metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C7×D4).32D4, (C7×Q8).32D4, C28.216(C2×D4), (C2×C28).453D4, C7⋊7(D4.7D4), C14.79C22≀C2, (C2×D4).205D14, D4⋊Dic7⋊42C2, Q8⋊Dic7⋊42C2, D4.14(C7⋊D4), (C2×Q8).174D14, Q8.14(C7⋊D4), C14.113(C4○D8), C28.48D4⋊27C2, C28.55D4⋊19C2, (C2×C28).485C23, (C22×C14).116D4, (C22×C4).165D14, C23.33(C7⋊D4), (D4×C14).246C22, C2.13(C24⋊D7), C4⋊Dic7.190C22, (Q8×C14).209C22, C2.31(D4.8D14), C2.23(D4.9D14), C14.125(C8.C22), (C22×C28).211C22, (C2×Dic14).139C22, (C2×C4○D4).7D7, C4.63(C2×C7⋊D4), (C2×D4.D7)⋊24C2, (C14×C4○D4).7C2, (C2×C7⋊Q16)⋊24C2, (C2×C14).568(C2×D4), (C2×C7⋊C8).179C22, (C2×C4).226(C7⋊D4), (C2×C4).569(C22×D7), C22.225(C2×C7⋊D4), SmallGroup(448,773)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C7×D4).32D4
G = < a,b,c,d,e | a7=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=dbd-1=ebe-1=b-1, dcd-1=ece-1=bc, ede-1=d-1 >
Subgroups: 532 in 152 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, C2×C14, C2×C14, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C7⋊C8, Dic14, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, D4.7D4, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, D4.D7, C7⋊Q16, C23.D7, C2×Dic14, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C28.55D4, D4⋊Dic7, Q8⋊Dic7, C28.48D4, C2×D4.D7, C2×C7⋊Q16, C14×C4○D4, (C7×D4).32D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, D14, C22≀C2, C4○D8, C8.C22, C7⋊D4, C22×D7, D4.7D4, C2×C7⋊D4, D4.8D14, D4.9D14, C24⋊D7, (C7×D4).32D4
►On 224 points
Generators in S224
(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 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)(113 127 120 134)(114 128 121 135)(115 129 122 136)(116 130 123 137)(117 131 124 138)(118 132 125 139)(119 133 126 140)(141 155 148 162)(142 156 149 163)(143 157 150 164)(144 158 151 165)(145 159 152 166)(146 160 153 167)(147 161 154 168)(169 183 176 190)(170 184 177 191)(171 185 178 192)(172 186 179 193)(173 187 180 194)(174 188 181 195)(175 189 182 196)(197 211 204 218)(198 212 205 219)(199 213 206 220)(200 214 207 221)(201 215 208 222)(202 216 209 223)(203 217 210 224)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 39)(26 40)(27 41)(28 42)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 141)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 204)(170 205)(171 206)(172 207)(173 208)(174 209)(175 210)(176 197)(177 198)(178 199)(179 200)(180 201)(181 202)(182 203)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(1 202 29 174)(2 201 30 173)(3 200 31 172)(4 199 32 171)(5 198 33 170)(6 197 34 169)(7 203 35 175)(8 209 36 181)(9 208 37 180)(10 207 38 179)(11 206 39 178)(12 205 40 177)(13 204 41 176)(14 210 42 182)(15 216 43 188)(16 215 44 187)(17 214 45 186)(18 213 46 185)(19 212 47 184)(20 211 48 183)(21 217 49 189)(22 223 50 195)(23 222 51 194)(24 221 52 193)(25 220 53 192)(26 219 54 191)(27 218 55 190)(28 224 56 196)(57 153 85 125)(58 152 86 124)(59 151 87 123)(60 150 88 122)(61 149 89 121)(62 148 90 120)(63 154 91 126)(64 146 92 118)(65 145 93 117)(66 144 94 116)(67 143 95 115)(68 142 96 114)(69 141 97 113)(70 147 98 119)(71 167 99 139)(72 166 100 138)(73 165 101 137)(74 164 102 136)(75 163 103 135)(76 162 104 134)(77 168 105 140)(78 160 106 132)(79 159 107 131)(80 158 108 130)(81 157 109 129)(82 156 110 128)(83 155 111 127)(84 161 112 133)
