direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×SD16⋊D7, SD16⋊9D14, C28.7C24, C56.35C23, Dic28⋊17C22, Dic14.3C23, C4.44(D4×D7), C7⋊C8.2C23, (C2×SD16)⋊5D7, (C4×D7).16D4, C28.82(C2×D4), (Q8×D7)⋊6C22, C4.7(C23×D7), (C14×SD16)⋊6C2, D14.51(C2×D4), (C2×C8).103D14, C8⋊D7⋊9C22, (C4×D7).4C23, D4.5(C22×D7), C7⋊Q16⋊6C22, (C7×D4).5C23, C8.11(C22×D7), (C2×Dic28)⋊26C2, (C2×D4).183D14, Q8.1(C22×D7), (C7×Q8).1C23, C14⋊2(C8.C22), D4.D7⋊10C22, (C2×Q8).150D14, Dic7.56(C2×D4), (C7×SD16)⋊9C22, (C22×D7).99D4, C22.140(D4×D7), (C2×C28).524C23, (C2×C56).117C22, (C2×Dic7).193D4, D4⋊2D7.4C22, C14.108(C22×D4), (D4×C14).165C22, (Q8×C14).147C22, (C2×Dic14).196C22, (C2×Q8×D7)⋊15C2, C2.81(C2×D4×D7), C7⋊2(C2×C8.C22), (C2×C8⋊D7)⋊5C2, (C2×D4.D7)⋊28C2, (C2×C7⋊Q16)⋊25C2, (C2×C14).397(C2×D4), (C2×C7⋊C8).180C22, (C2×C4×D7).157C22, (C2×D4⋊2D7).10C2, (C2×C4).613(C22×D7), SmallGroup(448,1213)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1156 in 258 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×8], C22, C22 [×8], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×16], D4 [×2], D4 [×5], Q8 [×2], Q8 [×11], C23 [×2], D7 [×2], C14, C14 [×2], C14 [×2], C2×C8, C2×C8, M4(2) [×4], SD16 [×4], SD16 [×4], Q16 [×8], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×9], C4○D4 [×6], Dic7 [×2], Dic7 [×4], C28 [×2], C28 [×2], D14 [×2], D14 [×2], C2×C14, C2×C14 [×4], C2×M4(2), C2×SD16, C2×SD16, C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C7⋊C8 [×2], C56 [×2], Dic14 [×4], Dic14 [×6], C4×D7 [×4], C4×D7 [×4], C2×Dic7, C2×Dic7 [×6], C7⋊D4 [×4], C2×C28, C2×C28, C7×D4 [×2], C7×D4, C7×Q8 [×2], C7×Q8, C22×D7, C22×C14, C2×C8.C22, C8⋊D7 [×4], Dic28 [×4], C2×C7⋊C8, D4.D7 [×4], C7⋊Q16 [×4], C2×C56, C7×SD16 [×4], C2×Dic14 [×2], C2×Dic14, C2×C4×D7, C2×C4×D7, D4⋊2D7 [×4], D4⋊2D7 [×2], Q8×D7 [×4], Q8×D7 [×2], C22×Dic7, C2×C7⋊D4, D4×C14, Q8×C14, C2×C8⋊D7, C2×Dic28, SD16⋊D7 [×8], C2×D4.D7, C2×C7⋊Q16, C14×SD16, C2×D4⋊2D7, C2×Q8×D7, C2×SD16⋊D7
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8.C22 [×2], C22×D4, C22×D7 [×7], C2×C8.C22, D4×D7 [×2], C23×D7, SD16⋊D7 [×2], C2×D4×D7, C2×SD16⋊D7
Generators and relations
G = < a,b,c,d,e | a2=b8=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b3, bd=db, ebe=b5, cd=dc, ece=b4c, ede=d-1 >
►On 224 points
Generators in S224
(1 196)(2 197)(3 198)(4 199)(5 200)(6 193)(7 194)(8 195)(9 180)(10 181)(11 182)(12 183)(13 184)(14 177)(15 178)(16 179)(17 174)(18 175)(19 176)(20 169)(21 170)(22 171)(23 172)(24 173)(25 221)(26 222)(27 223)(28 224)(29 217)(30 218)(31 219)(32 220)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 113)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 216)(58 209)(59 210)(60 211)(61 212)(62 213)(63 214)(64 215)(65 202)(66 203)(67 204)(68 205)(69 206)(70 207)(71 208)(72 201)(73 127)(74 128)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 145)(89 189)(90 190)(91 191)(92 192)(93 185)(94 186)(95 187)(96 188)(97 155)(98 156)(99 157)(100 158)(101 159)(102 160)(103 153)(104 154)(105 163)(106 164)(107 165)(108 166)(109 167)(110 168)(111 161)(112 162)
