direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C42.7C22, (C4×C8)⋊3C14, (C4×C56)⋊8C2, C4⋊C8⋊14C14, C4⋊C4.6C28, C8⋊C4⋊8C14, C22⋊C8.8C14, C22⋊C4.3C28, C14.47(C8○D4), C23.11(C2×C28), C42.61(C2×C14), C28.352(C4○D4), (C2×C56).327C22, (C4×C28).247C22, (C2×C28).989C23, C42⋊C2.8C14, C22.46(C22×C28), C14.61(C42⋊C2), (C22×C28).416C22, (C7×C4⋊C8)⋊33C2, C2.6(C7×C8○D4), (C7×C4⋊C4).18C4, (C7×C8⋊C4)⋊22C2, C4.50(C7×C4○D4), (C2×C8).51(C2×C14), (C2×C4).27(C2×C28), (C2×C28).200(C2×C4), (C7×C22⋊C8).17C2, (C7×C22⋊C4).10C4, (C22×C4).35(C2×C14), (C22×C14).22(C2×C4), C2.12(C7×C42⋊C2), (C2×C4).157(C22×C14), (C7×C42⋊C2).22C2, (C2×C14).239(C22×C4), SmallGroup(448,841)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C42.7C22
G = < a,b,c,d,e | a7=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe=bc2, cd=dc, ce=ec, ede=b2c2d >
Subgroups: 130 in 96 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C28, C28, C2×C14, C2×C14, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C56, C2×C28, C2×C28, C2×C28, C22×C14, C42.7C22, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C22×C28, C4×C56, C7×C8⋊C4, C7×C22⋊C8, C7×C4⋊C8, C7×C42⋊C2, C7×C42.7C22
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, C22×C4, C4○D4, C28, C2×C14, C42⋊C2, C8○D4, C2×C28, C22×C14, C42.7C22, C22×C28, C7×C4○D4, C7×C42⋊C2, C7×C8○D4, C7×C42.7C22
►On 224 points
Generators in S224
(1 145 33 137 25 129 17)(2 146 34 138 26 130 18)(3 147 35 139 27 131 19)(4 148 36 140 28 132 20)(5 149 37 141 29 133 21)(6 150 38 142 30 134 22)(7 151 39 143 31 135 23)(8 152 40 144 32 136 24)(9 64 168 56 160 48 125)(10 57 161 49 153 41 126)(11 58 162 50 154 42 127)(12 59 163 51 155 43 128)(13 60 164 52 156 44 121)(14 61 165 53 157 45 122)(15 62 166 54 158 46 123)(16 63 167 55 159 47 124)(65 169 193 81 185 73 177)(66 170 194 82 186 74 178)(67 171 195 83 187 75 179)(68 172 196 84 188 76 180)(69 173 197 85 189 77 181)(70 174 198 86 190 78 182)(71 175 199 87 191 79 183)(72 176 200 88 192 80 184)(89 117 217 105 209 97 201)(90 118 218 106 210 98 202)(91 119 219 107 211 99 203)(92 120 220 108 212 100 204)(93 113 221 109 213 101 205)(94 114 222 110 214 102 206)(95 115 223 111 215 103 207)(96 116 224 112 216 104 208)
(1 95 171 45)(2 42 172 92)(3 89 173 47)(4 44 174 94)(5 91 175 41)(6 46 176 96)(7 93 169 43)(8 48 170 90)(9 82 218 40)(10 37 219 87)(11 84 220 34)(12 39 221 81)(13 86 222 36)(14 33 223 83)(15 88 224 38)(16 35 217 85)(17 207 67 157)(18 154 68 204)(19 201 69 159)(20 156 70 206)(21 203 71 153)(22 158 72 208)(23 205 65 155)(24 160 66 202)(25 215 75 165)(26 162 76 212)(27 209 77 167)(28 164 78 214)(29 211 79 161)(30 166 80 216)(31 213 73 163)(32 168 74 210)(49 133 99 183)(50 180 100 130)(51 135 101 177)(52 182 102 132)(53 129 103 179)(54 184 104 134)(55 131 97 181)(56 178 98 136)(57 141 107 191)(58 188 108 138)(59 143 109 185)(60 190 110 140)(61 137 111 187)(62 192 112 142)(63 139 