(1 125 8 118)(2 124 9 117)(3 123 10 116)(4 122 11 115)(5 121 12 114)(6 120 13 113)(7 126 14 119)(15 139 22 132)(16 138 23 131)(17 137 24 130)(18 136 25 129)(19 135 26 128)(20 134 27 127)(21 140 28 133)(29 153 36 146)(30 152 37 145)(31 151 38 144)(32 150 39 143)(33 149 40 142)(34 148 41 141)(35 154 42 147)(43 167 50 160)(44 166 51 159)(45 165 52 158)(46 164 53 157)(47 163 54 156)(48 162 55 155)(49 168 56 161)(57 181 64 174)(58 180 65 173)(59 179 66 172)(60 178 67 171)(61 177 68 170)(62 176 69 169)(63 182 70 175)(71 195 78 188)(72 194 79 187)(73 193 80 186)(74 192 81 185)(75 191 82 184)(76 190 83 183)(77 196 84 189)(85 209 92 202)(86 208 93 201)(87 207 94 200)(88 206 95 199)(89 205 96 198)(90 204 97 197)(91 210 98 203)(99 223 106 216)(100 222 107 215)(101 221 108 214)(102 220 109 213)(103 219 110 212)(104 218 111 211)(105 224 112 217)
G:=sub<Sym(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), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(113,127,120,134)(114,128,121,135)(115,129,122,136)(116,130,123,137)(117,131,124,138)(118,132,125,139)(119,133,126,140)(141,155,148,162)(142,156,149,163)(143,157,150,164)(144,158,151,165)(145,159,152,166)(146,160,153,167)(147,161,154,168)(169,183,176,190)(170,184,177,191)(171,185,178,192)(172,186,179,193)(173,187,180,194)(174,188,181,195)(175,189,182,196)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,202,29,174)(2,201,30,173)(3,200,31,172)(4,199,32,171)(5,198,33,170)(6,197,34,169)(7,203,35,175)(8,209,36,181)(9,208,37,180)(10,207,38,179)(11,206,39,178)(12,205,40,177)(13,204,41,176)(14,210,42,182)(15,216,43,188)(16,215,44,187)(17,214,45,186)(18,213,46,185)(19,212,47,184)(20,211,48,183)(21,217,49,189)(22,223,50,195)(23,222,51,194)(24,221,52,193)(25,220,53,192)(26,219,54,191)(27,218,55,190)(28,224,56,196)(57,153,85,125)(58,152,86,124)(59,151,87,123)(60,150,88,122)(61,149,89,121)(62,148,90,120)(63,154,91,126)(64,146,92,118)(65,145,93,117)(66,144,94,116)(67,143,95,115)(68,142,96,114)(69,141,97,113)(70,147,98,119)(71,167,99,139)(72,166,100,138)(73,165,101,137)(74,164,102,136)(75,163,103,135)(76,162,104,134)(77,168,105,140)(78,160,106,132)(79,159,107,131)(80,158,108,130)(81,157,109,129)(82,156,110,128)(83,155,111,127)(84,161,112,133), (1,125,8,118)(2,124,9,117)(3,123,10,116)(4,122,11,115)(5,121,12,114)(6,120,13,113)(7,126,14,119)(15,139,22,132)(16,138,23,131)(17,137,24,130)(18,136,25,129)(19,135,26,128)(20,134,27,127)(21,140,28,133)(29,153,36,146)(30,152,37,145)(31,151,38,144)(32,150,39,143)(33,149,40,142)(34,148,41,141)(35,154,42,147)(43,167,50,160)(44,166,51,159)(45,165,52,158)(46,164,53,157)(47,163,54,156)(48,162,55,155)(49,168,56,161)(57,181,64,174)(58,180,65,173)(59,179,66,172)(60,178,67,171)(61,177,68,170)(62,176,69,169)(63,182,70,175)(71,195,78,188)(72,194,79,187)(73,193,80,186)(74,192,81,185)(75,191,82,184)(76,190,83,183)(77,196,84,189)(85,209,92,202)(86,208,93,201)(87,207,94,200)(88,206,95,199)(89,205,96,198)(90,204,97,197)(91,210,98,203)(99,223,106,216)(100,222,107,215)(101,221,108,214)(102,220,109,213)(103,219,110,212)(104,218,111,211)(105,224,112,217)>;
