direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C8⋊D7, C56⋊12C23, C28.66C24, (C2×C8)⋊36D14, C7⋊C8⋊11C23, (C22×C8)⋊12D7, C8⋊10(C22×D7), (C22×C56)⋊18C2, (C2×C56)⋊48C22, C14⋊1(C2×M4(2)), (C2×C14)⋊6M4(2), (C23×D7).8C4, C23.66(C4×D7), C4.65(C23×D7), C7⋊1(C22×M4(2)), C14.29(C23×C4), (C4×D7).33C23, (C2×C28).879C23, C28.145(C22×C4), D14.20(C22×C4), (C22×C4).470D14, Dic7.21(C22×C4), (C22×Dic7).17C4, (C22×C28).567C22, (C2×C4×D7).22C4, C4.120(C2×C4×D7), (C22×C7⋊C8)⋊22C2, (C2×C7⋊C8)⋊46C22, C2.30(D7×C22×C4), C22.75(C2×C4×D7), (C4×D7).35(C2×C4), (C2×C4).187(C4×D7), (D7×C22×C4).23C2, (C2×C28).257(C2×C4), (C2×C4×D7).301C22, (C22×D7).67(C2×C4), (C2×C4).823(C22×D7), (C22×C14).102(C2×C4), (C2×C14).155(C22×C4), (C2×Dic7).104(C2×C4), SmallGroup(448,1190)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1124 in 298 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C4, C4 [×3], C4 [×4], C22 [×7], C22 [×16], C7, C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×22], C23, C23 [×10], D7 [×4], C14, C14 [×6], C2×C8 [×6], C2×C8 [×6], M4(2) [×16], C22×C4, C22×C4 [×13], C24, Dic7 [×4], C28, C28 [×3], D14 [×4], D14 [×12], C2×C14 [×7], C22×C8, C22×C8, C2×M4(2) [×12], C23×C4, C7⋊C8 [×4], C56 [×4], C4×D7 [×16], C2×Dic7 [×6], C2×C28 [×6], C22×D7 [×6], C22×D7 [×4], C22×C14, C22×M4(2), C8⋊D7 [×16], C2×C7⋊C8 [×6], C2×C56 [×6], C2×C4×D7 [×12], C22×Dic7, C22×C28, C23×D7, C2×C8⋊D7 [×12], C22×C7⋊C8, C22×C56, D7×C22×C4, C22×C8⋊D7
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, M4(2) [×4], C22×C4 [×14], C24, D14 [×7], C2×M4(2) [×6], C23×C4, C4×D7 [×4], C22×D7 [×7], C22×M4(2), C8⋊D7 [×4], C2×C4×D7 [×6], C23×D7, C2×C8⋊D7 [×6], D7×C22×C4, C22×C8⋊D7
Generators and relations
G = < a,b,c,d,e | a2=b2=c8=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c5, ede=d-1 >
►On 224 points
Generators in S224
(1 200)(2 193)(3 194)(4 195)(5 196)(6 197)(7 198)(8 199)(9 169)(10 170)(11 171)(12 172)(13 173)(14 174)(15 175)(16 176)(17 149)(18 150)(19 151)(20 152)(21 145)(22 146)(23 147)(24 148)(25 217)(26 218)(27 219)(28 220)(29 221)(30 222)(31 223)(32 224)(33 209)(34 210)(35 211)(36 212)(37 213)(38 214)(39 215)(40 216)(41 133)(42 134)(43 135)(44 136)(45 129)(46 130)(47 131)(48 132)(49 120)(50 113)(51 114)(52 115)(53 116)(54 117)(55 118)(56 119)(57 205)(58 206)(59 207)(60 208)(61 201)(62 202)(63 203)(64 204)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 127)(72 128)(73 137)(74 138)(75 139)(76 140)(77 141)(78 142)(79 143)(80 144)(81 164)(82 165)(83 166)(84 167)(85 168)(86 161)(87 162)(88 163)(89 179)(90 180)(91 181)(92 182)(93 183)(94 184)(95 177)(96 178)(97 159)(98 160)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(105 187)(106 188)(107 189)(108 190)(109 191)(110 192)(111 185)(112 186)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 201)(10 202)(11 203)(12 204)(13 205)(14 206)(15 207)(16 208)(17 209)(18 210)(19 211)(20 212)(21 213)(22 214)(23 215)(24 216)(33 149)(34 150)(35 151)(36 152)(37 145)(38 146)(39 147)(40 148)(41 157)(42 158)(43 159)(44 160)(45 153)(46 154)(47 155)(48 156)(49 165)(50 166)(51 167)(52 168)(53 161)(54 162)(55 163)(56 164)(57 173)(58 174)(59 175)(60 176)(61 169)(62 170)(63 171)(64 172)(65 181)(66 182)(67 183)(68 184)(69 177)(70 178)(71 179)(72 180)(73 189)(74 190)(75 191)(76 192)(77 185)(78 186)(79 187)(80 188)(81 119)(82 120)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 127)(90 128)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 135)(98 136)(99 129)(100 130)(101 131)(102 132)(103 133)(104 134)(105 143)(106 144)(107 137)(108 138)(109 139)(110 140)(111 141)(112 142)(193 221)(194 222)(195 223)(196 224)(197 217)(198 218)(199 219)(200 220)
(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 54 57 158 183 145 191)(2 55 58 159 184 146 192)(3 56 59 160 177 147 185)(4 49 60 153 178 148 186)(5 50 61 154 179 149 187)(6 51 62 155 180 150 188)(7 52 63 156 181 151 189)(8 53 64 157 182 152 190)(9 130 127 209 143 224 83)(10 131 128 210 144 217 84)(11 132 121 211 137 218 85)(12 133 122 212 138 219 86)(13 134 123 213 139 220 87)(14 135 124 214 140 221 88)(15 136 125 215 141 222 81)(16 129 126 216 142 223 82)(17 105 196 113 201 100 89)(18 106 197 114 202 101 90)(19 107 198 115 203 102 91)(20 108 199 116 204 103 92)(21 109 200 117 205 104 93)(22 110 193 118 206 97 94)(23 111 194 119 207 98 95)(24 112 195 120 208 99 96)(25 167 170 47 72 34 80)(26 168 171 48 65 35 73)(27 161 172 41 66 36 74)(28 162 173 42 67 37 75)(29 163 174 43 68 38 76)(30 164 175 44 69 39 77)(31 165 176 45 70 40 78)(32 166 169 46 71 33 79)
(1 139)(2 144)(3 141)(4 138)(5 143)(6 140)(7 137)(8 142)(9 179)(10 184)(11 181)(12 178)(13 183)(14 180)(15 177)(16 182)(17 166)(18 163)(19 168)(20 165)(21 162)(22 167)(23 164)(24 161)(25 110)(26 107)(27 112)(28 109)(29 106)(30 111)(31 108)(32 105)(33 113)(34 118)(35 115)(36 120)(37 117)(38 114)(39 119)(40 116)(41 99)(42 104)(43 101)(44 98)(45 103)(46 100)(47 97)(48 102)(49 212)(50 209)(51 214)(52 211)(53 216)(54 213)(55 210)(56 215)(57 123)(58 128)(59 125)(60 122)(61 127)(62 124)(63 121)(64 126)(65 203)(66 208)(67 205)(68 202)(69 207)(70 204)(71 201)(72 206)(73 198)(74 195)(75 200)(76 197)(77 194)(78 199)(79 196)(80 193)(81 147)(82 152)(83 149)(84 146)(85 151)(86 148)(87 145)(88 150)(89 169)(90 174)(91 171)(92 176)(93 173)(94 170)(95 175)(96 172)(129 157)(130 154)(131 159)(132 156)(133 153)(134 158)(135 155)(136 160)(185 222)(186 219)(187 224)(188 221)(189 218)(190 223)(191 220)(192 217)
G:=sub<Sym(224)| (1,200)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,169)(10,170)(11,171)(12,172)(13,173)(14,174)(15,175)(16,176)(17,149)(18,150)(19,151)(20,152)(21,145)(22,146)(23,147)(24,148)(25,217)(26,218)(27,219)(28,220)(29,221)(30,222)(31,223)(32,224)(33,209)(34,210)(35,211)(36,212)(37,213)(38,214)(39,215)(40,216)(41,133)(42,134)(43,135)(44,136)(45,129)(46,130)(47,131)(48,132)(49,120)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,205)(58,206)(59,207)(60,208)(61,201)(62,202)(63,203)(64,204)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(81,164)(82,165)(83,166)(84,167)(85,168)(86,161)(87,162)(88,163)(89,179)(90,180)(91,181)(92,182)(93,183)(94,184)(95,177)(96,178)(97,159)(98,160)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,185)(112,186), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,201)(10,202)(11,203)(12,204)(13,205)(14,206)(15,207)(16,208)(17,209)(18,210)(19,211)(20,212)(21,213)(22,214)(23,215)(24,216)(33,149)(34,150)(35,151)(36,152)(37,145)(38,146)(39,147)(40,148)(41,157)(42,158)(43,159)(44,160)(45,153)(46,154)(47,155)(48,156)(49,165)(50,166)(51,167)(52,168)(53,161)(54,162)(55,163)(56,164)(57,173)(58,174)(59,175)(60,176)(61,169)(62,170)(63,171)(64,172)(65,181)(66,182)(67,183)(68,184)(69,177)(70,178)(71,179)(72,180)(73,189)(74,190)(75,191)(76,192)(77,185)(78,186)(79,187)(80,188)(81,119)(82,120)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,127)(90,128)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,135)(98,136)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,143)(106,144)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(193,221)(194,222)(195,223)(196,224)(197,217)(198,218)(199,219)(200,220), (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), (1,54,57,158,183,145,191)(2,55,58,159,184,146,192)(3,56,59,160,177,147,185)(4,49,60,153,178,148,186)(5,50,61,154,179,149,187)(6,51,62,155,180,150,188)(7,52,63,156,181,151,189)(8,53,64,157,182,152,190)(9,130,127,209,143,224,83)(10,131,128,210,144,217,84)(11,132,121,211,137,218,85)(12,133,122,212,138,219,86)(13,134,123,213,139,220,87)(14,135,124,214,140,221,88)(15,136,125,215,141,222,81)(16,129,126,216,142,223,82)(17,105,196,113,201,100,89)(18,106,197,114,202,101,90)(19,107,198,115,203,102,91)(20,108,199,116,204,103,92)(21,109,200,117,205,104,93)(22,110,193,118,206,97,94)(23,111,194,119,207,98,95)(24,112,195,120,208,99,96)(25,167,170,47,72,34,80)(26,168,171,48,65,35,73)(27,161,172,41,66,36,74)(28,162,173,42,67,37,75)(29,163,174,43,68,38,76)(30,164,175,44,69,39,77)(31,165,176,45,70,40,78)(32,166,169,46,71,33,79), (1,139)(2,144)(3,141)(4,138)(5,143)(6,140)(7,137)(8,142)(9,179)(10,184)(11,181)(12,178)(13,183)(14,180)(15,177)(16,182)(17,166)(18,163)(19,168)(20,165)(21,162)(22,167)(23,164)(24,161)(25,110)(26,107)(27,112)(28,109)(29,106)(30,111)(31,108)(32,105)(33,113)(34,118)(35,115)(36,120)(37,117)(38,114)(39,119)(40,116)(41,99)(42,104)(43,101)(44,98)(45,103)(46,100)(47,97)(48,102)(49,212)(50,209)(51,214)(52,211)(53,216)(54,213)(55,210)(56,215)(57,123)(58,128)(59,125)(60,122)(61,127)(62,124)(63,121)(64,126)(65,203)(66,208)(67,205)(68,202)(69,207)(70,204)(71,201)(72,206)(73,198)(74,195)(75,200)(76,197)(77,194)(78,199)(79,196)(80,193)(81,147)(82,152)(83,149)(84,146)(85,151)(86,148)(87,145)(88,150)(89,169)(90,174)(91,171)(92,176)(93,173)(94,170)(95,175)(96,172)(129,157)(130,154)(131,159)(132,156)(133,153)(134,158)(135,155)(136,160)(185,222)(186,219)(187,224)(188,221)(189,218)(190,223)(191,220)(192,217)>;
G:=Group( (1,200)(2,193)(3,194)(4,195)(5,196)(6,197)(7,198)(8,199)(9,169)(10,170)(11,171)(12,172)(13,173)(14,174)(15,175)(16,176)(17,149)(18,150)(19,151)(20,152)(21,145)(22,146)(23,147)(24,148)(25,217)(26,218)(27,219)(28,220)(29,221)(30,222)(31,223)(32,224)(33,209)(34,210)(35,211)(36,212)(37,213)(38,214)(39,215)(40,216)(41,133)(42,134)(43,135)(44,136)(45,129)(46,130)(47,131)(48,132)(49,120)(50,113)(51,114)(52,115)(53,116)(54,117)(55,118)(56,119)(57,205)(58,206)(59,207)(60,208)(61,201)(62,202)(63,203)(64,204)(65,121)(66,122)(67,123)(68,124)(69,125)(70,126)(71,127)(72,128)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(81,164)(82,165)(83,166)(84,167)(85,168)(86,161)(87,162)(88,163)(89,179)(90,180)(91,181)(92,182)(93,183)(94,184)(95,177)(96,178)(97,159)(98,160)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,185)(112,186), (1,28)(2,29)(3,30)(4,31)(5,32)(6,25)(7,26)(8,27)(9,201)(10,202)(11,203)(12,204)(13,205)(14,206)(15,207)(16,208)(17,209)(18,210)(19,211)(20,212)(21,213)(22,214)(23,215)(24,216)(33,149)(34,150)(35,151)(36,152)(37,145)(38,146)(39,147)(40,148)(41,157)(42,158)(43,159)(44,160)(45,153)(46,154)(47,155)(48,156)(49,165)(50,166)(51,167)(52,168)(53,161)(54,162)(55,163)(56,164)(57,173)(58,174)(59,175)(60,176)(61,169)(62,170)(63,171)(64,172)(65,181)(66,182)(67,183)(68,184)(69,177)(70,178)(71,179)(72,180)(73,189)(74,190)(75,191)(76,192)(77,185)(78,186)(79,187)(80,188)(81,119)(82,120)(83,113)(84,114)(85,115)(86,116)(87,117)(88,118)(89,127)(90,128)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,135)(98,136)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,143)(106,144)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(193,221)(194,222)(195,223)(196,224)(197,217)(198,218)(199,219)(200,220), (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), (1,54,57,158,183,145,191)(2,55,58,159,184,146,192)(3,56,59,160,177,147,185)(4,49,60,153,178,148,186)(5,50,61,154,179,149,187)(6,51,62,155,180,150,188)(7,52,63,156,181,151,189)(8,53,64,157,182,152,190)(9,130,127,209,143,224,83)(10,131,128,210,144,217,84)(11,132,121,211,137,218,85)(12,133,122,212,138,219,86)(13,134,123,213,139,220,87)(14,135,124,214,140,221,88)(15,136,125,215,141,222,81)(16,129,126,216,142,223,82)(17,105,196,113,201,100,89)(18,106,197,114,202,101,90)(19,107,198,115,203,102,91)(20,108,199,116,204,103,92)(21,109,200,117,205,104,93)(22,110,193,118,206,97,94)(23,111,194,119,207,98,95)(24,112,195,120,208,99,96)(25,167,170,47,72,34,80)(26,168,171,48,65,35,73)(27,161,172,41,66,36,74)(28,162,173,42,67,37,75)(29,163,174,43,68,38,76)(30,164,175,44,69,39,77)(31,165,176,45,70,40,78)(32,166,169,46,71,33,79), (1,139)(2,144)(3,141)(4,138)(5,143)(6,140)(7,137)(8,142)(9,179)(10,184)(11,181)(12,178)(13,183)(14,180)(15,177)(16,182)(17,166)(18,163)(19,168)(20,165)(21,162)(22,167)(23,164)(24,161)(25,110)(26,107)(27,112)(28,109)(29,106)(30,111)(31,108)(32,105)(33,113)(34,118)(35,115)(36,120)(37,117)(38,114)(39,119)(40,116)(41,99)(42,104)(43,101)(44,98)(45,103)(46,100)(47,97)(48,102)(49,212)(50,209)(51,214)(52,211)(53,216)(54,213)(55,210)(56,215)(57,123)(58,128)(59,125)(60,122)(61,127)(62,124)(63,121)(64,126)(65,203)(66,208)(67,205)(68,202)(69,207)(70,204)(71,201)(72,206)(73,198)(74,195)(75,200)(76,197)(77,194)(78,199)(79,196)(80,193)(81,147)(82,152)(83,149)(84,146)(85,151)(86,148)(87,145)(88,150)(89,169)(90,174)(91,171)(92,176)(93,173)(94,170)(95,175)(96,172)(129,157)(130,154)(131,159)(132,156)(133,153)(134,158)(135,155)(136,160)(185,222)(186,219)(187,224)(188,221)(189,218)(190,223)(191,220)(192,217) );
G=PermutationGroup([(1,200),(2,193),(3,194),(4,195),(5,196),(6,197),(7,198),(8,199),(9,169),(10,170),(11,171),(12,172),(13,173),(14,174),(15,175),(16,176),(17,149),(18,150),(19,151),(20,152),(21,145),(22,146),(23,147),(24,148),(25,217),(26,218),(27,219),(28,220),(29,221),(30,222),(31,223),(32,224),(33,209),(34,210),(35,211),(36,212),(37,213),(38,214),(39,215),(40,216),(41,133),(42,134),(43,135),(44,136),(45,129),(46,130),(47,131),(48,132),(49,120),(50,113),(51,114),(52,115),(53,116),(54,117),(55,118),(56,119),(57,205),(58,206),(59,207),(60,208),(61,201),(62,202),(63,203),(64,204),(65,121),(66,122),(67,123),(68,124),(69,125),(70,126),(71,127),(72,128),(73,137),(74,138),(75,139),(76,140),(77,141),(78,142),(79,143),(80,144),(81,164),(82,165),(83,166),(84,167),(85,168),(86,161),(87,162),(88,163),(89,179),(90,180),(91,181),(92,182),(93,183),(94,184),(95,177),(96,178),(97,159),(98,160),(99,153),(100,154),(101,155),(102,156),(103,157),(104,158),(105,187),(106,188),(107,189),(108,190),(109,191),(110,192),(111,185),(112,186)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,25),(7,26),(8,27),(9,201),(10,202),(11,203),(12,204),(13,205),(14,206),(15,207),(16,208),(17,209),(18,210),(19,211),(20,212),(21,213),(22,214),(23,215),(24,216),(33,149),(34,150),(35,151),(36,152),(37,145),(38,146),(39,147),(40,148),(41,157),(42,158),(43,159),(44,160),(45,153),(46,154),(47,155),(48,156),(49,165),(50,166),(51,167),(52,168),(53,161),(54,162),(55,163),(56,164),(57,173),(58,174),(59,175),(60,176),(61,169),(62,170),(63,171),(64,172),(65,181),(66,182),(67,183),(68,184),(69,177),(70,178),(71,179),(72,180),(73,189),(74,190),(75,191),(76,192),(77,185),(78,186),(79,187),(80,188),(81,119),(82,120),(83,113),(84,114),(85,115),(86,116),(87,117),(88,118),(89,127),(90,128),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(97,135),(98,136),(99,129),(100,130),(101,131),(102,132),(103,133),(104,134),(105,143),(106,144),(107,137),(108,138),(109,139),(110,140),(111,141),(112,142),(193,221),(194,222),(195,223),(196,224),(197,217),(198,218),(199,219),(200,220)], [(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)], [(1,54,57,158,183,145,191),(2,55,58,159,184,146,192),(3,56,59,160,177,147,185),(4,49,60,153,178,148,186),(5,50,61,154,179,149,187),(6,51,62,155,180,150,188),(7,52,63,156,181,151,189),(8,53,64,157,182,152,190),(9,130,127,209,143,224,83),(10,131,128,210,144,217,84),(11,132,121,211,137,218,85),(12,133,122,212,138,219,86),(13,134,123,213,139,220,87),(14,135,124,214,140,221,88),(15,136,125,215,141,222,81),(16,129,126,216,142,223,82),(17,105,196,113,201,100,89),(18,106,197,114,202,101,90),(19,107,198,115,203,102,91),(20,108,199,116,204,103,92),(21,109,200,117,205,104,93),(22,110,193,118,206,97,94),(23,111,194,119,207,98,95),(24,112,195,120,208,99,96),(25,167,170,47,72,34,80),(26,168,171,48,65,35,73),(27,161,172,41,66,36,74),(28,162,173,42,67,37,75),(29,163,174,43,68,38,76),(30,164,175,44,69,39,77),(31,165,176,45,70,40,78),(32,166,169,46,71,33,79)], [(1,139),(2,144),(3,141),(4,138),(5,143),(6,140),(7,137),(8,142),(9,179),(10,184),(11,181),(12,178),(13,183),(14,180),(15,177),(16,182),(17,166),(18,163),(19,168),(20,165),(21,162),(22,167),(23,164),(24,161),(25,110),(26,107),(27,112),(28,109),(29,106),(30,111),(31,108),(32,105),(33,113),(34,118),(35,115),(36,120),(37,117),(38,114),(39,119),(40,116),(41,99),(42,104),(43,101),(44,98),(45,103),(46,100),(47,97),(48,102),(49,212),(50,209),(51,214),(52,211),(53,216),(54,213),(55,210),(56,215),(57,123),(58,128),(59,125),(60,122),(61,127),(62,124),(63,121),(64,126),(65,203),(66,208),(67,205),(68,202),(69,207),(70,204),(71,201),(72,206),(73,198),(74,195),(75,200),(76,197),(77,194),(78,199),(79,196),(80,193),(81,147),(82,152),(83,149),(84,146),(85,151),(86,148),(87,145),(88,150),(89,169),(90,174),(91,171),(92,176),(93,173),(94,170),(95,175),(96,172),(129,157),(130,154),(131,159),(132,156),(133,153),(134,158),(135,155),(136,160),(185,222),(186,219),(187,224),(188,221),(189,218),(190,223),(191,220),(192,217)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 86 | 31 |
| 0 | 0 | 28 | 27 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 104 | 112 |
| 0 | 0 | 72 | 33 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 80 | 10 |
| 0 | 0 | 72 | 33 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,98,0,0,0,0,86,28,0,0,31,27],[1,0,0,0,0,1,0,0,0,0,104,72,0,0,112,33],[112,0,0,0,0,1,0,0,0,0,80,72,0,0,10,33] >;
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8P | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D7 | M4(2) | D14 | D14 | C4×D7 | C4×D7 | C8⋊D7 |
| kernel | C22×C8⋊D7 | C2×C8⋊D7 | C22×C7⋊C8 | C22×C56 | D7×C22×C4 | C2×C4×D7 | C22×Dic7 | C23×D7 | C22×C8 | C2×C14 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 1 | 1 | 12 | 2 | 2 | 3 | 8 | 18 | 3 | 18 | 6 | 48 |
In GAP, Magma, Sage, TeX
C_2^2\times C_8\rtimes D_7
% in TeX
G:=Group("C2^2xC8:D7"); // GroupNames label
G:=SmallGroup(448,1190);
// by ID
G=gap.SmallGroup(448,1190);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,1123,80,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^5,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