direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C42.78C22, (C4×C8)⋊6C14, (C4×C56)⋊11C2, Q8⋊C4⋊3C14, (C2×C28).366D4, C42.C2⋊2C14, D4⋊C4.1C14, C42.78(C2×C14), C4.4D4.5C14, C28.270(C4○D4), C14.130(C4○D8), (C2×C56).368C22, (C4×C28).362C22, (C2×C28).944C23, C22.109(D4×C14), C14.73(C4.4D4), (D4×C14).199C22, (Q8×C14).173C22, C2.17(C7×C4○D8), C4.15(C7×C4○D4), (C2×C4).56(C7×D4), C4⋊C4.19(C2×C14), (C2×C8).70(C2×C14), (C7×Q8⋊C4)⋊3C2, (C7×D4⋊C4).1C2, (C2×D4).22(C2×C14), (C2×C14).665(C2×D4), (C2×Q8).17(C2×C14), (C7×C42.C2)⋊19C2, C2.11(C7×C4.4D4), (C7×C4⋊C4).239C22, (C7×C4.4D4).14C2, (C2×C4).119(C22×C14), SmallGroup(448,896)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C42.78C22
G = < a,b,c,d,e | a7=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c2, be=eb, dcd=c-1, ce=ec, ede-1=b2cd >
Subgroups: 178 in 96 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C42.78C22, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, D4×C14, Q8×C14, C4×C56, C7×D4⋊C4, C7×Q8⋊C4, C7×C4.4D4, C7×C42.C2, C7×C42.78C22
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4.4D4, C4○D8, C7×D4, C22×C14, C42.78C22, D4×C14, C7×C4○D4, C7×C4.4D4, C7×C4○D8, C7×C42.78C22
►On 224 points
Generators in S224
(1 147 35 139 27 131 19)(2 148 36 140 28 132 20)(3 149 37 141 29 133 21)(4 150 38 142 30 134 22)(5 151 39 143 31 135 23)(6 152 40 144 32 136 24)(7 145 33 137 25 129 17)(8 146 34 138 26 130 18)(9 62 166 54 158 46 122)(10 63 167 55 159 47 123)(11 64 168 56 160 48 124)(12 57 161 49 153 41 125)(13 58 162 50 154 42 126)(14 59 163 51 155 43 127)(15 60 164 52 156 44 128)(16 61 165 53 157 45 121)(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)(89 116 217 105 209 97 201)(90 117 218 106 210 98 202)(91 118 219 107 211 99 203)(92 119 220 108 212 100 204)(93 120 221 109 213 101 205)(94 113 222 110 214 102 206)(95 114 223 111 215 103 207)(96 115 224 112 216 104 208)
(1 96 169 42)(2 89 170 43)(3 90 171 44)(4 91 172 45)(5 92 173 46)(6 93 174 47)(7 94 175 48)(8 95 176 41)(9 39 220 85)(10 40 221 86)(11 33 222 87)(12 34 223 88)(13 35 224 81)(14 36 217 82)(15 37 218 83)(16 38 219 84)(17 206 71 160)(18 207 72 153)(19 208 65 154)(20 201 66 155)(21 202 67 156)(22 203 68 157)(23 204 69 158)(24 205 70 159)(25 214 79 168)(26 215 80 161)(27 216 73 162)(28 209 74 163)(29 210 75 164)(30 211 76 165)(31 212 77 166)(32 213 78 167)(49 130 103 184)(50 131 104 177)(51 132 97 178)(52 133 98 179)(53 134 99 180)(54 135 100 181)(55 136 101 182)(56 129 102 183)(57 138 111 192)(58 139 112 185)(59 140 105 186)(60 141 106 187)(61 142 107 188)(62 143 108 189)(63 144 109 190)(64 137 110 191)(113 199 124 145)(114 200 125 146)(115 193 126 147)(116 194 127 148)(117 195 128 149)(118 196 121 150)(119 197 122 151)(120 198 123 152)
(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)
(2 176)(3 7)(4 174)(6 172)(8 170)(9 224)(10 12)(11 222)(13 220)(14 16)(15 218)(17 21)(18 66)(20 72)(22 70)(24 68)(25 29)(26 74)(28 80)(30 78)(32 76)(33 37)(34 82)(36 88)(38 86)(40 84)(41 47)(42 92)(43 45)(44 90)(46 96)(48 94)(49 55)(50 100)(51 53)(52 98)(54 104)(56 102)(57 63)(58 108)(59 61)(60 106)(62 112)(64 110)(67 71)(75 79)(83 87)(89 91)(93 95)(97 99)(101 103)(105 107)(109 111)(113 124)(114 120)(115 122)(116 118)(117 128)(119 126)(121 127)(123 125)(129 133)(130 178)(132 184)(134 182)(136 180)(137 141)(138 186)(140 192)(142 190)(144 188)(145 149)(146 194)(148 200)(150 198)(152 196)(153 159)(154 204)(155 157)(156 202)(158 208)(160 206)(161 167)(162 212)(163 165)(164 210)(166 216)(168 214)(171 175)(179 183)(187 191)(195 199)(201 203)(205 207)(209 211)(213 215)(217 219)(221 223)
