direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C8⋊D7, C56⋊12C23, C28.66C24, (C2×C8)⋊36D14, C7⋊C8⋊11C23, C8⋊10(C22×D7), (C22×C8)⋊12D7, (C22×C56)⋊18C2, (C2×C56)⋊48C22, (C2×C14)⋊6M4(2), C14⋊1(C2×M4(2)), (C23×D7).8C4, C4.65(C23×D7), C23.66(C4×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, (C22×Dic7).17C4, Dic7.21(C22×C4), (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), (C2×C14).155(C22×C4), (C22×C14).102(C2×C4), (C2×Dic7).104(C2×C4), SmallGroup(448,1190)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C8⋊D7
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 >
Subgroups: 1124 in 298 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, Dic7, C28, C28, D14, D14, C2×C14, C22×C8, C22×C8, C2×M4(2), C23×C4, C7⋊C8, C56, C4×D7, C2×Dic7, C2×C28, C22×D7, C22×D7, C22×C14, C22×M4(2), C8⋊D7, C2×C7⋊C8, C2×C56, C2×C4×D7, C22×Dic7, C22×C28, C23×D7, C2×C8⋊D7, C22×C7⋊C8, C22×C56, D7×C22×C4, C22×C8⋊D7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, M4(2), C22×C4, C24, D14, C2×M4(2), C23×C4, C4×D7, C22×D7, C22×M4(2), C8⋊D7, C2×C4×D7, C23×D7, C2×C8⋊D7, D7×C22×C4, C22×C8⋊D7
►On 224 points
Generators in S224
(1 200)(2 193)(3 194)(4 195)(5 196)(6 197)(7 198)(8 199)(9 184)(10 177)(11 178)(12 179)(13 180)(14 181)(15 182)(16 183)(17 163)(18 164)(19 165)(20 166)(21 167)(22 168)(23 161)(24 162)(25 217)(26 218)(27 219)(28 220)(29 221)(30 222)(31 223)(32 224)(33 118)(34 119)(35 120)(36 113)(37 114)(38 115)(39 116)(40 117)(41 133)(42 134)(43 135)(44 136)(45 129)(46 130)(47 131)(48 132)(49 211)(50 212)(51 213)(52 214)(53 215)(54 216)(55 209)(56 210)(57 128)(58 121)(59 122)(60 123)(61 124)(62 125)(63 126)(64 127)(65 206)(66 207)(67 208)(68 201)(69 202)(70 203)(71 204)(72 205)(73 137)(74 138)(75 139)(76 140)(77 141)(78 142)(79 143)(80 144)(81 150)(82 151)(83 152)(84 145)(85 146)(86 147)(87 148)(88 149)(89 172)(90 173)(91 174)(92 175)(93 176)(94 169)(95 170)(96 171)(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 47 57 179 161 145 191)(2 48 58 180 162 146 192)(3 41 59 181 163 147 185)(4 42 60 182 164 148 186)(5 43 61 183 165 149 187)(6 44 62 184 166 150 188)(7 45 63 177 167 151 189)(8 46 64 178 168 152 190)(9 20 81 106 197 136 125)(10 21 82 107 198 129 126)(11 22 83 108 199 130 127)(12 23 84 109 200 131 128)(13 24 85 110 193 132 121)(14 17 86 111 194 133 122)(15 18 87 112 195 134 123)(16 19 88 105 196 135 124)(25 160 170 68 50 34 80)(26 153 171 69 51 35 73)(27 154 172 70 52 36 74)(28 155 173 71 53 37 75)(29 156 174 72 54 38 76)(30 157 175 65 55 39 77)(31 158 176 66 56 40 78)(32 159 169 67 49 33 79)(89 203 214 113 138 219 100)(90 204 215 114 139 220 101)(91 205 216 115 140 221 102)(92 206 209 116 141 222 103)(93 207 210 117 142 223 104)(94 208 211 118 143 224 97)(95 201 212 119 144 217 98)(96 202 213 120 137 218 99)
(1 139)(2 144)(3 141)(4 138)(5 143)(6 140)(7 137)(8 142)(9 72)(10 69)(11 66)(12 71)(13 68)(14 65)(15 70)(16 67)(17 175)(18 172)(19 169)(20 174)(21 171)(22 176)(23 173)(24 170)(25 110)(26 107)(27 112)(28 109)(29 106)(30 111)(31 108)(32 105)(33 135)(34 132)(35 129)(36 134)(37 131)(38 136)(39 133)(40 130)(41 116)(42 113)(43 118)(44 115)(45 120)(46 117)(47 114)(48 119)(49 124)(50 121)(51 126)(52 123)(53 128)(54 125)(55 122)(56 127)(57 215)(58 212)(59 209)(60 214)(61 211)(62 216)(63 213)(64 210)(73 198)(74 195)(75 200)(76 197)(77 194)(78 199)(79 196)(80 193)(81 156)(82 153)(83 158)(84 155)(85 160)(86 157)(87 154)(88 159)(89 164)(90 161)(91 166)(92 163)(93 168)(94 165)(95 162)(96 167)(97 149)(98 146)(99 151)(100 148)(101 145)(102 150)(103 147)(104 152)(177 202)(178 207)(179 204)(180 201)(181 206)(182 203)(183 208)(184 205)(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)(9,184)(10,177)(11,178)(12,179)(13,180)(14,181)(15,182)(16,183)(17,163)(18,164)(19,165)(20,166)(21,167)(22,168)(23,161)(24,162)(25,217)(26,218)(27,219)(28,220)(29,221)(30,222)(31,223)(32,224)(33,118)(34,119)(35,120)(36,113)(37,114)(38,115)(39,116)(40,117)(41,133)(42,134)(43,135)(44,136)(45,129)(46,130)(47,131)(48,132)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(55,209)(56,210)(57,128)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,206)(66,207)(67,208)(68,201)(69,202)(70,203)(71,204)(72,205)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(81,150)(82,151)(83,152)(84,145)(85,146)(86,147)(87,148)(88,149)(89,172)(90,173)(91,174)(92,175)(93,176)(94,169)(95,170)(96,171)(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), (1,47,57,179,161,145,191)(2,48,58,180,162,146,192)(3,41,59,181,163,147,185)(4,42,60,182,164,148,186)(5,43,61,183,165,149,187)(6,44,62,184,166,150,188)(7,45,63,177,167,151,189)(8,46,64,178,168,152,190)(9,20,81,106,197,136,125)(10,21,82,107,198,129,126)(11,22,83,108,199,130,127)(12,23,84,109,200,131,128)(13,24,85,110,193,132,121)(14,17,86,111,194,133,122)(15,18,87,112,195,134,123)(16,19,88,105,196,135,124)(25,160,170,68,50,34,80)(26,153,171,69,51,35,73)(27,154,172,70,52,36,74)(28,155,173,71,53,37,75)(29,156,174,72,54,38,76)(30,157,175,65,55,39,77)(31,158,176,66,56,40,78)(32,159,169,67,49,33,79)(89,203,214,113,138,219,100)(90,204,215,114,139,220,101)(91,205,216,115,140,221,102)(92,206,209,116,141,222,103)(93,207,210,117,142,223,104)(94,208,211,118,143,224,97)(95,201,212,119,144,217,98)(96,202,213,120,137,218,99), (1,139)(2,144)(3,141)(4,138)(5,143)(6,140)(7,137)(8,142)(9,72)(10,69)(11,66)(12,71)(13,68)(14,65)(15,70)(16,67)(17,175)(18,172)(19,169)(20,174)(21,171)(22,176)(23,173)(24,170)(25,110)(26,107)(27,112)(28,109)(29,106)(30,111)(31,108)(32,105)(33,135)(34,132)(35,129)(36,134)(37,131)(38,136)(39,133)(40,130)(41,116)(42,113)(43,118)(44,115)(45,120)(46,117)(47,114)(48,119)(49,124)(50,121)(51,126)(52,123)(53,128)(54,125)(55,122)(56,127)(57,215)(58,212)(59,209)(60,214)(61,211)(62,216)(63,213)(64,210)(73,198)(74,195)(75,200)(76,197)(77