(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)
(1 223)(2 218)(3 221)(4 224)(5 219)(6 222)(7 217)(8 220)(9 71)(10 66)(11 69)(12 72)(13 67)(14 70)(15 65)(16 68)(17 59)(18 62)(19 57)(20 60)(21 63)(22 58)(23 61)(24 64)(25 198)(26 193)(27 196)(28 199)(29 194)(30 197)(31 200)(32 195)(33 81)(34 84)(35 87)(36 82)(37 85)(38 88)(39 83)(40 86)(41 98)(42 101)(43 104)(44 99)(45 102)(46 97)(47 100)(48 103)(49 108)(50 111)(51 106)(52 109)(53 112)(54 107)(55 110)(56 105)(73 90)(74 93)(75 96)(76 91)(77 94)(78 89)(79 92)(80 95)(113 151)(114 146)(115 149)(116 152)(117 147)(118 150)(119 145)(120 148)(121 188)(122 191)(123 186)(124 189)(125 192)(126 187)(127 190)(128 185)(129 156)(130 159)(131 154)(132 157)(133 160)(134 155)(135 158)(136 153)(137 166)(138 161)(139 164)(140 167)(141 162)(142 165)(143 168)(144 163)(169 211)(170 214)(171 209)(172 212)(173 215)(174 210)(175 213)(176 216)(177 207)(178 202)(179 205)(180 208)(181 203)(182 206)(183 201)(184 204)
(1 43 61 167 149 179 189)(2 44 62 168 150 180 190)(3 45 63 161 151 181 191)(4 46 64 162 152 182 192)(5 47 57 163 145 183 185)(6 48 58 164 146 184 186)(7 41 59 165 147 177 187)(8 42 60 166 148 178 188)(9 90 197 132 213 110 85)(10 91 198 133 214 111 86)(11 92 199 134 215 112 87)(12 93 200 135 216 105 88)(13 94 193 136 209 106 81)(14 95 194 129 210 107 82)(15 96 195 130 211 108 83)(16 89 196 131 212 109 84)(17 142 117 207 126 217 98)(18 143 118 208 127 218 99)(19 144 119 201 128 219 100)(20 137 120 202 121 220 101)(21 138 113 203 122 221 102)(22 139 114 204 123 222 103)(23 140 115 205 124 223 104)(24 141 116 206 125 224 97)(25 160 170 50 40 66 76)(26 153 171 51 33 67 77)(27 154 172 52 34 68 78)(28 155 173 53 35 69 79)(29 156 174 54 36 70 80)(30 157 175 55 37 71 73)(31 158 176 56 38 72 74)(32 159 169 49 39 65 75)
(1 189)(2 186)(3 191)(4 188)(5 185)(6 190)(7 187)(8 192)(9 136)(10 133)(11 130)(12 135)(13 132)(14 129)(15 134)(16 131)(17 113)(18 118)(19 115)(20 120)(21 117)(22 114)(23 119)(24 116)(25 80)(26 77)(27 74)(28 79)(29 76)(30 73)(31 78)(32 75)(33 171)(34 176)(35 173)(36 170)(37 175)(38 172)(39 169)(40 174)(41 177)(42 182)(43 179)(44 184)(45 181)(46 178)(47 183)(48 180)(50 54)(52 56)(57 145)(58 150)(59 147)(60 152)(61 149)(62 146)(63 151)(64 148)(65 159)(66 156)(67 153)(68 158)(69 155)(70 160)(71 157)(72 154)(81 213)(82 210)(83 215)(84 212)(85 209)(86 214)(87 211)(88 216)(89 196)(90 193)(91 198)(92 195)(93 200)(94 197)(95 194)(96 199)(97 206)(98 203)(99 208)(100 205)(101 202)(102 207)(103 204)(104 201)(106 110)(108 112)(121 220)(122 217)(123 222)(124 219)(125 224)(126 221)(127 218)(128 223)(138 142)(140 144)(162 166)(164 168)
