direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D7×Q16, C28.10C24, C56.33C23, Dic28⋊15C22, Dic14.6C23, C14⋊2(C2×Q16), C4.46(D4×D7), C7⋊2(C22×Q16), (C14×Q16)⋊7C2, (C4×D7).30D4, C28.85(C2×D4), C7⋊C8.22C23, D14.65(C2×D4), (C2×C8).246D14, (C7×Q16)⋊8C22, C7⋊Q16⋊8C22, C4.10(C23×D7), C8.39(C22×D7), (C2×Dic28)⋊20C2, Q8.4(C22×D7), (C7×Q8).4C23, (Q8×D7).4C22, (C2×C56).98C22, (C2×Q8).152D14, Dic7.13(C2×D4), (C8×D7).15C22, (C4×D7).27C23, C22.142(D4×D7), (C2×C28).527C23, (C2×Dic7).123D4, (C22×D7).112D4, C14.111(C22×D4), (Q8×C14).149C22, (C2×Dic14).198C22, (D7×C2×C8).6C2, C2.84(C2×D4×D7), (C2×Q8×D7).8C2, (C2×C7⋊Q16)⋊27C2, (C2×C14).400(C2×D4), (C2×C7⋊C8).285C22, (C2×C4×D7).259C22, (C2×C4).615(C22×D7), SmallGroup(448,1216)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1124 in 258 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×6], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×17], Q8 [×4], Q8 [×16], C23, D7 [×4], C14, C14 [×2], C2×C8, C2×C8 [×5], Q16 [×4], Q16 [×12], C22×C4 [×3], C2×Q8 [×2], C2×Q8 [×16], Dic7 [×2], Dic7 [×4], C28 [×2], C28 [×4], D14 [×6], C2×C14, C22×C8, C2×Q16, C2×Q16 [×11], C22×Q8 [×2], C7⋊C8 [×2], C56 [×2], Dic14 [×4], Dic14 [×10], C4×D7 [×4], C4×D7 [×8], C2×Dic7, C2×Dic7 [×2], C2×C28, C2×C28 [×2], C7×Q8 [×4], C7×Q8 [×2], C22×D7, C22×Q16, C8×D7 [×4], Dic28 [×4], C2×C7⋊C8, C7⋊Q16 [×8], C2×C56, C7×Q16 [×4], C2×Dic14 [×2], C2×Dic14 [×2], C2×C4×D7, C2×C4×D7 [×2], Q8×D7 [×8], Q8×D7 [×4], Q8×C14 [×2], D7×C2×C8, C2×Dic28, D7×Q16 [×8], C2×C7⋊Q16 [×2], C14×Q16, C2×Q8×D7 [×2], C2×D7×Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, Q16 [×4], C2×D4 [×6], C24, D14 [×7], C2×Q16 [×6], C22×D4, C22×D7 [×7], C22×Q16, D4×D7 [×2], C23×D7, D7×Q16 [×2], C2×D4×D7, C2×D7×Q16
Generators and relations
G = < a,b,c,d,e | a2=b7=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►On 224 points
Generators in S224
(1 164)(2 165)(3 166)(4 167)(5 168)(6 161)(7 162)(8 163)(9 156)(10 157)(11 158)(12 159)(13 160)(14 153)(15 154)(16 155)(17 150)(18 151)(19 152)(20 145)(21 146)(22 147)(23 148)(24 149)(25 141)(26 142)(27 143)(28 144)(29 137)(30 138)(31 139)(32 140)(33 111)(34 112)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 89)(49 98)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 97)(57 213)(58 214)(59 215)(60 216)(61 209)(62 210)(63 211)(64 212)(65 221)(66 222)(67 223)(68 224)(69 217)(70 218)(71 219)(72 220)(73 193)(74 194)(75 195)(76 196)(77 197)(78 198)(79 199)(80 200)(81 201)(82 202)(83 203)(84 204)(85 205)(86 206)(87 207)(88 208)(113 184)(114 177)(115 178)(116 179)(117 180)(118 181)(119 182)(120 183)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 169)(129 185)(130 186)(131 187)(132 188)(133 189)(134 190)(135 191)(136 192)