105 189)(64 186 106 144)(113 193 128 151)(114 148 121 198)(115 195 122 145)(116 150 123 200)(117 197 124 147)(118 152 125 194)(119 199 126 149)(120 146 127 196)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)(161 163 165 167)(162 164 166 168)(169 171 173 175)(170 172 174 176)(177 179 181 183)(178 180 182 184)(185 187 189 191)(186 188 190 192)(193 195 197 199)(194 196 198 200)(201 203 205 207)(202 204 206 208)(209 211 213 215)(210 212 214 216)(217 219 221 223)(218 220 222 224)
(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)
(2 176)(4 170)(6 172)(8 174)(9 218)(10 14)(11 220)(12 16)(13 222)(15 224)(18 72)(20 66)(22 68)(24 70)(26 80)(28 74)(30 76)(32 78)(34 88)(36 82)(38 84)(40 86)(41 45)(42 92)(43 47)(44 94)(46 96)(48 90)(49 53)(50 100)(51 55)(52 102)(54 104)(56 98)(57 61)(58 108)(59 63)(60 110)(62 112)(64 106)(89 93)(91 95)(97 101)(99 103)(105 109)(107 111)(113 117)(114 121)(115 119)(116 123)(118 125)(120 127)(122 126)(124 128)(130 184)(132 178)(134 180)(136 182)(138 192)(140 186)(142 188)(144 190)(146 200)(148 194)(150 196)(152 198)(153 157)(154 204)(155 159)(156 206)(158 208)(160 202)(161 165)(162 212)(163 167)(164 214)(166 216)(168 210)(201 205)(203 207)(209 213)(211 215)(217 221)(219 223)
G:=sub<Sym(224)| (1,145,33,137,25,129,17)(2,146,34,138,26,130,18)(3,147,35,139,27,131,19)(4,148,36,140,28,132,20)(5,149,37,141,29,133,21)(6,150,38,142,30,134,22)(7,151,39,143,31,135,23)(8,152,40,144,32,136,24)(9,64,168,56,160,48,125)(10,57,161,49,153,41,126)(11,58,162,50,154,42,127)(12,59,163,51,155,43,128)(13,60,164,52,156,44,121)(14,61,165,53,157,45,122)(15,62,166,54,158,46,123)(16,63,167,55,159,47,124)(65,169,193,81,185,73,177)(66,170,194,82,186,74,178)(67,171,195,83,187,75,179)(68,172,196,84,188,76,180)(69,173,197,85,189,77,181)(70,174,198,86,190,78,182)(71,175,199,87,191,79,183)(72,176,200,88,192,80,184)(89,117,217,105,209,97,201)(90,118,218,106,210,98,202)(91,119,219,107,211,99,203)(92,120,220,108,212,100,204)(93,113,221,109,213,101,205)(94,114,222,110,214,102,206)(95,115,223,111,215,103,207)(96,116,224,112,216,104,208), (1,95,171,45)(2,42,172,92)(3,89,173,47)(4,44,174,94)(5,91,175,41)(6,46,176,96)(7,93,169,43)(8,48,170,90)(9,82,218,40)(10,37,219,87)(11,84,220,34)(12,39,221,81)(13,86,222,36)(14,33,223,83)(15,88,224,38)(16,35,217,85)(17,207,67,157)(18,154,68,204)(19,201,69,159)(20,156,70,206)(21,203,71,153)(22,158,72,208)(23,205,65,155)(24,160,66,202)(25,215,75,165)(26,162,76,212)(27,209,77,167)(28,164,78,214)(29,211,79,161)(30,166,80,216)(31,213,73,163)(32,168,74,210)(49,133,99,183)(50,180,100,130)(51,135,101,177)(52,182,102,132)(53,129,103,179)(54,184,104,134)(55,131,97,181)(56,178,98,136)(57,141,107,191)(58,188,108,138)(59,143,109,185)(60,190,110,140)(61,137,111,187)(62,192,112,142)(63,139,105,189)(64,186,106,144)(113,193,128,151)(114,148,121,198)(115,195,122,145)(116,150,123,200)(117,197,124,147)(118,152,125,