,194)(78,199)(79,196)(80,193)(81,156)(82,153)(83,158)(84,155)(85,160)(86,157)(87,154)(88,159)(89,164)(90,161)(91,166)(92,163)(93,168)(94,165)(95,162)(96,167)(97,149)(98,146)(99,151)(100,148)(101,145)(102,150)(103,147)(104,152)(177,202)(178,207)(179,204)(180,201)(181,206)(182,203)(183,208)(184,205)(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)(9,184)(10,177)(11,178)(12,179)(13,180)(14,181)(15,182)(16,183)(17,163)(18,164)(19,165)(20,166)(21,167)(22,168)(23,161)(24,162)(25,217)(26,218)(27,219)(28,220)(29,221)(30,222)(31,223)(32,224)(33,118)(34,119)(35,120)(36,113)(37,114)(38,115)(39,116)(40,117)(41,133)(42,134)(43,135)(44,136)(45,129)(46,130)(47,131)(48,132)(49,211)(50,212)(51,213)(52,214)(53,215)(54,216)(55,209)(56,210)(57,128)(58,121)(59,122)(60,123)(61,124)(62,125)(63,126)(64,127)(65,206)(66,207)(67,208)(68,201)(69,202)(70,203)(71,204)(72,205)(73,137)(74,138)(75,139)(76,140)(77,141)(78,142)(79,143)(80,144)(81,150)(82,151)(83,152)(84,145)(85,146)(86,147)(87,148)(88,149)(89,172)(90,173)(91,174)(92,175)(93,176)(94,169)(95,170)(96,171)(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), (1,47,57,179,161,145,191)(2,48,58,180,162,146,192)(3,41,59,181,163,147,185)(4,42,60,182,164,148,186)(5,43,61,183,165,149,187)(6,44,62,184,166,150,188)(7,45,63,177,167,151,189)(8,46,64,178,168,152,190)(9,20,81,106,197,136,125)(10,21,82,107,198,129,126)(11,22,83,108,199,130,127)(12,23,84,109,200,131,128)(13,24,85,110,193,132,121)(14,17,86,111,194,133,122)(15,18,87,112,195,134,123)(16,19,88,105,196,135,124)(25,160,170,68,50,34,80)(26,153,171,69,51,35,73)(27,154,172,70,52,36,74)(28,155,173,71,53,37,75)(29,156,174,72,54,38,76)(30,157,175,65,55,39,77)(31,158,176,66,56,40,78)(32,159,169,67,49,33,79)(89,203,214,113,138,219,100)(90,204,215,114,139,220,101)(91,205,216,115,140,221,102)(92,206,209,116,141,222,103)(93,207,210,117,142,223,104)(94,208,211,118,143,224,97)(95,201,212,119,144,217,98)(96,202,213,120,137,218,99), (1,139)(2,144)(3,141)(4,138)(5,143)(6,140)(7,137)(8,142)(9,72)(10,69)(11,66)(12,71)(13,68)(14,65)(15,70)(16,67)(17,175)(18,172)(19,169)(20,174)(21,171)(22,176)(23,173)(24,170)(25,110)(26,107)(27,112)(28,109)(29,106)(30,111)(31,108)(32,105)(33,135)(34,132)(35,129)(36,134)(37,131)(38,136)(39,133)(40,130)(41,116)(42,113)(43,118)(44,115)(45,120)(46,117)(47,114)(48,119)(49,124)(50,121)(51,126)(52,123)(53,128)(54,125)(55,122)(56,127)(57,215)(58,212)(59,209)(60,214)(61,211)(62,216)(63,213)(64,210)(73,198)(74,195)(75,200)(76,197)(77,194)(78,199)(79,196)(80,193)(81,156)(82,153)(83,158)(84,155)(85,160)(86,157)(87,154)(88,159)(89,164)(90,161)(91,166)(92,163)(93,168)(94,165)(95,162)(96,167)(97,149)(98,146)(99,151)(100,148)(101,145)(102,150)(103,147)(104,152)(177,202)(178,207)(179,204)(180,201)(181,206)(182,203)(183,208)(184,205)(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),(9,184),(10,177),(11,178),(12,179),(13,180),(14,181),(15,182),(16,183),(17,163),(18,164),(19,165),(20,166),(21,167),(22,168),(23,161),(24,162),(25,217),(26,218),(27,219),(28,220),(29,221),(30,222),(31,223),(32,224),(33,118),(34,119),(35,120),(36,113),(37,114),(38,115),(39,116),(40,117),(41,133),(42,134),(43,135),(44,136),(45,129),(46,130),(47,131),(48,132),(49,211),(50,212),(51,213),(52,214),(53,215),(54,216),(55,209),(56,210),(57,128),(58,121),(59,122),(60,123),(61,124),(62,125),(63,126),(64,127),(65,206),(66,207),(67,208),(68,201),(69,202),(70,203),(71,204),(72,205),(73,137),(74,138),(75,139),(76,140),(77,141),(78,142),(79,143),(80,144),(81,150),(82,151),(83,152),(84,145),(85,146),(86,147),(87,148),(88,149),(89,172),(90,173),(91,174),(92,175),(93,176),(94,169),(95,170),(96,171),(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)], [(1,47,57,179,161,145,191),(2,48,58,180,162,146,192),(3,41,59,181,163,147,185),(4,42,60,182,164,148,186),(5,43,61,183,165,149,187),(6,44,62,184,166,150,188),(7,45,63,177,167,151,189),(8,46,64,178,168,152,190),(9,20,81,106,197,136,125),(10,21,82,107,198,129,126),(11,22,83,108,199,130,127),(12,23,84,109,200,131,128),(13,24,85,110,193,132,121),(14,17,86,111,194,133,122),(15,18,87,112,195,134,123),(16,19,88,105,196,135,124),(25,160,170,68,50,34,80),(26,153,171,69,51,35,73),(27,154,172,70,52,36,74),(28,155,173,71,53,37,75),(29,156,174,72,54,38,76),(30,157,175,65,55,39,77),(31,158,176,66,56,40,78),(32,159,169,67,49,33,79),(89,203,214,113,138,219,100),(90,204,215,114,139,220,101),(91,205,216,115,140,221,102),(92,206,209,116,141,222,103),(93,207,210,117,142,223,104),(94,208,211,118,143,224,97),(95,201,212,119,144,217,98),(96,202,213,120,137,218,99)], [(1,139),(2,144),(3,141),(4,138),(5,143),(6,140),(7,137),(8,142),(9,72),(10,69),(11,66),(12,71),(13,68),(14,65),(15,70),(16,67),(17,175),(18,172),(19,169),(20,174),(21,171),(22,176),(23,173),(24,170),(25,110),(26,107),(27,112),(28,109),(29,106),(30,111),(31,108),(32,105),(33,135),(34,132),(35,129),(36,134),(37,131),(38,136),(39,133),(40,130),(41,116),(42,113),(43,118),(44,115),(45,120),(46,117),(47,114),(48,119),(49,124),(50,121),(51,126),(52,123),(53,128),(54,125),(55,122),(56,127),(57,215),(58,212),(59,209),(60,214),(61,211),(62,216),(63,213),(64,210),(73,198),(74,195),(75,200),(76,197),(77,194),(78,199),(79,196),(80,193),(81,156),(82,153),(83,158),(84,155),(85,160),(86,157),(87,154),(88,159),(89,164),(90,161),(91,166),(92,163),(93,168),(94,165),(95,162),(96,167),(97,149),(98,146),(99,151),(100,148),(101,145),(102,150),(103,147),(104,152),(177,202),(178,207),(179,204),(180,201),(181,206),(182,203),(183,208),(184,205),(185,222),(186,219),(187,224),(188,221),(189,218),(190,223),(191,220),(192,217)]])
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 |
Matrix representation of C22×C8⋊D7 ►in 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] >;
C22×C8⋊D7 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