(1 194 16 17 210 92 103)(2 195 9 18 211 93 104)(3 196 10 19 212 94 97)(4 197 11 20 213 95 98)(5 198 12 21 214 96 99)(6 199 13 22 215 89 100)(7 200 14 23 216 90 101)(8 193 15 24 209 91 102)(25 136 112 117 128 221 203)(26 129 105 118 121 222 204)(27 130 106 119 122 223 205)(28 131 107 120 123 224 206)(29 132 108 113 124 217 207)(30 133 109 114 125 218 208)(31 134 110 115 126 219 201)(32 135 111 116 127 220 202)(33 179 176 72 82 140 191)(34 180 169 65 83 141 192)(35 181 170 66 84 142 185)(36 182 171 67 85 143 186)(37 183 172 68 86 144 187)(38 184 173 69 87 137 188)(39 177 174 70 88 138 189)(40 178 175 71 81 139 190)(41 52 162 80 153 148 60)(42 53 163 73 154 149 61)(43 54 164 74 155 150 62)(44 55 165 75 156 151 63)(45 56 166 76 157 152 64)(46 49 167 77 158 145 57)(47 50 168 78 159 146 58)(48 51 161 79 160 147 59)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 97)(8 98)(9 215)(10 216)(11 209)(12 210)(13 211)(14 212)(15 213)(16 214)(17 21)(18 22)(19 23)(20 24)(25 207)(26 208)(27 201)(28 202)(29 203)(30 204)(31 205)(32 206)(33 172)(34 173)(35 174)(36 175)(37 176)(38 169)(39 170)(40 171)(41 76)(42 77)(43 78)(44 79)(45 80)(46 73)(47 74)(48 75)(49 163)(50 164)(51 165)(52 166)(53 167)(54 168)(55 161)(56 162)(57 154)(58 155)(59 156)(60 157)(61 158)(62 159)(63 160)(64 153)(65 188)(66 189)(67 190)(68 191)(69 192)(70 185)(71 186)(72 187)(81 143)(82 144)(83 137)(84 138)(85 139)(86 140)(87 141)(88 142)(89 195)(90 196)(91 197)(92 198)(93 199)(94 200)(95 193)(96 194)(105 125)(106 126)(107 127)(108 128)(109 121)(110 122)(111 123)(112 124)(113 117)(114 118)(115 119)(116 120)(129 218)(130 219)(131 220)(132 221)(133 222)(134 223)(135 224)(136 217)(145 149)(146 150)(147 151)(148 152)(177 181)(178 182)(179 183)(180 184)
(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)
(1 144 5 140)(2 143 6 139)(3 142 7 138)(4 141 8 137)(9 36 13 40)(10 35 14 39)(11 34 15 38)(12 33 16 37)(17 183 21 179)(18 182 22 178)(19 181 23 177)(20 180 24 184)(25 163 29 167)(26 162 30 166)(27 161 31 165)(28 168 32 164)(41 218 45 222)(42 217 46 221)(43 224 47 220)(44 223 48 219)(49 203 53 207)(50 202 54 206)(51 201 55 205)(52 208 56 204)(57 128 61 124)(58 127 62 123)(59 126 63 122)(60 125 64 121)(65 91 69 95)(66 90 70 94)(67 89 71 93)(68 96 72 92)(73 132 77 136)(74 131 78 135)(75 130 79 134)(76 129 80 133)(81 104 85 100)(82 103 86 99)(83 102 87 98)(84 101 88 97)(105 153 109 157)(106 160 110 156)(107 159 111 155)(108 158 112 154)(113 145 117 149)(114 152 118 148)(115 151 119 147)(116 150 120 146)(169 209 173 213)(170 216 174 212)(171 215 175 211)(172 214 176 210)(185 200 189 196)(186 199 190 195)(187 198 191 194)(188 197 192 193)
