,54)(28,71,85,39)(113,169,153,216)(114,182,154,201)(115,195,155,214)(116,180,156,199)(117,193,157,212)(118,178,158,197)(119,191,159,210)(120,176,160,223)(121,189,161,208)(122,174,162,221)(123,187,163,206)(124,172,164,219)(125,185,165,204)(126,170,166,217)(127,183,167,202)(128,196,168,215)(129,181,141,200)(130,194,142,213)(131,179,143,198)(132,192,144,211)(133,177,145,224)(134,190,146,209)(135,175,147,222)(136,188,148,207)(137,173,149,220)(138,186,150,205)(139,171,151,218)(140,184,152,203)>;
G:=Group( (1,100)(2,87)(3,102)(4,89)(5,104)(6,91)(7,106)(8,93)(9,108)(10,95)(11,110)(12,97)(13,112)(14,99)(15,86)(16,101)(17,88)(18,103)(19,90)(20,105)(21,92)(22,107)(23,94)(24,109)(25,96)(26,111)(27,98)(28,85)(29,167)(30,154)(31,141)(32,156)(33,143)(34,158)(35,145)(36,160)(37,147)(38,162)(39,149)(40,164)(41,151)(42,166)(43,153)(44,168)(45,155)(46,142)(47,157)(48,144)(49,159)(50,146)(51,161)(52,148)(53,163)(54,150)(55,165)(56,152)(57,123)(58,138)(59,125)(60,140)(61,127)(62,114)(63,129)(64,116)(65,131)(66,118)(67,133)(68,120)(69,135)(70,122)(71,137)(72,124)(73,139)(74,126)(75,113)(76,128)(77,115)(78,130)(79,117)(80,132)(81,119)(82,134)(83,121)(84,136)(169,216)(170,203)(171,218)(172,205)(173,220)(174,207)(175,222)(176,209)(177,224)(178,211)(179,198)(180,213)(181,200)(182,215)(183,202)(184,217)(185,204)(186,219)(187,206)(188,221)(189,208)(190,223)(191,210)(192,197)(193,212)(194,199)(195,214)(196,201), (1,100)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74)(113,167)(114,168)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,153)(128,154)(129,155)(130,156)(131,157)(132,158)(133,159)(134,160)(135,161)(136,162)(137,163)(138,164)(139,165)(140,166)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,211)(179,212)(180,213)(181,214)(182,215)(183,216)(184,217)(185,218)(186,219)(187,220)(188,221)(189,222)(190,223)(191,224)(192,197)(193,198)(194,199)(195,200)(196,201), (1,188)(2,189)(3,190)(4,191)(5,192)(6,193)(7,194)(8,195)(9,196)(10,169)(11,170)(12,171)(13,172)(14,173)(15,174)(16,175)(17,176)(18,177)(19,178)(20,179)(21,180)(22,181)(23,182)(24,183)(25,184)(26,185)(27,186)(28,187)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,141)(78,142)(79,143)(80,144)(81,145)(82,146)(83,147)(84,148)(85,206)(86,207)(87,208)(88,209)(89,210)(90,211)(91,212)(92,213)(93,214)(94,215)(95,216)(96,217)(97,218)(98,219)(99,220)(100,221)(101,222)(102,223)(103,224)(104,197)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205), (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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(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), (1,84,86,52)(2,69,87,37)(3,82,88,50)(4,67,89,35)(5,80,90,48)(6,65,91,33)(7,78,92,46)(8,63,93,31)(9,76,94,44)(10,61,95,29)(11,74,96,42)(12,59,97,55)(13,72,98,40)(14,57,99,53)(15,70,100,38)(16,83,101,51)(17,68,102,36)(18,81,103,49)(19,66,104,34)(20,79,105,47)(21,64,106,32)(22,77,107,45)(23,62,108,30)(24,75,109,43)(25,60,110,56)(26,73,111,41)(27,58,112,54)(28,71,85,39)(113,169,153,216)(114,182,154,201)(115,195,155,214)(116,180,156,199)(117,193,157,212)(118,178,158,197)(119,191,159,210)(120,176,160,223)(121,189,161,208)(122,174,162,221)(123,187,163,206)(124,172,164,219)(125,185,165,204)(126,170,166,217)(127,183,167,202)(128,196,168,215)(129,181,141,200)(130,194,142,213)(131,179,143,198)(132,192,144,211)(133,177,145,224)(134,190,146,209)(135,175,147,222)(136,188,148,207)(137,173,149,220)(138,186,150,205)(139,171,151,218)(140,184,152,203) );
G=PermutationGroup([[(1,100),(2,87),(3,102),(4,89),(5,104),(6,91),(7,106),(8,93),(9,108),(10,95),(11,110),(12,97),(13,112),(14,99),(15,86),(16,101),(17,88),(18,103),(19,90),(20,105),(21,92),(22,107),(23,94),(24,109),(25,96),(26,111),(27,98),(28,85),(29,167),(30,154),(31,141),(32,156),(33,143),(34,158),(35,145),(36,160),(37,147),(38,162),(39,149),(40,164),(41,151),(42,166),(43,153),(44,168),(45,155),(46,142),(47,157),(48,144),(49,159),(50,146),(51,161),(52,148),(53,163),(54,150),(55,165),(56,152),(57,123),(58,138),(59,125),(60,140),(61,127),(62,114),(63,129),(64,116),(65,131),(66,118),(67,133),(68,120),(69,135),(70,122),(71,137),(72,124),(73,139),(74,126),(75,113),(76,128),(77,115),(78,130),(79,117),(80,132),(81,119),(82,134),(83,121),(84,136),(169,216),(170,203),(171,218),(172,205),(173,220),(174,207),(175,222),(176,209),(177,224),(178,211),(179,198),(180,213),(181,200),(182,215),(183,202),(184,217),(185,204),(186,219),(187,206),(188,221),(189,208),(190,223),(191,210),(192,197),(193,212),(194,199),(195,214),(196,201)], [(1,100),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,82),(37