direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C42.6C22, C4⋊C8⋊12C14, C4⋊C4.5C28, C4.76(D4×C14), C28.61(C4⋊C4), (C2×C28).75Q8, C4.23(Q8×C14), (C2×C28).521D4, C28.481(C2×D4), C22⋊C4.2C28, (C22×C8).7C14, C42.6(C2×C14), C28.129(C2×Q8), C14.46(C8○D4), C23.17(C2×C28), (C22×C56).13C2, (C2×C56).325C22, (C2×C28).986C23, (C4×C28).246C22, C42⋊C2.5C14, (C2×M4(2)).14C14, (C14×M4(2)).32C2, C22.44(C22×C28), (C22×C28).586C22, (C7×C4⋊C8)⋊31C2, C4.12(C7×C4⋊C4), C2.5(C7×C8○D4), (C7×C4⋊C4).17C4, C2.10(C14×C4⋊C4), C14.66(C2×C4⋊C4), C22.8(C7×C4⋊C4), (C2×C4).17(C7×Q8), (C2×C8).63(C2×C14), (C2×C4).24(C2×C28), (C2×C4).124(C7×D4), (C7×C22⋊C4).9C4, (C2×C14).25(C4⋊C4), (C2×C28).197(C2×C4), (C22×C14).84(C2×C4), (C2×C4).154(C22×C14), (C2×C14).236(C22×C4), (C22×C4).115(C2×C14), (C7×C42⋊C2).19C2, SmallGroup(448,832)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C42.6C22
G = < a,b,c,d,e | a7=b4=c4=1, d2=c, e2=b2c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe-1=bc2, cd=dc, ce=ec, ede-1=b2c2d >
Subgroups: 146 in 114 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C28, C28, C28, C2×C14, C2×C14, C2×C14, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C56, C2×C28, C2×C28, C22×C14, C42.6C22, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C7×M4(2), C22×C28, C7×C4⋊C8, C7×C42⋊C2, C22×C56, C14×M4(2), C7×C42.6C22
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C23, C14, C4⋊C4, C22×C4, C2×D4, C2×Q8, C28, C2×C14, C2×C4⋊C4, C8○D4, C2×C28, C7×D4, C7×Q8, C22×C14, C42.6C22, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, C14×C4⋊C4, C7×C8○D4, C7×C42.6C22
►On 224 points
Generators in S224
(1 151 39 143 31 135 23)(2 152 40 144 32 136 24)(3 145 33 137 25 129 17)(4 146 34 138 26 130 18)(5 147 35 139 27 131 19)(6 148 36 140 28 132 20)(7 149 37 141 29 133 21)(8 150 38 142 30 134 22)(9 122 58 162 50 154 42)(10 123 59 163 51 155 43)(11 124 60 164 52 156 44)(12 125 61 165 53 157 45)(13 126 62 166 54 158 46)(14 127 63 167 55 159 47)(15 128 64 168 56 160 48)(16 121 57 161 49 153 41)(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 43 171 95)(2 92 172 48)(3 45 173 89)(4 94 174 42)(5 47 175 91)(6 96 176 44)(7 41 169 93)(8 90 170 46)(9 146 114 198)(10 195 115 151)(11 148 116 200)(12 197 117 145)(13 150 118 194)(14 199 119 147)(15 152 120 196)(16 193 113 149)(17 157 69 201)(18 206 70 154)(19 159 71 203)(20 208 72 156)(21 153 65 205)(22 202 66 158)(23 155 67 207)(24 204 68 160)(25 165 77 209)(26 214 78 162)(27 167 79 211)(28 216 80 164)(29 161 73 213)(30 210 74 166)(31 163 75 215)(32 212 76 168)(33 125 85 217)(34 222 86 122)(35 127 87 219)(36 224 88 124)(37 121 81 221)(38 218 82 126)(39 123 83 223)(40 220 84 128)(49 177 101 133)(50 130 102 182)(51 179 103 135)(52 132 104 184)(53 181 97 129)(54 134 98 178)(55 183 99 131)(56 136 100 180)(57 185 109 141)(58 138 110 190)(59 187 111 143)(60 140 112 192)(61 189 105 137)(62 142 106 186)(63 191 107 139)(64 144 108 188)
(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)