G:=sub<Sym(224)| (1,196)(2,197)(3,198)(4,199)(5,200)(6,193)(7,194)(8,195)(9,180)(10,181)(11,182)(12,183)(13,184)(14,177)(15,178)(16,179)(17,174)(18,175)(19,176)(20,169)(21,170)(22,171)(23,172)(24,173)(25,221)(26,222)(27,223)(28,224)(29,217)(30,218)(31,219)(32,220)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,113)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,216)(58,209)(59,210)(60,211)(61,212)(62,213)(63,214)(64,215)(65,202)(66,203)(67,204)(68,205)(69,206)(70,207)(71,208)(72,201)(73,127)(74,128)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,145)(89,189)(90,190)(91,191)(92,192)(93,185)(94,186)(95,187)(96,188)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,153)(104,154)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,161)(112,162), (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), (1,223)(2,218)(3,221)(4,224)(5,219)(6,222)(7,217)(8,220)(9,71)(10,66)(11,69)(12,72)(13,67)(14,70)(15,65)(16,68)(17,59)(18,62)(19,57)(20,60)(21,63)(22,58)(23,61)(24,64)(25,198)(26,193)(27,196)(28,199)(29,194)(30,197)(31,200)(32,195)(33,81)(34,84)(35,87)(36,82)(37,85)(38,88)(39,83)(40,86)(41,98)(42,101)(43,104)(44,99)(45,102)(46,97)(47,100)(48,103)(49,108)(50,111)(51,106)(52,109)(53,112)(54,107)(55,110)(56,105)(73,90)(74,93)(75,96)(76,91)(77,94)(78,89)(79,92)(80,95)(113,151)(114,146)(115,149)(116,152)(117,147)(118,150)(119,145)(120,148)(121,188)(122,191)(123,186)(124,189)(125,192)(126,187)(127,190)(128,185)(129,156)(130,159)(131,154)(132,157)(133,160)(134,155)(135,158)(136,153)(137,166)(138,161)(139,164)(140,167)(141,162)(142,165)(143,168)(144,163)(169,211)(170,214)(171,209)(172,212)(173,215)(174,210)(175,213)(176,216)(177,207)(178,202)(179,205)(180,208)(181,203)(182,206)(183,201)(184,204), (1,43,61,167,149,179,189)(2,44,62,168,150,180,190)(3,45,63,161,151,181,191)(4,46,64,162,152,182,192)(5,47,57,163,145,183,185)(6,48,58,164,146,184,186)(7,41,59,165,147,177,187)(8,42,60,166,148,178,188)(9,90,197,132,213,110,85)(10,91,198,133,214,111,86)(11,92,199,134,215,112,87)(12,93,200,135,216,105,88)(13,94,193,136,209,106,81)(14,95,194,129,210,107,82)(15,96,195,130,211,108,83)(16,89,196,131,212,109,84)(17,142,117,207,126,217,98)(18,143,118,208,127,218,99)(19,144,119,201,128,219,100)(20,137,120,202,121,220,101)(21,138,113,203,122,221,102)(22,139,114,204,123,222,103)(23,140,115,205,124,223,104)(24,141,116,206,125,224,97)(25,160,170,50,40,66,76)(26,153,171,51,33,67,77)(27,154,172,52,34,68,78)(28,155,173,53,35,69,79)(29,156,174,54,36,70,80)(30,157,175,55,37,71,73)(31,158,176,56,38,72,74)(32,159,169,49,39,65,75), (1,189)(2,186)(3,191)(4,188)(5,185)(6,190)(7,187)(8,192)(9,136)(10,133)(11,130)(12,135)(13,132)(14,129)(15,134)(16,131)(17,113)(18,118)(19,115)(20,120)(21,117)(22,114)(23,119)(24,116)(25,80)(26,77)(27,74)(28,79)(29,76)(30,73)(31,78)(32,75)(33,171)(34,176)(35,173)(36,170)(37,175)(38,172)(39,169)(40,174)(41,177)(42,182)(43,179)(44,184)(45,181)(46,178)(47,183)(48,180)(50,54)(52,56)(57,145)(58,150)(59,147)(60,152)(61,149)(62,146)(63,151)(64,148)(65,159)(66,156)(67,153)(68,158)(69,155)(70,160)(71,157)(72,154)(81,213)(82,210)(83,215)(84,212)(85,209)(86,214)(87,211)(88,216)(89,196)(90,193)(91,198)(92,195)(93,200)(94,197)(95,194)(96,199)(97,206)(98,203)(99,208)(100,205)(101,202)(102,207)(103,204)(104,201)(106,110)(108,112)(121,220)(122,217)(123,222)(124,219)(125,224)(126,221)(127,218)(128,223)(138,142)(140,144)(162,166)(164,168)>;
G:=Group( (1,196)(2,197)(3,198)(4,199)(5,200)(6,193)(7,194)(8,195)(9,180)(10,181)(11,182)(12,183)(13,184)(14,177)(15,178)(16,179)(17,174)(18,175)(19,176)(20,169)(21,170)(22,171)(23,172)(24,173)(25,221)(26,222)(27,223)(28,224)(29,217)(30,218)(31,219)(32,220)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,113)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,216)(58,209)(59,210)(60,211)(61,212)(62,213)(63,214)(64,215)(65,202)(66,203)(67,204)(68,205)(69,206)(70,207)(71,208)(72,201)(73,127)(74,128)(75,121)(76,122)(77,123)(78,124)(79,125)(80,126)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,145)(89,189)(90,190)(91,191)(92,192)(93,185)(94,186)(95,187)(96,188)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,153)(104,154)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,161)(112,162), (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), (1,223)(2,218)(3,221)(4,224)(5,219)(6,222)(7,217)(8,220)(9,71)(10,66)(11,69)(12,72)(13,67)(14,70)(15,65)(16,68)(17,59)(18,62)(19,57)(20,60)(21,63)(22,58)(23,61)(24,64)(25,198)(26,193)(27,196)(28,199)(29,194)(30,197)(31,200)(32,195)(33,81)(34,84)(35,87)(36,82)(37,85)(38,88)(39,83)(40,86)(41,98)(42,101)(43,104)(44,99)(45,102)(46,97)(47,100)(48,103)(49,108)(50,111)(51,106)(52,109)(53,112)(54,107)(55,110)(56,105)(73,90)(74,93)(75,96)(76,91)(77,94)(78,89)(79,92)(80,95)(113,151)(114,146)(115,149)(116,152)(117,147)(118,150)(119,145)(120,148)(121,188)(122,191)(123,186)(124,189)(125,192)(126,187)(127,190)(128,185)(129,156)(130,159)(131,154)(132,157)(133,160)(134,155)(135,158)(136,153)(137,166)(138,161)(139,164)(140,167)(141,162)(142,165)(143,168)(144,163)(169,211)(170,214)(171,209)(172,212)(173,215)(174,210)(175,213)(176,216)(177,207)(178,202)(179,205)(180,208)(181,203)(182,206)(183,201)(184,204), (1,43,61,167,149,179,189)(2,44,62,168,150,180,190)(3,45,63,161,151,181,191)(4,46,64,162,152,182,192)(5,47,57,163,145,183,185)(6,48,58,164,146,184,186)(7,41,59,165,147,177,187)(8,42,60,166,148,178,188)(9,90,197,132,213,110,85)(10,91,198,133,214,111,86)(11,92,199,134,215,112,87)(12,93,200,135,216,105,88)(13,94,193,136,209,106,81)(14,95,194,129,210,107,82)(15,96,195,130,211,108,83)(16,89,196,131,212,109,84)(17,142,117,207,126,217,98)(18,143,118,208,127,218,99)(19,144,119,201,128,219,100)(20,137,120,202,121,220,101)(21,138,113,203,122,221,102)(22,139,114,204,123,222,103)(23,140,115,205,124,223,104)(24,141,116,206,125,224,97)(25,160,170,50,40,66,76)(26,153,171,51,33,67,77)(27,154,172,52,34,68,78)(28,155,173,53,35,69,79)(29,156,174,54,36,70,80)(30,157,175,55,37,71,73)(31,158,176,56,38,72,74)(32,159,169,49,39,65,75), (1,189)(2,186)(3,191)(4,188)(5,185)(6,190)(7,187)(8,192)(9,136)(10,133)(11,130)(12,135)(13,132)(14,129)(15,134)(16,131)(17,113)(18,118)(19,115)(20,120)(21,117)(22,114)(23,119)(24,116)(25,80)(26,77)(27,74)(28,79)(29,76)(30,73)(31,78)(32,75)(33,171)(34,176)(35,173)(36,170)(37,175)(38,172)(39,169)(40,174)(41,177)(42,182)(43,179)(44,184)(45,181)(46,178)(47,183)(48,180)(50,54)(52,56)(57,145)(58,150)(59,147)(60,152)(61,149)(62,146)(63,151)(64,148)(65,159)(66,156)(67,153)(68,158)(69,155)(70,160)(71,157)(72,154)(81,213)(82,210)(83,215)(84,212)(85,209)(86,214)(87,211)(88,216)(89,196)(90,193)(91,198)(92,195)(93,200)(94,197)(95,194)(96,199)(97,206)(98,203)(99,208)(100,205)(101,202)(102,207)(103,204)(104,201)(106,110)(108,112)(121,220)(122,217)(123,222)(124,219)(125,224)(126,221)(127,218)(128,223)(138,142)(140,144)(162,166)(164,168) );