,83),(38,84),(39,57),(40,58),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,69),(52,70),(53,71),(54,72),(55,73),(56,74),(113,167),(114,168),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146),(121,147),(122,148),(123,149),(124,150),(125,151),(126,152),(127,153),(128,154),(129,155),(130,156),(131,157),(132,158),(133,159),(134,160),(135,161),(136,162),(137,163),(138,164),(139,165),(140,166),(169,202),(170,203),(171,204),(172,205),(173,206),(174,207),(175,208),(176,209),(177,210),(178,211),(179,212),(180,213),(181,214),(182,215),(183,216),(184,217),(185,218),(186,219),(187,220),(188,221),(189,222),(190,223),(191,224),(192,197),(193,198),(194,199),(195,200),(196,201)], [(1,188),(2,189),(3,190),(4,191),(5,192),(6,193),(7,194),(8,195),(9,196),(10,169),(11,170),(12,171),(13,172),(14,173),(15,174),(16,175),(17,176),(18,177),(19,178),(20,179),(21,180),(22,181),(23,182),(24,183),(25,184),(26,185),(27,186),(28,187),(29,113),(30,114),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,121),(38,122),(39,123),(40,124),(41,125),(42,126),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,137),(54,138),(55,139),(56,140),(57,149),(58,150),(59,151),(60,152),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,161),(70,162),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,141),(78,142),(79,143),(80,144),(81,145),(82,146),(83,147),(84,148),(85,206),(86,207),(87,208),(88,209),(89,210),(90,211),(91,212),(92,213),(93,214),(94,215),(95,216),(96,217),(97,218),(98,219),(99,220),(100,221),(101,222),(102,223),(103,224),(104,197),(105,198),(106,199),(107,200),(108,201),(109,202),(110,203),(111,204),(112,205)], [(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(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)], [(1,84,86,52),(2,69,87,37),(3,82,88,50),(4,67,89,35),(5,80,90,48),(6,65,91,33),(7,78,92,46),(8,63,93,31),(9,76,94,44),(10,61,95,29),(11,74,96,42),(12,59,97,55),(13,72,98,40),(14,57,99,53),(15,70,100,38),(16,83,101,51),(17,68,102,36),(18,81,103,49),(19,66,104,34),(20,79,105,47),(21,64,106,32),(22,77,107,45),(23,62,108,30),(24,75,109,43),(25,60,110,56),(26,73,111,41),(27,58,112,54),(28,71,85,39),(113,169,153,216),(114,182,154,201),(115,195,155,214),(116,180,156,199),(117,193,157,212),(118,178,158,197),(119,191,159,210),(120,176,160,223),(121,189,161,208),(122,174,162,221),(123,187,163,206),(124,172,164,219),(125,185,165,204),(126,170,166,217),(127,183,167,202),(128,196,168,215),(129,181,141,200),(130,194,142,213),(131,179,143,198),(132,192,144,211),(133,177,145,224),(134,190,146,209),(135,175,147,222),(136,188,148,207),(137,173,149,220),(138,186,150,205),(139,171,151,218),(140,184,152,203)]])
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 | ··· | 14AS | 28A | ··· | 28L |
| 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 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | C4○D4 | D14 | Dic7 | D14 | C7⋊D4 | D4×D7 | D4⋊2D7 |
| kernel | C24.19D14 | C2×C4×Dic7 | C2×C4⋊Dic7 | C2×C23.D7 | D4×C2×C14 | D4×C14 | C2×Dic7 | C2×C28 | C22×D4 | C2×C14 | C22×C4 | C2×D4 | C24 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 1 | 4 | 1 | 8 | 4 | 4 | 3 | 4 | 3 | 12 | 6 | 24 | 6 | 6 |
Matrix representation of C24.19D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 10 | 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 | 2 | 1 |
| 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 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 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 20 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 25 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 3 | 25 |
| 26 | 11 | 0 | 0 | 0 | 0 |
| 7 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 18 | 0 | 0 |
| 0 | 0 | 19 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [28,10,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,2,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[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],[16,20,0,0,0,0,0,20,0,0,0,0,0,0,16,25,0,0,0,0,0,20,0,0,0,0,0,0,4,3,0,0,0,0,4,25],[26,7,0,0,0,0,11,3,0,0,0,0,0,0,11,19,0,0,0,0,18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C24.19D14 in GAP, Magma, Sage, TeX
C_2^4._{19}D_{14} % in TeX
G:=Group("C2^4.19D14"); // GroupNames label
G:=SmallGroup(448,755);
// by ID
G=gap.SmallGroup(448,755);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,422,387,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=d,f^2=d*b=b*d,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*c*d,b*c=c*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=e^13>;
// generators/relations