(1 95 175 47)(2 48 176 96)(3 89 169 41)(4 42 170 90)(5 91 171 43)(6 44 172 92)(7 93 173 45)(8 46 174 94)(9 194 118 146)(10 147 119 195)(11 196 120 148)(12 149 113 197)(13 198 114 150)(14 151 115 199)(15 200 116 152)(16 145 117 193)(17 201 65 153)(18 154 66 202)(19 203 67 155)(20 156 68 204)(21 205 69 157)(22 158 70 206)(23 207 71 159)(24 160 72 208)(25 209 73 161)(26 162 74 210)(27 211 75 163)(28 164 76 212)(29 213 77 165)(30 166 78 214)(31 215 79 167)(32 168 80 216)(33 217 81 121)(34 122 82 218)(35 219 83 123)(36 124 84 220)(37 221 85 125)(38 126 86 222)(39 223 87 127)(40 128 88 224)(49 129 97 177)(50 178 98 130)(51 131 99 179)(52 180 100 132)(53 133 101 181)(54 182 102 134)(55 135 103 183)(56 184 104 136)(57 137 105 185)(58 186 106 138)(59 139 107 187)(60 188 108 140)(61 141 109 189)(62 190 110 142)(63 143 111 191)(64 192 112 144)
G:=sub<Sym(224)| (1,151,39,143,31,135,23)(2,152,40,144,32,136,24)(3,145,33,137,25,129,17)(4,146,34,138,26,130,18)(5,147,35,139,27,131,19)(6,148,36,140,28,132,20)(7,149,37,141,29,133,21)(8,150,38,142,30,134,22)(9,122,58,162,50,154,42)(10,123,59,163,51,155,43)(11,124,60,164,52,156,44)(12,125,61,165,53,157,45)(13,126,62,166,54,158,46)(14,127,63,167,55,159,47)(15,128,64,168,56,160,48)(16,121,57,161,49,153,41)(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), (1,43,171,95)(2,92,172,48)(3,45,173,89)(4,94,174,42)(5,47,175,91)(6,96,176,44)(7,41,169,93)(8,90,170,46)(9,146,114,198)(10,195,115,151)(11,148,116,200)(12,197,117,145)(13,150,118,194)(14,199,119,147)(15,152,120,196)(16,193,113,149)(17,157,69,201)(18,206,70,154)(19,159,71,203)(20,208,72,156)(21,153,65,205)(22,202,66,158)(23,155,67,207)(24,204,68,160)(25,165,77,209)(26,214,78,162)(27,167,79,211)(28,216,80,164)(29,161,73,213)(30,210,74,166)(31,163,75,215)(32,212,76,168)(33,125,85,217)(34,222,86,122)(35,127,87,219)(36,224,88,124)(37,121,81,221)(38,218,82,126)(39,123,83,223)(40,220,84,128)(49,177,101,133)(50,130,102,182)(51,179,103,135)(52,132,104,184)(53,181,97,129)(54,134,98,178)(55,183,99,131)(56,136,100,180)(57,185,109,141)(58,138,110,190)(59,187,111,143)(60,140,112,192)(61,189,105,137)(62,142,106,186)(63,191,107,139)(64,144,108,188), (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), (1,95,175,47)(2,48,176,96)(3,89,169,41)(4,42,170,90)(5,91,171,43)(6,44,172,92)(7,93,173,45)(8,46,174,94)(9,194,118,146)(10,147,119,195)(11,196,120,148)(12,149,113,197)(13,198,114,150)(14,151,115,199)(15,200,116,152)(16,145,117,193)(17,201,65,153)(18,154,66,202)(19,203,67,155)(20,156,68,204)(21,205,69,157)(22,158,70,206)(23,207,71,159)(24,160,72,208)(25,209,73,161)(26,162,74,210)(27,211,75,163)(28,164,76,212)(29,213,77,165)(30,166,78,214)(31,215,79,167)(32,168,80,216)(33,217,81,121)(34,122,82,218)(35,219,83,123)(36,124,84,220)(37,221,85,125)(38,126,86,222)(39,223,87,127)(40,128,88,224)(49,129,97,177)(50,178,98,130)(51,131,99,179)(52,180,100,132)(53,133,101,181)(54,182,102,134)(55,135,103,183)(56,184,104,136)(57,137,105,185)(58,186,106,138)(59,139,107,187)(60,188,108,140)(61,141,109,189)(62,190,110,142)(63,143,111,191)(64,192,112,144)>;