G=PermutationGroup([(1,196),(2,197),(3,198),(4,199),(5,200),(6,193),(7,194),(8,195),(9,180),(10,181),(11,182),(12,183),(13,184),(14,177),(15,178),(16,179),(17,174),(18,175),(19,176),(20,169),(21,170),(22,171),(23,172),(24,173),(25,221),(26,222),(27,223),(28,224),(29,217),(30,218),(31,219),(32,220),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,113),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,216),(58,209),(59,210),(60,211),(61,212),(62,213),(63,214),(64,215),(65,202),(66,203),(67,204),(68,205),(69,206),(70,207),(71,208),(72,201),(73,127),(74,128),(75,121),(76,122),(77,123),(78,124),(79,125),(80,126),(81,146),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,145),(89,189),(90,190),(91,191),(92,192),(93,185),(94,186),(95,187),(96,188),(97,155),(98,156),(99,157),(100,158),(101,159),(102,160),(103,153),(104,154),(105,163),(106,164),(107,165),(108,166),(109,167),(110,168),(111,161),(112,162)], [(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)], [(1,223),(2,218),(3,221),(4,224),(5,219),(6,222),(7,217),(8,220),(9,71),(10,66),(11,69),(12,72),(13,67),(14,70),(15,65),(16,68),(17,59),(18,62),(19,57),(20,60),(21,63),(22,58),(23,61),(24,64),(25,198),(26,193),(27,196),(28,199),(29,194),(30,197),(31,200),(32,195),(33,81),(34,84),(35,87),(36,82),(37,85),(38,88),(39,83),(40,86),(41,98),(42,101),(43,104),(44,99),(45,102),(46,97),(47,100),(48,103),(49,108),(50,111),(51,106),(52,109),(53,112),(54,107),(55,110),(56,105),(73,90),(74,93),(75,96),(76,91),(77,94),(78,89),(79,92),(80,95),(113,151),(114,146),(115,149),(116,152),(117,147),(118,150),(119,145),(120,148),(121,188),(122,191),(123,186),(124,189),(125,192),(126,187),(127,190),(128,185),(129,156),(130,159),(131,154),(132,157),(133,160),(134,155),(135,158),(136,153),(137,166),(138,161),(139,164),(140,167),(141,162),(142,165),(143,168),(144,163),(169,211),(170,214),(171,209),(172,212),(173,215),(174,210),(175,213),(176,216),(177,207),(178,202),(179,205),(180,208),(181,203),(182,206),(183,201),(184,204)], [(1,43,61,167,149,179,189),(2,44,62,168,150,180,190),(3,45,63,161,151,181,191),(4,46,64,162,152,182,192),(5,47,57,163,145,183,185),(6,48,58,164,146,184,186),(7,41,59,165,147,177,187),(8,42,60,166,148,178,188),(9,90,197,132,213,110,85),(10,91,198,133,214,111,86),(11,92,199,134,215,112,87),(12,93,200,135,216,105,88),(13,94,193,136,209,106,81),(14,95,194,129,210,107,82),(15,96,195,130,211,108,83),(16,89,196,131,212,109,84),(17,142,117,207,126,217,98),(18,143,118,208,127,218,99),(19,144,119,201,128,219,100),(20,137,120,202,121,220,101),(21,138,113,203,122,221,102),(22,139,114,204,123,222,103),(23,140,115,205,124,223,104),(24,141,116,206,125,224,97),(25,160,170,50,40,66,76),(26,153,171,51,33,67,77),(27,154,172,52,34,68,78),(28,155,173,53,35,69,79),(29,156,174,54,36,70,80),(30,157,175,55,37,71,73),(31,158,176,56,38,72,74),(32,159,169,49,39,65,75)], [(1,189),(2,186),(3,191),(4,188),(5,185),(6,190),(7,187),(8,192),(9,136),(10,133),(11,130),(12,135),(13,132),(14,129),(15,134),(16,131),(17,113),(18,118),(19,115),(20,120),(21,117),(22,114),(23,119),(24,116),(25,80),(26,77),(27,74),(28,79),(29,76),(30,73),(31,78),(32,75),(33,171),(34,176),(35,173),(36,170),(37,175),(38,172),(39,169),(40,174),(41,177),(42,182),(43,179),(44,184),(45,181),(46,178),(47,183),(48,180),(50,54),(52,56),(57,145),(58,150),(59,147),(60,152),(61,149),(62,146),(63,151),(64,148),(65,159),(66,156),(67,153),(68,158),(69,155),(70,160),(71,157),(72,154),(81,213),(82,210),(83,215),(84,212),(85,209),(86,214),(87,211),(88,216),(89,196),(90,193),(91,198),(92,195),(93,200),(94,197),(95,194),(96,199),(97,206),(98,203),(99,208),(100,205),(101,202),(102,207),(103,204),(104,201),(106,110),(108,112),(121,220),(122,217),(123,222),(124,219),(125,224),(126,221),(127,218),(128,223),(138,142),(140,144),(162,166),(164,168)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 60 | 18 | 0 | 0 | 0 | 0 |
| 95 | 53 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 37 | 27 |
| 0 | 0 | 0 | 0 | 12 | 76 |
| 0 | 0 | 38 | 43 | 37 | 27 |
| 0 | 0 | 107 | 75 | 12 | 76 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 86 | 20 | 34 | 92 |
| 0 | 0 | 34 | 27 | 66 | 79 |
| 0 | 0 | 103 | 66 | 27 | 93 |
| 0 | 0 | 67 | 10 | 79 | 86 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 112 | 0 | 0 |
| 0 | 0 | 100 | 89 | 0 | 0 |
| 0 | 0 | 0 | 0 | 103 | 112 |
| 0 | 0 | 0 | 0 | 100 | 89 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 25 | 0 | 0 |
| 0 | 0 | 100 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 103 | 25 |
| 0 | 0 | 0 | 0 | 100 | 10 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[60,95,0,0,0,0,18,53,0,0,0,0,0,0,0,0,38,107,0,0,0,0,43,75,0,0,37,12,37,12,0,0,27,76,27,76],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,86,34,103,67,0,0,20,27,66,10,0,0,34,66,27,79,0,0,92,79,93,86],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,100,0,0,0,0,112,89,0,0,0,0,0,0,103,100,0,0,0,0,112,89],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,100,0,0,0,0,25,10,0,0,0,0,0,0,103,100,0,0,0,0,25,10] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 14 | 14 | 2 | 2 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | C8.C22 | D4×D7 | D4×D7 | SD16⋊D7 |
| kernel | C2×SD16⋊D7 | C2×C8⋊D7 | C2×Dic28 | SD16⋊D7 | C2×D4.D7 | C2×C7⋊Q16 | C14×SD16 | C2×D4⋊2D7 | C2×Q8×D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 3 | 12 | 3 | 3 | 2 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_2\times SD_{16}\rtimes D_7 % in TeX
G:=Group("C2xSD16:D7"); // GroupNames label
G:=SmallGroup(448,1213);
// by ID
G=gap.SmallGroup(448,1213);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,1123,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^3,b*d=d*b,e*b*e=b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations
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