G:=Group( (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), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(113,127,120,134)(114,128,121,135)(115,129,122,136)(116,130,123,137)(117,131,124,138)(118,132,125,139)(119,133,126,140)(141,155,148,162)(142,156,149,163)(143,157,150,164)(144,158,151,165)(145,159,152,166)(146,160,153,167)(147,161,154,168)(169,183,176,190)(170,184,177,191)(171,185,178,192)(172,186,179,193)(173,187,180,194)(174,188,181,195)(175,189,182,196)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,39)(26,40)(27,41)(28,42)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210)(176,197)(177,198)(178,199)(179,200)(180,201)(181,202)(182,203)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (1,202,29,174)(2,201,30,173)(3,200,31,172)(4,199,32,171)(5,198,33,170)(6,197,34,169)(7,203,35,175)(8,209,36,181)(9,208,37,180)(10,207,38,179)(11,206,39,178)(12,205,40,177)(13,204,41,176)(14,210,42,182)(15,216,43,188)(16,215,44,187)(17,214,45,186)(18,213,46,185)(19,212,47,184)(20,211,48,183)(21,217,49,189)(22,223,50,195)(23,222,51,194)(24,221,52,193)(25,220,53,192)(26,219,54,191)(27,218,55,190)(28,224,56,196)(57,153,85,125)(58,152,86,124)(59,151,87,123)(60,150,88,122)(61,149,89,121)(62,148,90,120)(63,154,91,126)(64,146,92,118)(65,145,93,117)(66,144,94,116)(67,143,95,115)(68,142,96,114)(69,141,97,113)(70,147,98,119)(71,167,99,139)(72,166,100,138)(73,165,101,137)(74,164,102,136)(75,163,103,135)(76,162,104,134)(77,168,105,140)(78,160,106,132)(79,159,107,131)(80,158,108,130)(81,157,109,129)(82,156,110,128)(83,155,111,127)(84,161,112,133), (1,125,8,118)(2,124,9,117)(3,123,10,116)(4,122,11,115)(5,121,12,114)(6,120,13,113)(7,126,14,119)(15,139,22,132)(16,138,23,131)(17,137,24,130)(18,136,25,129)(19,135,26,128)(20,134,27,127)(21,140,28,133)(29,153,36,146)(30,152,37,145)(31,151,38,144)(32,150,39,143)(33,149,40,142)(34,148,41,141)(35,154,42,147)(43,167,50,160)(44,166,51,159)(45,165,52,158)(46,164,53,157)(47,163,54,156)(48,162,55,155)(49,168,56,161)(57,181,64,174)(58,180,65,173)(59,179,66,172)(60,178,67,171)(61,177,68,170)(62,176,69,169)(63,182,70,175)(71,195,78,188)(72,194,79,187)(73,193,80,186)(74,192,81,185)(75,191,82,184)(76,190,83,183)(77,196,84,189)(85,209,92,202)(86,208,93,201)(87,207,94,200)(88,206,95,199)(89,205,96,198)(90,204,97,197)(91,210,98,203)(99,223,106,216)(100,222,107,215)(101,221,108,214)(102,220,109,213)(103,219,110,212)(104,218,111,211)(105,224,112,217) );
G=PermutationGroup([[(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)], [(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21),(29,50,36,43),(30,51,37,44),(31,52,38,45),(32,53,39,46),(33,54,40,47),(34,55,41,48),(35,56,42,49),(57,78,64,71),(58,79,65,72),(59,80,66,73),(60,81,67,74),(61,82,68,75),(62,83,69,76),(63,84,70,77),(85,106,92,99),(86,107,93,100),(87,108,94,101),(88,109,95,102),(89,110,96,103),(90,111,97,104),(91,112,98,105),(113,127,120,134),(114,128,121,135),(115,129,122,136),(116,130,123,137),(117,131,124,138),(118,132,125,139),(119,133,126,140),(141,155,148,162),(142,156,149,163),(143,157,150,164),(144,158,151,165),(145,159,152,166),(146,160,153,167),(147,161,154,168),(169,183,176,190),(170,184,177,191),(171,185,178,192),(172,186,179,193),(173,187,180,194),(174,188,181,195),(175,189,182,196),(197,211,204,218),(198,212,205,219),(199,213,206,220),(200,214,207,221),(201,215,208,222),(202,216,209,223),(203,217,210,224)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,39),(26,40),(27,41),(28,42),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(113,148),(114,149),(115,150),(116,151),(117,152),(118,153),(119,154),(120,141),(121,142),(122,143),(123,144),(124,145),(125,146),(126,147),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,204),(170,205),(171,206),(172,207),(173,208),(174,209),(175,210),(176,197),(177,198),(178,199),(179,200),(180,201),(181,202),