194)(119,199,126,149)(120,146,127,196), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224), (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), (2,176)(4,170)(6,172)(8,174)(9,218)(10,14)(11,220)(12,16)(13,222)(15,224)(18,72)(20,66)(22,68)(24,70)(26,80)(28,74)(30,76)(32,78)(34,88)(36,82)(38,84)(40,86)(41,45)(42,92)(43,47)(44,94)(46,96)(48,90)(49,53)(50,100)(51,55)(52,102)(54,104)(56,98)(57,61)(58,108)(59,63)(60,110)(62,112)(64,106)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111)(113,117)(114,121)(115,119)(116,123)(118,125)(120,127)(122,126)(124,128)(130,184)(132,178)(134,180)(136,182)(138,192)(140,186)(142,188)(144,190)(146,200)(148,194)(150,196)(152,198)(153,157)(154,204)(155,159)(156,206)(158,208)(160,202)(161,165)(162,212)(163,167)(164,214)(166,216)(168,210)(201,205)(203,207)(209,213)(211,215)(217,221)(219,223)>;
G:=Group( (1,145,33,137,25,129,17)(2,146,34,138,26,130,18)(3,147,35,139,27,131,19)(4,148,36,140,28,132,20)(5,149,37,141,29,133,21)(6,150,38,142,30,134,22)(7,151,39,143,31,135,23)(8,152,40,144,32,136,24)(9,64,168,56,160,48,125)(10,57,161,49,153,41,126)(11,58,162,50,154,42,127)(12,59,163,51,155,43,128)(13,60,164,52,156,44,121)(14,61,165,53,157,45,122)(15,62,166,54,158,46,123)(16,63,167,55,159,47,124)(65,169,193,81,185,73,177)(66,170,194,82,186,74,178)(67,171,195,83,187,75,179)(68,172,196,84,188,76,180)(69,173,197,85,189,77,181)(70,174,198,86,190,78,182)(71,175,199,87,191,79,183)(72,176,200,88,192,80,184)(89,117,217,105,209,97,201)(90,118,218,106,210,98,202)(91,119,219,107,211,99,203)(92,120,220,108,212,100,204)(93,113,221,109,213,101,205)(94,114,222,110,214,102,206)(95,115,223,111,215,103,207)(96,116,224,112,216,104,208), (1,95,171,45)(2,42,172,92)(3,89,173,47)(4,44,174,94)(5,91,175,41)(6,46,176,96)(7,93,169,43)(8,48,170,90)(9,82,218,40)(10,37,219,87)(11,84,220,34)(12,39,221,81)(13,86,222,36)(14,33,223,83)(15,88,224,38)(16,35,217,85)(17,207,67,157)(18,154,68,204)(19,201,69,159)(20,156,70,206)(21,203,71,153)(22,158,72,208)(23,205,65,155)(24,160,66,202)(25,215,75,165)(26,162,76,212)(27,209,77,167)(28,164,78,214)(29,211,79,161)(30,166,80,216)(31,213,73,163)(32,168,74,210)(49,133,99,183)(50,180,100,130)(51,135,101,177)(52,182,102,132)(53,129,103,179)(54,184,104,134)(55,131,97,181)(56,178,98,136)(57,141,107,191)(58,188,108,138)(59,143,109,185)(60,190,110,140)(61,137,111,187)(62,192,112,142)(63,139,105,189)(64,186,106,144)(113,193,128,151)(114,148,121,198)(115,195,122,145)(116,150,123,200)(117,197,124,147)(118,152,125,194)(119,199,126,149)(120,146,127,196), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96)(97,99,101,103)(98,100,102,104)(105,107,109,111)(106,108,110,112)(113,115,117,119)(114,116,118,120)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,139,141,143)(138,140,142,144)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,163,165,167)(162,164,166,168)(169,171,173,175)(170,172