G:=sub<Sym(224)| (1,164)(2,165)(3,166)(4,167)(5,168)(6,161)(7,162)(8,163)(9,156)(10,157)(11,158)(12,159)(13,160)(14,153)(15,154)(16,155)(17,150)(18,151)(19,152)(20,145)(21,146)(22,147)(23,148)(24,149)(25,141)(26,142)(27,143)(28,144)(29,137)(30,138)(31,139)(32,140)(33,111)(34,112)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,89)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,97)(57,213)(58,214)(59,215)(60,216)(61,209)(62,210)(63,211)(64,212)(65,221)(66,222)(67,223)(68,224)(69,217)(70,218)(71,219)(72,220)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(113,184)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,169)(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)(136,192), (1,194,16,17,210,92,103)(2,195,9,18,211,93,104)(3,196,10,19,212,94,97)(4,197,11,20,213,95,98)(5,198,12,21,214,96,99)(6,199,13,22,215,89,100)(7,200,14,23,216,90,101)(8,193,15,24,209,91,102)(25,136,112,117,128,221,203)(26,129,105,118,121,222,204)(27,130,106,119,122,223,205)(28,131,107,120,123,224,206)(29,132,108,113,124,217,207)(30,133,109,114,125,218,208)(31,134,110,115,126,219,201)(32,135,111,116,127,220,202)(33,179,176,72,82,140,191)(34,180,169,65,83,141,192)(35,181,170,66,84,142,185)(36,182,171,67,85,143,186)(37,183,172,68,86,144,187)(38,184,173,69,87,137,188)(39,177,174,70,88,138,189)(40,178,175,71,81,139,190)(41,52,162,80,153,148,60)(42,53,163,73,154,149,61)(43,54,164,74,155,150,62)(44,55,165,75,156,151,63)(45,56,166,76,157,152,64)(46,49,167,77,158,145,57)(47,50,168,78,159,146,58)(48,51,161,79,160,147,59), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,215)(10,216)(11,209)(12,210)(13,211)(14,212)(15,213)(16,214)(17,21)(18,22)(19,23)(20,24)(25,207)(26,208)(27,201)(28,202)(29,203)(30,204)(31,205)(32,206)(33,172)(34,173)(35,174)(36,175)(37,176)(38,169)(39,170)(40,171)(41,76)(42,77)(43,78)(44,79)(45,80)(46,73)(47,74)(48,75)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,161)(56,162)(57,154)(58,155)(59,156)(60,157)(61,158)(62,159)(63,160)(64,153)(65,188)(66,189)(67,190)(68,191)(69,192)(70,185)(71,186)(72,187)(81,143)(82,144)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(89,195)(90,196)(91,197)(92,198)(93,199)(94,200)(95,193)(96,194)(105,125)(106,126)(107,127)(108,128)(109,121)(110,122)(111,123)(112,124)(113,117)(114,118)(115,119)(116,120)(129,218)(130,219)(131,220)(132,221)(133,222)(134,223)(135,224)(136,217)(145,149)(146,150)(147,151)(148,152)(177,181)(178,182)(179,183)(180,184), (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), (1,144,5,140)(2,143,6,139)(3,142,7,138)(4,141,8,137)(9,36,13,40)(10,35,14,39)(11,34,15,38)(12,33,16,37)(17,183,21,179)(18,182,22,178)(19,181,23,177)(20,180,24,184)(25,163,29,167)(26,162,30,166)(27,161,31,165)(28,168,32,164)(41,218,45,222)(42,217,46,221)(43,224,47,220)(44,223,48,219)(49,203,53,207)(50,202,54,206)(51,201,55,205)(52,208,56,204)(57,128,61,124)(58,127,62,123)(59,126,63,122)(60,125,64,121)(65,91,69,95)(66,90,70,94)(67,89,71,93)(68,96,72,92)(73,132,77,136)(74,131,78,135)(75,130,79,134)(76,129,80,133)(81,104,85,100)(82,103,86,99)(83,102,87,98)(84,101,88,97)(105,153,109,157)(106,160,110,156)(107,159,111,155)(108,158,112,154)(113,145,117,149)(114,152,118,148)(115,151,119,147)(116,150,120,146)(169,209,173,213)(170,216,174,212)(171,215,175,211)(172,214,176,210)(185,200,189,196)(186,199,190,195)(187,198,191,194)(188,197,192,193)>;