G:=Group( (1,151,39,143,31,135,23)(2,152,40,144,32,136,24)(3,145,33,137,25,129,17)(4,146,34,138,26,130,18)(5,147,35,139,27,131,19)(6,148,36,140,28,132,20)(7,149,37,141,29,133,21)(8,150,38,142,30,134,22)(9,122,58,162,50,154,42)(10,123,59,163,51,155,43)(11,124,60,164,52,156,44)(12,125,61,165,53,157,45)(13,126,62,166,54,158,46)(14,127,63,167,55,159,47)(15,128,64,168,56,160,48)(16,121,57,161,49,153,41)(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), (1,43,171,95)(2,92,172,48)(3,45,173,89)(4,94,174,42)(5,47,175,91)(6,96,176,44)(7,41,169,93)(8,90,170,46)(9,146,114,198)(10,195,115,151)(11,148,116,200)(12,197,117,145)(13,150,118,194)(14,199,119,147)(15,152,120,196)(16,193,113,149)(17,157,69,201)(18,206,70,154)(19,159,71,203)(20,208,72,156)(21,153,65,205)(22,202,66,158)(23,155,67,207)(24,204,68,160)(25,165,77,209)(26,214,78,162)(27,167,79,211)(28,216,80,164)(29,161,73,213)(30,210,74,166)(31,163,75,215)(32,212,76,168)(33,125,85,217)(34,222,86,122)(35,127,87,219)(36,224,88,124)(37,121,81,221)(38,218,82,126)(39,123,83,223)(40,220,84,128)(49,177,101,133)(50,130,102,182)(51,179,103,135)(52,132,104,184)(53,181,97,129)(54,134,98,178)(55,183,99,131)(56,136,100,180)(57,185,109,141)(58,138,110,190)(59,187,111,143)(60,140,112,192)(61,189,105,137)(62,142,106,186)(63,191,107,139)(64,144,108,188), (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), (1,95,175,47)(2,48,176,96)(3,89,169,41)(4,42,170,90)(5,91,171,43)(6,44,172,92)(7,93,173,45)(8,46,174,94)(9,194,118,146)(10,147,119,195)(11,196,120,148)(12,149,113,197)(13,198,114,150)(14,151,115,199)(15,200,116,152)(16,145,117,193)(17,201,65,153)(18,154,66,202)(19,203,67,155)(20,156,68,204)(21,205,69,157)(22,158,70,206)(23,207,71,159)(24,160,72,208)(25,209,73,161)(26,162,74,210)(27,211,75,163)(28,164,76,212)(29,213,77,165)(30,166,78,214)(31,215,79,167)(32,168,80,216)(33,217,81,121)(34,122,82,218)(35,219,83,123)(36,124,84,220)(37,221,85,125)(38,126,86,222)(39,223,87,127)(40,128,88,224)(49,129,97,177)(50,178,98,130)(51,131,99,179)(52,180,100,132)(53,133,101,181)(54,182,102,134)(55,135,103,183)(56,184,104,136)(57,137,105,185)(58,186,106,138)(59,139,107,187)(60,188,108,140)(61,141,109,189)(62,190,110,142)(63,143,111,191)(64,192,112,144) );
G=PermutationGroup([[(1,151,39,143,31,135,23),(2,152,40,144,32,136,24),(3,145,33,137,25,129,17),(4,146,34,138,26,130,18),(5,147,35,139,27,131,19),(6,148,36,140,28,132,20),(7,149,37,141,29,133,21),(8,150,38,142,30,134,22),(9,122,58,162,50,154,42),(10,123,59,163,51,155,43),(11,124,60,164,52,156,44),(12,125,61,165,53,157,45),(13,126,62,166,54,158,46),(14,127,63,167,55,159,47),(15,128,64,168,56,160,48),(16,121,57,161,49,153,41),(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)], [(1,43,171,95),(2,92,172,48),(3,45,173,89),(4,94,174,42),(5,47,175,91),(6,96,176,44),(7,41,169,93),(8,90,170,46),(9,146,114,198),(10,195,115,151),(11,148,116,200),(12,197,117,145),(13,150,118,194),(14,199,119,147),(15,152,120,196),(16,193,113,149),(17,157,69,201),(18,206,70,154),(19,159,71,203),(20,208,72,156),(21,153,65,205),(22,202,66,158),(23,155,67,207),(24,204,68,160),(25,165,77,209),(26,214,78,162),(27,167,79,211),(28,216,80,164),(29,161,73,213),(30,210,74,166),(31,163,75,215),(32,212,76,168),(33,125,85,217),(34,222,86,122),(35,127,87,219),(36,224,88,124),(37,121,81,221),(38,218,82,126),(39,123,83,223),(40,220,84,128),(49,177,101,133),(50,130,102,182),(51,179,103,135),(52,132,104,184),(53,181,97,129),(54,134,98,178),(55,183,99,131),(56,136,100,180),(57,185,109,141),(58,138,110,190),(59,187,111,143),(60,140,112,192),(61,189,105,137),(62,142,106,186),(63,191,107,139),(64,144,108,188)], [(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)], [(1,95,175,47),(2,48,176,96),(3,89,169,41),(4,42,170,90),(5,91,171,43),(6,44,172,92),(7,93,173,45),(8,46,174,94),(9,194,118,146),(10,147,119,195),(11,196,120,148),(12,149,113,197),(13,198,114,150),(14,151,115,199),(15,200,116,152),(16,145,117,193),(17,201,65,153),(18,154,66,202),(19,203,67,155),(20,156,68,204),(21,205,69,157),(22,158,70,206),(23,207,71,159),(24,160,72,208),(25,209,73,161),(26,162,74,210),(27,211,75,163),(28,164,76,212),(29,213,77,165),(30,166,78,214),(31,215,79,167),(32,168,80,216),(33,217,81,121),(34,122,82,218),(35,219,83,123),(36,124,84,220),(37,221,85,125),(38,126,86,222),(39,223,87,127),(40,128,88,224),(49,129,97,177),(50,178,98,130),(51,131,99,179),(52,180,100,132),(53,133,101,181),(54,182,102,134),(55,135,103,183),(56,184,104,136),(57,137,105,185),(58,186,106,138),(59,139,107,187),(60,188,108,140),(61,141,109,189),(62,190,110,142),(63,143,111,191),(64,192,112,144)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | ··· | 7F | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AJ | 28AK | ··· | 28BH | 56A | ··· | 56AV | 56AW | ··· | 56BT |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | C28 | D4 | Q8 | C8○D4 | C7×D4 | C7×Q8 | C7×C8○D4 |
| kernel | C7×C42.6C22 | C7×C4⋊C8 | C7×C42⋊C2 | C22×C56 | C14×M4(2) | C7×C22⋊C4 | C7×C4⋊C4 | C42.6C22 | C4⋊C8 | C42⋊C2 | C22×C8 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×C28 | C2×C28 | C14 | C2×C4 | C2×C4 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 6 | 24 | 6 | 6 | 6 | 24 | 24 | 2 | 2 | 8 | 12 | 12 | 48 |
Matrix representation of C7×C42.6C22 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 41 | 2 | 0 | 0 |
| 64 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 44 | 0 | 0 | 0 |
| 4 | 69 | 0 | 0 |
| 0 | 0 | 69 | 0 |
| 0 | 0 | 0 | 69 |
| 72 | 111 | 0 | 0 |
| 50 | 41 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,109,0,0,0,0,109],[41,64,0,0,2,72,0,0,0,0,0,112,0,0,1,0],[15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[44,4,0,0,0,69,0,0,0,0,69,0,0,0,0,69],[72,50,0,0,111,41,0,0,0,0,0,1,0,0,1,0] >;
C7×C42.6C22 in GAP, Magma, Sage, TeX
C_7\times C_4^2._6C_2^2
% in TeX
G:=Group("C7xC4^2.6C2^2"); // GroupNames label
G:=SmallGroup(448,832);
// by ID
G=gap.SmallGroup(448,832);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,400,2403,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^4=1,d^2=c,e^2=b^2*c^2,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^-1=b*c^2,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*c^2*d>;
// generators/relations