,174,176)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,195,197,199)(194,196,198,200)(201,203,205,207)(202,204,206,208)(209,211,213,215)(210,212,214,216)(217,219,221,223)(218,220,222,224), (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), (2,176)(4,170)(6,172)(8,174)(9,218)(10,14)(11,220)(12,16)(13,222)(15,224)(18,72)(20,66)(22,68)(24,70)(26,80)(28,74)(30,76)(32,78)(34,88)(36,82)(38,84)(40,86)(41,45)(42,92)(43,47)(44,94)(46,96)(48,90)(49,53)(50,100)(51,55)(52,102)(54,104)(56,98)(57,61)(58,108)(59,63)(60,110)(62,112)(64,106)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111)(113,117)(114,121)(115,119)(116,123)(118,125)(120,127)(122,126)(124,128)(130,184)(132,178)(134,180)(136,182)(138,192)(140,186)(142,188)(144,190)(146,200)(148,194)(150,196)(152,198)(153,157)(154,204)(155,159)(156,206)(158,208)(160,202)(161,165)(162,212)(163,167)(164,214)(166,216)(168,210)(201,205)(203,207)(209,213)(211,215)(217,221)(219,223) );
G=PermutationGroup([[(1,145,33,137,25,129,17),(2,146,34,138,26,130,18),(3,147,35,139,27,131,19),(4,148,36,140,28,132,20),(5,149,37,141,29,133,21),(6,150,38,142,30,134,22),(7,151,39,143,31,135,23),(8,152,40,144,32,136,24),(9,64,168,56,160,48,125),(10,57,161,49,153,41,126),(11,58,162,50,154,42,127),(12,59,163,51,155,43,128),(13,60,164,52,156,44,121),(14,61,165,53,157,45,122),(15,62,166,54,158,46,123),(16,63,167,55,159,47,124),(65,169,193,81,185,73,177),(66,170,194,82,186,74,178),(67,171,195,83,187,75,179),(68,172,196,84,188,76,180),(69,173,197,85,189,77,181),(70,174,198,86,190,78,182),(71,175,199,87,191,79,183),(72,176,200,88,192,80,184),(89,117,217,105,209,97,201),(90,118,218,106,210,98,202),(91,119,219,107,211,99,203),(92,120,220,108,212,100,204),(93,113,221,109,213,101,205),(94,114,222,110,214,102,206),(95,115,223,111,215,103,207),(96,116,224,112,216,104,208)], [(1,95,171,45),(2,42,172,92),(3,89,173,47),(4,44,174,94),(5,91,175,41),(6,46,176,96),(7,93,169,43),(8,48,170,90),(9,82,218,40),(10,37,219,87),(11,84,220,34),(12,39,221,81),(13,86,222,36),(14,33,223,83),(15,88,224,38),(16,35,217,85),(17,207,67,157),(18,154,68,204),(19,201,69,159),(20,156,70,206),(21,203,71,153),(22,158,72,208),(23,205,65,155),(24,160,66,202),(25,215,75,165),(26,162,76,212),(27,209,77,167),(28,164,78,214),(29,211,79,161),(30,166,80,216),(31,213,73,163),(32,168,74,210),(49,133,99,183),(50,180,100,130),(51,135,101,177),(52,182,102,132),(53,129,103,179),(54,184,104,134),(55,131,97,181),(56,178,98,136),(57,141,107,191),(58,188,108,138),(59,143,109,185),(60,190,110,140),(61,137,111,187),(62,192,112,142),(63,139,105,189),(64,186,106,144),(113,193,128,151),(114,148,121,198),(115,195,122,145),(116,150,123,200),(117,197,124,147),(118,152,125,194),(119,199,126,149),(120,146,127,196)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96),(97,99,101,103),(98,100,102,104),(105,107,109,111),(106,108,110,112),(113,115,117,119),