(182,203),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(1,202,29,174),(2,201,30,173),(3,200,31,172),(4,199,32,171),(5,198,33,170),(6,197,34,169),(7,203,35,175),(8,209,36,181),(9,208,37,180),(10,207,38,179),(11,206,39,178),(12,205,40,177),(13,204,41,176),(14,210,42,182),(15,216,43,188),(16,215,44,187),(17,214,45,186),(18,213,46,185),(19,212,47,184),(20,211,48,183),(21,217,49,189),(22,223,50,195),(23,222,51,194),(24,221,52,193),(25,220,53,192),(26,219,54,191),(27,218,55,190),(28,224,56,196),(57,153,85,125),(58,152,86,124),(59,151,87,123),(60,150,88,122),(61,149,89,121),(62,148,90,120),(63,154,91,126),(64,146,92,118),(65,145,93,117),(66,144,94,116),(67,143,95,115),(68,142,96,114),(69,141,97,113),(70,147,98,119),(71,167,99,139),(72,166,100,138),(73,165,101,137),(74,164,102,136),(75,163,103,135),(76,162,104,134),(77,168,105,140),(78,160,106,132),(79,159,107,131),(80,158,108,130),(81,157,109,129),(82,156,110,128),(83,155,111,127),(84,161,112,133)], [(1,125,8,118),(2,124,9,117),(3,123,10,116),(4,122,11,115),(5,121,12,114),(6,120,13,113),(7,126,14,119),(15,139,22,132),(16,138,23,131),(17,137,24,130),(18,136,25,129),(19,135,26,128),(20,134,27,127),(21,140,28,133),(29,153,36,146),(30,152,37,145),(31,151,38,144),(32,150,39,143),(33,149,40,142),(34,148,41,141),(35,154,42,147),(43,167,50,160),(44,166,51,159),(45,165,52,158),(46,164,53,157),(47,163,54,156),(48,162,55,155),(49,168,56,161),(57,181,64,174),(58,180,65,173),(59,179,66,172),(60,178,67,171),(61,177,68,170),(62,176,69,169),(63,182,70,175),(71,195,78,188),(72,194,79,187),(73,193,80,186),(74,192,81,185),(75,191,82,184),(76,190,83,183),(77,196,84,189),(85,209,92,202),(86,208,93,201),(87,207,94,200),(88,206,95,199),(89,205,96,198),(90,204,97,197),(91,210,98,203),(99,223,106,216),(100,222,107,215),(101,221,108,214),(102,220,109,213),(103,219,110,212),(104,218,111,211),(105,224,112,217)]])
79 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 56 | 56 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
79 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C7⋊D4 | C7⋊D4 | C7⋊D4 | C8.C22 | D4.8D14 | D4.9D14 |
| kernel | (C7×D4).32D4 | C28.55D4 | D4⋊Dic7 | Q8⋊Dic7 | C28.48D4 | C2×D4.D7 | C2×C7⋊Q16 | C14×C4○D4 | C2×C28 | C7×D4 | C7×Q8 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C14 | C2×C4 | D4 | Q8 | C23 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 3 | 3 | 3 | 3 | 4 | 6 | 12 | 12 | 6 | 1 | 6 | 6 |
Matrix representation of (C7×D4).32D4 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 106 | 0 |
| 0 | 0 | 0 | 16 |
| 112 | 8 | 0 | 0 |
| 28 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 8 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 51 | 22 | 0 | 0 |
| 77 | 62 | 0 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 0 |
| 26 | 9 | 0 | 0 |
| 88 | 87 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,106,0,0,0,0,16],[112,28,0,0,8,1,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,8,1,0,0,0,0,1,0,0,0,0,112],[51,77,0,0,22,62,0,0,0,0,0,1,0,0,112,0],[26,88,0,0,9,87,0,0,0,0,0,1,0,0,1,0] >;
(C7×D4).32D4 in GAP, Magma, Sage, TeX
(C_7\times D_4)._{32}D_4 % in TeX
G:=Group("(C7xD4).32D4"); // GroupNames label
G:=SmallGroup(448,773);
// by ID
G=gap.SmallGroup(448,773);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,253,254,184,1684,851,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=1,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations