direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4○D4×Dic7, D4⋊7(C2×Dic7), Q8⋊6(C2×Dic7), (Q8×Dic7)⋊33C2, (D4×Dic7)⋊45C2, (C2×D4).252D14, C14.50(C23×C4), C28.99(C22×C4), (C2×Q8).208D14, (C2×C14).312C24, (C2×C28).888C23, (C22×C4).388D14, C2.12(C23×Dic7), C4.21(C22×Dic7), C22.48(C23×D7), (D4×C14).274C22, C4⋊Dic7.391C22, (Q8×C14).241C22, C23.209(C22×D7), C23.21D14⋊36C2, C22.3(C22×Dic7), (C22×C28).294C22, (C22×C14).238C23, (C2×Dic7).291C23, (C4×Dic7).303C22, C23.D7.134C22, (C22×Dic7).235C22, C7⋊5(C4×C4○D4), (C7×C4○D4)⋊5C4, C2.7(D7×C4○D4), (C2×C28)⋊16(C2×C4), (C7×D4)⋊21(C2×C4), (C7×Q8)⋊19(C2×C4), (C2×C4×Dic7)⋊14C2, (C2×C4)⋊8(C2×Dic7), (C2×C4○D4).14D7, (C14×C4○D4).13C2, C14.214(C2×C4○D4), (C2×C14).30(C22×C4), (C2×C4).637(C22×D7), SmallGroup(448,1279)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 916 in 310 conjugacy classes, 195 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×10], C22, C22 [×6], C22 [×6], C7, C2×C4, C2×C4 [×15], C2×C4 [×20], D4 [×12], Q8 [×4], C23 [×3], C14, C14 [×2], C14 [×6], C42 [×10], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic7 [×4], Dic7 [×6], C28 [×8], C2×C14, C2×C14 [×6], C2×C14 [×6], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C2×Dic7 [×8], C2×Dic7 [×12], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×C14 [×3], C4×C4○D4, C4×Dic7, C4×Dic7 [×9], C4⋊Dic7 [×6], C23.D7 [×6], C22×Dic7 [×6], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C2×C4×Dic7 [×3], C23.21D14 [×3], D4×Dic7 [×6], Q8×Dic7 [×2], C14×C4○D4, C4○D4×Dic7
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C4○D4 [×4], C24, Dic7 [×8], D14 [×7], C23×C4, C2×C4○D4 [×2], C2×Dic7 [×28], C22×D7 [×7], C4×C4○D4, C22×Dic7 [×14], C23×D7, D7×C4○D4 [×2], C23×Dic7, C4○D4×Dic7
Generators and relations
G = < a,b,c,d,e | a4=c2=d14=1, b2=a2, e2=d7, ab=ba, ac=ca, ad=da, ae=ea, cbc=a2b, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►On 224 points
Generators in S224
(1 15 93 144)(2 16 94 145)(3 17 95 146)(4 18 96 147)(5 19 97 148)(6 20 98 149)(7 21 85 150)(8 22 86 151)(9 23 87 152)(10 24 88 153)(11 25 89 154)(12 26 90 141)(13 27 91 142)(14 28 92 143)(29 184 198 71)(30 185 199 72)(31 186 200 73)(32 187 201 74)(33 188 202 75)(34 189 203 76)(35 190 204 77)(36 191 205 78)(37 192 206 79)(38 193 207 80)(39 194 208 81)(40 195 209 82)(41 196 210 83)(42 183 197 84)(43 70 168 181)(44 57 155 182)(45 58 156 169)(46 59 157 170)(47 60 158 171)(48 61 159 172)(49 62 160 173)(50 63 161 174)(51 64 162 175)(52 65 163 176)(53 66 164 177)(54 67 165 178)(55 68 166 179)(56 69 167 180)(99 138 121 212)(100 139 122 213)(101 140 123 214)(102 127 124 215)(103 128 125 216)(104 129 126 217)(105 130 113 218)(106 131 114 219)(107 132 115 220)(108 133 116 221)(109 134 117 222)(110 135 118 223)(111 136 119 224)(112 137 120 211)
(1 15 93 144)(2 16 94 145)(3 17 95 146)(4 18 96 147)(5 19 97 148)(6 20 98 149)(7 21 85 150)(8 22 86 151)(9 23 87 152)(10 24 88 153)(11 25 89 154)(12 26 90 141)(13 27 91 142)(14 28 92 143)(29 184 198 71)(30 185 199 72)(31 186 200 73)(32 187 201 74)(33 188 202 75)(34 189 203 76)(35 190 204 77)(36 191 205 78)(37 192 206 79)(38 193 207 80)(39 194 208 81)(40 195 209 82)(41 196 210 83)(42 183 197 84)(43 181 168 70)(44 182 155 57)(45 169 156 58)(46 170 157 59)(47 171 158 60)(48 172 159 61)(49 173 160 62)(50 174 161 63)(51 175 162 64)(52 176 163 65)(53 177 164 66)(54 178 165 67)(55 179 166 68)(56 180 167 69)(99 212 121 138)(100 213 122 139)(101 214 123 140)(102 215 124 127)(103 216 125 128)(104 217 126 129)(105 218 113 130)(106 219 114 131)(107 220 115 132)(108 221 116 133)(109 222 117 134)(110 223 118 135)(111 224 119 136)(112 211 120 137)
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 118)(16 119)(17 120)(18 121)(19 122)(20 123)(21 124)(22 125)(23 126)(24 113)(25 114)(26 115)(27 116)(28 117)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(49 83)(50 84)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(85 215)(86 216)(87 217)(88 218)(89 219)(90 220)(91 221)(92 222)(93 223)(94 224)(95 211)(96 212)(97 213)(98 214)(99 147)(100 148)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 141)(108 142)(109 143)(110 144)(111 145)(112 146)(155 191)(156 192)(157 193)(158 194)(159 195)(160 196)(161 183)(162 184)(163 185)(164 186)(165 187)(166 188)(167 189)(168 190)(169 206)(170 207)(171 208)(172 209)(173 210)(174 197)(175 198)(176 199)(177 200)(178 201)(179 202)(180 203)(181 204)(182 205)
(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 35 8 42)(2 34 9 41)(3 33 10 40)(4 32 11 39)(5 31 12 38)(6 30 13 37)(7 29 14 36)(15 190 22 183)(16 189 23 196)(17 188 24 195)(18 187 25 194)(19 186 26 193)(20 185 27 192)(21 184 28 191)(43 103 50 110)(44 102 51 109)(45 101 52 108)(46 100 53 107)(47 99 54 106)(48 112 55 105)(49 111 56 104)(57 127 64 134)(58 140 65 133)(59 139 66 132)(60 138 67 131)(61 137 68 130)(62 136 69 129)(63 135 70 128)(71 143 78 150)(72 142 79 149)(73 141 80 148)(74 154 81 147)(75 153 82 146)(76 152 83 145)(77 151 84 144)(85 198 92 205)(86 197 93 204)(87 210 94 203)(88 209 95 202)(89 208 96 201)(90 207 97 200)(91 206 98 199)(113 159 120 166)(114 158 121 165)(115 157 122 164)(116 156 123 163)(117 155 124 162)(118 168 125 161)(119 167 126 160)(169 214 176 221)(170 213 177 220)(171 212 178 219)(172 211 179 218)(173 224 180 217)(174 223 181 216)(175 222 182 215)
G:=sub<Sym(224)| (1,15,93,144)(2,16,94,145)(3,17,95,146)(4,18,96,147)(5,19,97,148)(6,20,98,149)(7,21,85,150)(8,22,86,151)(9,23,87,152)(10,24,88,153)(11,25,89,154)(12,26,90,141)(13,27,91,142)(14,28,92,143)(29,184,198,71)(30,185,199,72)(31,186,200,73)(32,187,201,74)(33,188,202,75)(34,189,203,76)(35,190,204,77)(36,191,205,78)(37,192,206,79)(38,193,207,80)(39,194,208,81)(40,195,209,82)(41,196,210,83)(42,183,197,84)(43,70,168,181)(44,57,155,182)(45,58,156,169)(46,59,157,170)(47,60,158,171)(48,61,159,172)(49,62,160,173)(50,63,161,174)(51,64,162,175)(52,65,163,176)(53,66,164,177)(54,67,165,178)(55,68,166,179)(56,69,167,180)(99,138,121,212)(100,139,122,213)(101,140,123,214)(102,127,124,215)(103,128,125,216)(104,129,126,217)(105,130,113,218)(106,131,114,219)(107,132,115,220)(108,133,116,221)(109,134,117,222)(110,135,118,223)(111,136,119,224)(112,137,120,211), (1,15,93,144)(2,16,94,145)(3,17,95,146)(4,18,96,147)(5,19,97,148)(6,20,98,149)(7,21,85,150)(8,22,86,151)(9,23,87,152)(10,24,88,153)(11,25,89,154)(12,26,90,141)(13,27,91,142)(14,28,92,143)(29,184,198,71)(30,185,199,72)(31,186,200,73)(32,187,201,74)(33,188,202,75)(34,189,203,76)(35,190,204,77)(36,191,205,78)(37,192,206,79)(38,193,207,80)(39,194,208,81)(40,195,209,82)(41,196,210,83)(42,183,197,84)(43,181,168,70)(44,182,155,57)(45,169,156,58)(46,170,157,59)(47,171,158,60)(48,172,159,61)(49,173,160,62)(50,174,161,63)(51,175,162,64)(52,176,163,65)(53,177,164,66)(54,178,165,67)(55,179,166,68)(56,180,167,69)(99,212,121,138)(100,213,122,139)(101