G:=Group( (1,164)(2,165)(3,166)(4,167)(5,168)(6,161)(7,162)(8,163)(9,156)(10,157)(11,158)(12,159)(13,160)(14,153)(15,154)(16,155)(17,150)(18,151)(19,152)(20,145)(21,146)(22,147)(23,148)(24,149)(25,141)(26,142)(27,143)(28,144)(29,137)(30,138)(31,139)(32,140)(33,111)(34,112)(35,105)(36,106)(37,107)(38,108)(39,109)(40,110)(41,90)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,89)(49,98)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,97)(57,213)(58,214)(59,215)(60,216)(61,209)(62,210)(63,211)(64,212)(65,221)(66,222)(67,223)(68,224)(69,217)(70,218)(71,219)(72,220)(73,193)(74,194)(75,195)(76,196)(77,197)(78,198)(79,199)(80,200)(81,201)(82,202)(83,203)(84,204)(85,205)(86,206)(87,207)(88,208)(113,184)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,169)(129,185)(130,186)(131,187)(132,188)(133,189)(134,190)(135,191)(136,192), (1,194,16,17,210,92,103)(2,195,9,18,211,93,104)(3,196,10,19,212,94,97)(4,197,11,20,213,95,98)(5,198,12,21,214,96,99)(6,199,13,22,215,89,100)(7,200,14,23,216,90,101)(8,193,15,24,209,91,102)(25,136,112,117,128,221,203)(26,129,105,118,121,222,204)(27,130,106,119,122,223,205)(28,131,107,120,123,224,206)(29,132,108,113,124,217,207)(30,133,109,114,125,218,208)(31,134,110,115,126,219,201)(32,135,111,116,127,220,202)(33,179,176,72,82,140,191)(34,180,169,65,83,141,192)(35,181,170,66,84,142,185)(36,182,171,67,85,143,186)(37,183,172,68,86,144,187)(38,184,173,69,87,137,188)(39,177,174,70,88,138,189)(40,178,175,71,81,139,190)(41,52,162,80,153,148,60)(42,53,163,73,154,149,61)(43,54,164,74,155,150,62)(44,55,165,75,156,151,63)(45,56,166,76,157,152,64)(46,49,167,77,158,145,57)(47,50,168,78,159,146,58)(48,51,161,79,160,147,59), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,97)(8,98)(9,215)(10,216)(11,209)(12,210)(13,211)(14,212)(15,213)(16,214)(17,21)(18,22)(19,23)(20,24)(25,207)(26,208)(27,201)(28,202)(29,203)(30,204)(31,205)(32,206)(33,172)(34,173)(35,174)(36,175)(37,176)(38,169)(39,170)(40,171)(41,76)(42,77)(43,78)(44,79)(45,80)(46,73)(47,74)(48,75)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,161)(56,162)(57,154)(58,155)(59,156)(60,157)(61,158)(62,159)(63,160)(64,153)(65,188)(66,189)(67,190)(68,191)(69,192)(70,185)(71,186)(72,187)(81,143)(82,144)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(89,195)(90,196)(91,197)(92,198)(93,199)(94,200)(95,193)(96,194)(105,125)(106,126)(107,127)(108,128)(109,121)(110,122)(111,123)(112,124)(113,117)(114,118)(115,119)(116,120)(129,218)(130,219)(131,220)(132,221)(133,222)(134,223)(135,224)(136,217)(145,149)(146,150)(147,151)(148,152)(177,181)(178,182)(179,183)(180,184), (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), (1,144,5,140)(2,143,6,139)(3,142,7,138)(4,141,8,137)(9,36,13,40)(10,35,14,39)(11,34,15,38)(12,33,16,37)(17,183,21,179)(18,182,22,178)(19,181,23,177)(20,180,24,184)(25,163,29,167)(26,162,30,166)(27