(114,116,118,120),(121,123,125,127),(122,124,126,128),(129,131,133,135),(130,132,134,136),(137,139,141,143),(138,140,142,144),(145,147,149,151),(146,148,150,152),(153,155,157,159),(154,156,158,160),(161,163,165,167),(162,164,166,168),(169,171,173,175),(170,172,174,176),(177,179,181,183),(178,180,182,184),(185,187,189,191),(186,188,190,192),(193,195,197,199),(194,196,198,200),(201,203,205,207),(202,204,206,208),(209,211,213,215),(210,212,214,216),(217,219,221,223),(218,220,222,224)], [(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)], [(2,176),(4,170),(6,172),(8,174),(9,218),(10,14),(11,220),(12,16),(13,222),(15,224),(18,72),(20,66),(22,68),(24,70),(26,80),(28,74),(30,76),(32,78),(34,88),(36,82),(38,84),(40,86),(41,45),(42,92),(43,47),(44,94),(46,96),(48,90),(49,53),(50,100),(51,55),(52,102),(54,104),(56,98),(57,61),(58,108),(59,63),(60,110),(62,112),(64,106),(89,93),(91,95),(97,101),(99,103),(105,109),(107,111),(113,117),(114,121),(115,119),(116,123),(118,125),(120,127),(122,126),(124,128),(130,184),(132,178),(134,180),(136,182),(138,192),(140,186),(142,188),(144,190),(146,200),(148,194),(150,196),(152,198),(153,157),(154,204),(155,159),(156,206),(158,208),(160,202),(161,165),(162,212),(163,167),(164,214),(166,216),(168,210),(201,205),(203,207),(209,213),(211,215),(217,221),(219,223)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 7A | ··· | 7F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14R | 14S | ··· | 14X | 28A | ··· | 28X | 28Y | ··· | 28AV | 28AW | ··· | 28BN | 56A | ··· | 56AV | 56AW | ··· | 56BT |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | C28 | C4○D4 | C8○D4 | C7×C4○D4 | C7×C8○D4 |
| kernel | C7×C42.7C22 | C4×C56 | C7×C8⋊C4 | C7×C22⋊C8 | C7×C4⋊C8 | C7×C42⋊C2 | C7×C22⋊C4 | C7×C4⋊C4 | C42.7C22 | C4×C8 | C8⋊C4 | C22⋊C8 | C4⋊C8 | C42⋊C2 | C22⋊C4 | C4⋊C4 | C28 | C14 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 6 | 24 | 24 | 4 | 8 | 24 | 48 |
Matrix representation of C7×C42.7C22 ►in GL4(𝔽113) generated by
| 106 | 0 | 0 | 0 |
| 0 | 106 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 98 | 1 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 98 | 0 | 0 | 0 |
| 2 | 15 | 0 | 0 |
| 0 | 0 | 69 | 0 |
| 0 | 0 | 0 | 69 |
| 1 | 15 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,30,0,0,0,0,30],[98,0,0,0,1,15,0,0,0,0,0,1,0,0,112,0],[112,0,0,0,0,112,0,0,0,0,15,0,0,0,0,15],[98,2,0,0,0,15,0,0,0,0,69,0,0,0,0,69],[1,0,0,0,15,112,0,0,0,0,1,0,0,0,0,112] >;
C7×C42.7C22 in GAP, Magma, Sage, TeX
C_7\times C_4^2._7C_2^2
% in TeX
G:=Group("C7xC4^2.7C2^2"); // GroupNames label
G:=SmallGroup(448,841);
// by ID
G=gap.SmallGroup(448,841);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,2403,310,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^4=e^2=1,d^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1*c^2,e*b*e=b*c^2,c*d=d*c,c*e=e*c,e*d*e=b^2*c^2*d>;
// generators/relations