,214,123,140)(102,215,124,127)(103,216,125,128)(104,217,126,129)(105,218,113,130)(106,219,114,131)(107,220,115,132)(108,221,116,133)(109,222,117,134)(110,223,118,135)(111,224,119,136)(112,211,120,137), (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,118)(16,119)(17,120)(18,121)(19,122)(20,123)(21,124)(22,125)(23,126)(24,113)(25,114)(26,115)(27,116)(28,117)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(85,215)(86,216)(87,217)(88,218)(89,219)(90,220)(91,221)(92,222)(93,223)(94,224)(95,211)(96,212)(97,213)(98,214)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(155,191)(156,192)(157,193)(158,194)(159,195)(160,196)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190)(169,206)(170,207)(171,208)(172,209)(173,210)(174,197)(175,198)(176,199)(177,200)(178,201)(179,202)(180,203)(181,204)(182,205), (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), (1,35,8,42)(2,34,9,41)(3,33,10,40)(4,32,11,39)(5,31,12,38)(6,30,13,37)(7,29,14,36)(15,190,22,183)(16,189,23,196)(17,188,24,195)(18,187,25,194)(19,186,26,193)(20,185,27,192)(21,184,28,191)(43,103,50,110)(44,102,51,109)(45,101,52,108)(46,100,53,107)(47,99,54,106)(48,112,55,105)(49,111,56,104)(57,127,64,134)(58,140,65,133)(59,139,66,132)(60,138,67,131)(61,137,68,130)(62,136,69,129)(63,135,70,128)(71,143,78,150)(72,142,79,149)(73,141,80,148)(74,154,81,147)(75,153,82,146)(76,152,83,145)(77,151,84,144)(85,198,92,205)(86,197,93,204)(87,210,94,203)(88,209,95,202)(89,208,96,201)(90,207,97,200)(91,206,98,199)(113,159,120,166)(114,158,121,165)(115,157,122,164)(116,156,123,163)(117,155,124,162)(118,168,125,161)(119,167,126,160)(169,214,176,221)(170,213,177,220)(171,212,178,219)(172,211,179,218)(173,224,180,217)(174,223,181,216)(175,222,182,215)>;
G:=Group( (1,15,93,144)(2,16,94,145)(3,17,95,146)(4,18,96,147)(5,19,97,148)(6,20,98,149)(7,21,85,150)(8,22,86,151)(9,23,87,152)(10,24,88,153)(11,25,89,154)(12,26,90,141)(13,27,91,142)(14,28,92,143)(29,184,198,71)(30,185,199,72)(31,186,200,73)(32,187,201,74)(33,188,202,75)(34,189,203,76)(35,190,204,77)(36,191,205,78)(37,192,206,79)(38,193,207,80)(39,194,208,81)(40,195,209,82)(41,196,210,83)(42,183,197,84)(43,70,168,181)(44,57,155,182)(45,58,156,169)(46,59,157,170)(47,60,158,171)(48,61,159,172)(49,62,160,173)(50,63,161,174)(51,64,162,175)(52,65,163,176)(53,66,164,177)(54,67,165,178)(55,68,166,179)(56,69,167,180)(99,138,121,212)(100,139,122,213)(101,140,123,214)(102,127,124,215)(103,128,125,216)(104,129,126,217)(105,130,113,218)(106,131,114,219)(107,132,115,220)(108,133,116,221)(109,134,117,222)(110,135,118,223)(111,136,119,224)(112,137,120,211), (1,15,93,144)(2,16,94,145)(3,17,95,146)(4,18,96,147)(5,19,97,148)(6,20,98,149)(7,21,85,150)(8,22,86,151)(9,23,87,152)(10,24,88,153)(11,25,89,154)(12,26,90,141)(13,27,91,142)(14,28,92,143)(29,184,198,71)(30,185,199,72)(31,186,200,73)(32,187,201,74)(33,188,202,75)(34,189,203,76)(35,190,204,77)(36,191,205,78)(37,192,206,79)(38,193,207,80)(39,194,208,81)(40,195,209,82)(41,196,210,83)(42,183,197,84)(43,181,168,70)(44,182,155,57)(45,169,156,58)(46,170,157,59)(47,171,158,60)(48,172,159,61)(49,173,160,62)(50,174,161,63)(51,175,162,64)(52,176,163,65)(53,177,164,66)(54,178,165,67)(55,179,166,68)(56,180,167,69)(99,212,121,138)(100,213,122,139)(101,214,123,140)(102,215,124,127)(103,216,125,128)(104,217,126,129)(105,218,113,130)(106,219,114,131)(107,220,115,132)(108,221,116,133)(109,222,117,134)(110,223,118,135)(111,224,119,136)(112,211,120,137), (1,135)(2,136)(3,137)(4,138)(5,139)(6,140)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,118)(16,119)(17,120)(18,121)(19,122)(20,123)(21,124)(22,125)(23,126)(24,113)(25,114)(26,115)(27,116)(28,117)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,77)(44,78)(45,79)(46,80)(47,81)(48,82)(49,83)(50,84)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(85,215)(86,216)(87,217)(88,218)(89,219)(90,220)(91,221)(92,222)(93,223)(94,224)(95,211)(96,212)(97,213)(98,214)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,141)(108,142)(109,143)(110,144)(111,145)(112,146)(155,191)(156,192)(157,193)(158,194)(159,195)(160,196)(161,183)(162,184)(163,185)(164,186)(165,187)(166,188)(167,189)(168,190)(169,206)(170,207)(171,208)(172,209)(173,210)(174,197)(175,198)(176,199)(177,200)(178,201)(179,202)(180,203)(181,204)(182,205), (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), (1,35,8,42)(2,34,9,41)(3,33,10,40)(4,32,11,39)(5,31,12,38)(6,30,13,37)(7,29,14,36)(15,190,22,183)(16,189,23,196)(17,188,24,195)(18,187,25,194)(19,186,26,193)(20,185,27,192)(21,184,28,191)(43,103,50,110)(44,102,51,109)(45,101,52,108)(46,100,53,107)(47,99,54,106)(48,112,55,105)(49,111,56,104)(57,127,64,134)(58,140,65,133)(59,139,66,132)(60,138,67,131)(61,137,68,130)(62,136,69,129)(63,135,70,128)(71,143,78,150)(72,142,79,149)(73,141,80,148)(74,154,81,147)(75,153,82,146)(76,152,83,145)(77,151,84,144)(85,198,92,205)(86,197,93,204)(87,210,94,203)(88,209,95,202)(89,208,96,201)(90,207,97,200)(91,206,98,199)(113,159,120,166)(114,158,121,165)(115,157,122,164)(116,156,123,163)(117,155,124,162)(118,168,125,161)(119,167,126,160)(169,214,176,221)(170,213,177,220)(171,212,178,219)(172,211,179,218)(173,224,180,217)(174,223,181,216)(175,222,182,215) );
G=PermutationGroup([(1,15,93,144),(2,16,94,145),(3,17,95,146),(4,18,96,147),(5,19,97,148),(6,20,98,149),(7,21,85,150),(8,22,86,151),(9,23,87,152),(10,24,88,153),(11,25,89,154),(12,26,90,141),(13,27,91,142),(14,28,92,143),(29,184,198,71),(30,185,199,72),(31,186,200,73),(32,187,201,74),(33,188,202,75),(34,189,203,76),(35,190,204,77),(36,191,205,78),(37,192,206,79),(38,193,207,80),(39,194,208,81),(40,195,209,82),(41,196,210,83),(42,183,197,84),(43,70,168,181),(44,57,155,182),(45,58,156,169),(46,59,157,170),(47,60,158,171),(48,61,159,172),(49,62,160,173),(50,63,161,174),(51,64,162,175),(52,65,163,176),(53,66,164,177),(54,67,165,178),(55,68,166,179),(56,69,167,180),(99,138,121,212),(100,139,122,213),(101,140,123,214),(102,127,124,215),(103,128,125,216),(104,129,126,217),(105,130,113,218),(106,131,114,219),(107,132,115,220),(108,133,116,221),(109,134,117,222),(110,135,118,223),(111,136,119,224),(112,137,120,211)], [(1,15,93,144),(2,16,94,145),(3,17,95,146),(4,18,96,147),(5,19,97,148),(6,20,98,149),(7,21,85,150),(8,22,86,151),(9,23,87,152),(10,24,88,153),(11,25,89,154),(12,26,90,141),(13,27,91,142),(14,28,92,143),(29,184,198,71),(30,185,199,72),(31,186,200,73),(32,187,201,74),(33,188,202,75),(34,189,203,76),(35,190,204,77),(36,191,205,78),(37,192,206,79),(38,193,207,80),(39,194,208,81),(40,195,209,82),(41,196,210,83),(42,183,197,84),(43,181,168,70),(44,182,155,57),(45,169,156,58),(46,170,157,59),(47,171,158,60),(48,172,159,61),(49,173,160,62),(50,174,161,63),(51,175,162,64),(52,176,163,65),(53,177,164,66),(54,178,165,67),(55,179,166,68),(56,180,167,69),(99,212,121,138),(100,213,122,139),(101,214,123,140),(102,215,124,127),(103,216,125,128),(104,217,126,129),(105,218,113,130),(106,219,114,131),(107,220,115,132),(108,221,116,133),(109,222,117,134),(110,223,118,135),(111,224,119,136),(112,211,120,137)], [(1,135),(2,136),(3,137),(4,138),(5,139),(6,140),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,118),(16,119),(17,120),(18,121),(19,122),(20,123),(21,124),(22,125),(23,126),(24,113),(25,114),(26,115),(27,116),(28,117),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(43,77),(44,78),(45,79),(46,80),(47,81),(48,82),(49,83),(50,84),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(85,215),(86,216),(87,217),(88,218),(89,219),(90,220),(91,221),(92,222),(93,223),(94,224),(95,211),(96,212),(97,213),(98,214),(99,147),(100,148),(101,149),(102,150),(103,151),(104,152),(105,153),(106,154),(107,141),(108,142),(109,143),(110,144),(111,145),(112,146),(155,191),(156,192),(157,193),(158,194),(159,195),(160,196),(161,183),(162,184),(163,185),(164,186),(165,187),(166,188),(167,189),(168,190),(169,206),(170,207),(171,208),(172,209),(173,210),(174,197),(175,198),(176,199),(177,200),(178,201),(179,202),(180,203),(181,204),(182,205)], [(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)], [(1,35,8,42),(2,34,9,41),(3,33,10,40),(4,32,11,39),(5,31,12,38),(6,30,13,37),(7,29,14,36),(15,190,22,183),(16,189,23,196),(17,188,24,195),(18,187,25,194),(19,186,26,193),(20,185,27,192),(21,184,28,191),(43,103,50,110),(44,102,51,109),(45,101,52,108),(46,100,53,107),(47,99,54,106),(48,112,55,105),(49,111,56,104),(57,127,64,134),(58,140,65,133),(59,139,66,132),(60,138,67,131),(61,137,68,130),(62,136,69,129),(63,135,70,128),(71,143,78,150),(72,142,79,149),(73,141,80,148),(74,154,81,147),(75,153,82,146),(76,152,83,145),(77,151,84,144),(85,198,92,205),(86,197,93,204),(87,210,94,203),(88,209,95,202),(89,208,96,201),(90,207,97,200),(91,206,98,199),(113,159,120,166),(114,158,121,165),(115,157,122,164),(116,156,123,163),(117,155,124,162),(118,168,125,161),(119,167,126,160),(169,214,176,221),(170,213,177,220),(171,212,178,219),(172,211,179,218),(173,224,180,217),(174,223,181,216),(175,222,182,215)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 12 | 17 |
| 0 | 0 | 0 | 17 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 27 | 1 |
| 8 | 28 | 0 | 0 |
| 6 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 19 | 0 | 0 |
| 14 | 20 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,12,0,0,0,0,12],[28,0,0,0,0,28,0,0,0,0,12,0,0,0,17,17],[28,0,0,0,0,28,0,0,0,0,28,27,0,0,0,1],[8,6,0,0,28,3,0,0,0,0,1,0,0,0,0,1],[9,14,0,0,19,20,0,0,0,0,28,0,0,0,0,28] >;
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | ··· | 4R | 4S | ··· | 4AD | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | C4○D4 | D14 | D14 | D14 | Dic7 | D7×C4○D4 |
| kernel | C4○D4×Dic7 | C2×C4×Dic7 | C23.21D14 | D4×Dic7 | Q8×Dic7 | C14×C4○D4 | C7×C4○D4 | C2×C4○D4 | Dic7 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 2 | 1 | 16 | 3 | 8 | 9 | 9 | 3 | 24 | 12 |
In GAP, Magma, Sage, TeX
C_4\circ D_4\times Dic_7
% in TeX
G:=Group("C4oD4xDic7"); // GroupNames label
G:=SmallGroup(448,1279);
// by ID
G=gap.SmallGroup(448,1279);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,184,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^2=d^14=1,b^2=a^2,e^2=d^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=a^2*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