,161,31,165)(28,168,32,164)(41,218,45,222)(42,217,46,221)(43,224,47,220)(44,223,48,219)(49,203,53,207)(50,202,54,206)(51,201,55,205)(52,208,56,204)(57,128,61,124)(58,127,62,123)(59,126,63,122)(60,125,64,121)(65,91,69,95)(66,90,70,94)(67,89,71,93)(68,96,72,92)(73,132,77,136)(74,131,78,135)(75,130,79,134)(76,129,80,133)(81,104,85,100)(82,103,86,99)(83,102,87,98)(84,101,88,97)(105,153,109,157)(106,160,110,156)(107,159,111,155)(108,158,112,154)(113,145,117,149)(114,152,118,148)(115,151,119,147)(116,150,120,146)(169,209,173,213)(170,216,174,212)(171,215,175,211)(172,214,176,210)(185,200,189,196)(186,199,190,195)(187,198,191,194)(188,197,192,193) );
G=PermutationGroup([(1,164),(2,165),(3,166),(4,167),(5,168),(6,161),(7,162),(8,163),(9,156),(10,157),(11,158),(12,159),(13,160),(14,153),(15,154),(16,155),(17,150),(18,151),(19,152),(20,145),(21,146),(22,147),(23,148),(24,149),(25,141),(26,142),(27,143),(28,144),(29,137),(30,138),(31,139),(32,140),(33,111),(34,112),(35,105),(36,106),(37,107),(38,108),(39,109),(40,110),(41,90),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,89),(49,98),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,97),(57,213),(58,214),(59,215),(60,216),(61,209),(62,210),(63,211),(64,212),(65,221),(66,222),(67,223),(68,224),(69,217),(70,218),(71,219),(72,220),(73,193),(74,194),(75,195),(76,196),(77,197),(78,198),(79,199),(80,200),(81,201),(82,202),(83,203),(84,204),(85,205),(86,206),(87,207),(88,208),(113,184),(114,177),(115,178),(116,179),(117,180),(118,181),(119,182),(120,183),(121,170),(122,171),(123,172),(124,173),(125,174),(126,175),(127,176),(128,169),(129,185),(130,186),(131,187),(132,188),(133,189),(134,190),(135,191),(136,192)], [(1,194,16,17,210,92,103),(2,195,9,18,211,93,104),(3,196,10,19,212,94,97),(4,197,11,20,213,95,98),(5,198,12,21,214,96,99),(6,199,13,22,215,89,100),(7,200,14,23,216,90,101),(8,193,15,24,209,91,102),(25,136,112,117,128,221,203),(26,129,105,118,121,222,204),(27,130,106,119,122,223,205),(28,131,107,120,123,224,206),(29,132,108,113,124,217,207),(30,133,109,114,125,218,208),(31,134,110,115,126,219,201),(32,135,111,116,127,220,202),(33,179,176,72,82,140,191),(34,180,169,65,83,141,192),(35,181,170,66,84,142,185),(36,182,171,67,85,143,186),(37,183,172,68,86,144,187),(38,184,173,69,87,137,188),(39,177,174,70,88,138,189),(40,178,175,71,81,139,190),(41,52,162,80,153,148,60),(42,53,163,73,154,149,61),(43,54,164,74,155,150,62),(44,55,165,75,156,151,63),(45,56,166,76,157,152,64),(46,49,167,77,158,145,57),(47,50,168,78,159,146,58),(48,51,161,79,160,147,59)], [(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,97),(8,98),(9,215),(10,216),(11,209),(12,210),(13,211),(14,212),(15,213),(16,214),(17,21),(18,22),(19,23),(20,24),(25,207),(26,208),(27,201),(28,202),(29,203),(30,204),(31,205),(32,206),(33,172),(34,173),(35,174),(36,175),(37,176),(38,169),(39,170),(40,171),(41,76),(42,77),(43,78),(44,79),(45,80),(46,73),(47,74),(48,75),(49,163),(50,164),(51,165),(52,166),(53,167),(54,168),(55,161),(56,162),(57,154),(58,155),(59,156),(60,157),(61,158),(62,159),(63,160),(64,153),(65,188),(66,189),(67,190),(68,191),(69,192),(70,185),(71,186),(72,187),(81,143),(82,144),(83,137),(84,138),(85,139),(86,140),(87,141),(88,142),(89,195),(90,196),(91,197),(92,198),(93,199),(94,200),(95,193),(96,194),(105,125),(106,126),(107,127),(108,128),(109,121),(110,122),(111,123),(112,124),(113,117),(114,118),(115,119),(116,120),(129,218),(130,219),(131,220),(132,221),(133,222),(134,223),(135,224),(136,217),(145,149),(146,150),(147,151),(148,152),(177,181),(178,182),(179,183),(180,184)], [(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)], [(1,144,5,140),(2,143,6,139),(3,142,7,138),(4,141,8,137),(9,36,13,40),(10,35,14,39),(11,34,15,38),(12,33,16,37),(17,183,21,179),(18,182,22,178),(19,181,23,177),(20,180,24,184),(25,163,29,167),(26,162,30,166),(27,161,31,165),(28,168,32,164),(41,218,45,222),(42,217,46,221),(43,224,47,220),(44,223,48,219),(49,203,53,207),(50,202,54,206),(51,201,55,205),(52,208,56,204),(57,128,61,124),(58,127,62,123),(59,126,63,122),(60,125,64,121),(65,91,69,95),(66,90,70,94),(67,89,71,93),(68,96,72,92),(73,132,77,136),(74,131,78,135),(75,130,79,134),(76,129,80,133),(81,104,85,100),(82,103,86,99),(83,102,87,98),(84,101,88,97),(105,153,109,157),(106,160,110,156),(107,159,111,155),(108,158,112,154),(113,145,117,149),(114,152,118,148),(115,151,119,147),(116,150,120,146),(169,209,173,213),(170,216,174,212),(171,215,175,211),(172,214,176,210),(185,200,189,196),(186,199,190,195),(187,198,191,194),(188,197,192,193)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 104 | 1 | 0 | 0 |
| 41 | 33 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 33 | 112 | 0 | 0 |
| 71 | 80 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 51 | 88 |
| 0 | 0 | 104 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 110 |
| 0 | 0 | 112 | 83 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[104,41,0,0,1,33,0,0,0,0,1,0,0,0,0,1],[33,71,0,0,112,80,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,51,104,0,0,88,0],[1,0,0,0,0,1,0,0,0,0,30,112,0,0,110,83] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | Q16 | D14 | D14 | D14 | D4×D7 | D4×D7 | D7×Q16 |
| kernel | C2×D7×Q16 | D7×C2×C8 | C2×Dic28 | D7×Q16 | C2×C7⋊Q16 | C14×Q16 | C2×Q8×D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×Q16 | D14 | C2×C8 | Q16 | C2×Q8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 3 | 8 | 3 | 12 | 6 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_7\times Q_{16} % in TeX
G:=Group("C2xD7xQ16"); // GroupNames label
G:=SmallGroup(448,1216);
// by ID
G=gap.SmallGroup(448,1216);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^7=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,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
○
≀