(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)
G:=sub<Sym(224)| (1,147,35,139,27,131,19)(2,148,36,140,28,132,20)(3,149,37,141,29,133,21)(4,150,38,142,30,134,22)(5,151,39,143,31,135,23)(6,152,40,144,32,136,24)(7,145,33,137,25,129,17)(8,146,34,138,26,130,18)(9,62,166,54,158,46,122)(10,63,167,55,159,47,123)(11,64,168,56,160,48,124)(12,57,161,49,153,41,125)(13,58,162,50,154,42,126)(14,59,163,51,155,43,127)(15,60,164,52,156,44,128)(16,61,165,53,157,45,121)(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)(89,116,217,105,209,97,201)(90,117,218,106,210,98,202)(91,118,219,107,211,99,203)(92,119,220,108,212,100,204)(93,120,221,109,213,101,205)(94,113,222,110,214,102,206)(95,114,223,111,215,103,207)(96,115,224,112,216,104,208), (1,96,169,42)(2,89,170,43)(3,90,171,44)(4,91,172,45)(5,92,173,46)(6,93,174,47)(7,94,175,48)(8,95,176,41)(9,39,220,85)(10,40,221,86)(11,33,222,87)(12,34,223,88)(13,35,224,81)(14,36,217,82)(15,37,218,83)(16,38,219,84)(17,206,71,160)(18,207,72,153)(19,208,65,154)(20,201,66,155)(21,202,67,156)(22,203,68,157)(23,204,69,158)(24,205,70,159)(25,214,79,168)(26,215,80,161)(27,216,73,162)(28,209,74,163)(29,210,75,164)(30,211,76,165)(31,212,77,166)(32,213,78,167)(49,130,103,184)(50,131,104,177)(51,132,97,178)(52,133,98,179)(53,134,99,180)(54,135,100,181)(55,136,101,182)(56,129,102,183)(57,138,111,192)(58,139,112,185)(59,140,105,186)(60,141,106,187)(61,142,107,188)(62,143,108,189)(63,144,109,190)(64,137,110,191)(113,199,124,145)(114,200,125,146)(115,193,126,147)(116,194,127,148)(117,195,128,149)(118,196,121,150)(119,197,122,151)(120,198,123,152), (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), (2,176)(3,7)(4,174)(6,172)(8,170)(9,224)(10,12)(11,222)(13,220)(14,16)(15,218)(17,21)(18,66)(20,72)(22,70)(24,68)(25,29)(26,74)(28,80)(30,78)(32,76)(33,37)(34,82)(36,88)(38,86)(40,84)(41,47)(42,92)(43,45)(44,90)(46,96)(48,94)(49,55)(50,100)(51,53)(52,98)(54,104)(56,102)(57,63)(58,108)(59,61)(60,106)(62,112)(64,110)(67,71)(75,79)(83,87)(89,91)(93,95)(97,99)(101,103)(105,107)(109,111)(113,124)(114,120)(115,122)(116,118)(117,128)(119,126)(121,127)(123,125)(129,133)(130,178)(132,184)(134,182)(136,180)(137,141)(138,186)(140,192)(142,190)(144,188)(145,149)(146,194)(148,200)(150,198)(152,196)(153,159)(154,204)(155,157)(156,202)(158,208)(160,206)(161,167)(162,212)(163,165)(164,210)(166,216)(168,214)(171,175)(179,183)(187,191)(195,199)(201,203)(205,207)(209,211)(213,215)(217,219)(221,223), (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)>;
G:=Group( (1,147,35,139,27,131,19)(2,148,36,140,28,132,20)(3,149,37,141,29,133,21)(4,150,38,142,30,134,22)(5,151,39,143,31,135,23)(6,152,40,144,32,136,24)(7,145,33,137,25,129,17)(8,146,34,138,26,130,18)(9,62,166,54,158,46,122)(10,63,167,55,159,47,123)(11,64,168,56,160,48,124)(12,57,161,49,153,41,125)(13,58,162,50,154,42,126)(14,59,163,51,155,43,127)(15,60,164,52,156,44,128)(16,61,165,53,157,45,121)(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)(89,116,217,105,209,97,201)(90,117,218,106,210,98,202)(91,118,219,107,211,99,203)(92,119,220,108,212,100,204)(93,120,221,109,213,101,205)(94,113,222,110,214,102,206)(95,114,223,111,215,103,207)(96,115,224,112,216,104,208), (1,96,169,42)(2,89,170,43)(3,90,171,44)(4,91,172,45)(5,92,173,46)(6,93,174,47)(7,94,175,48)(8,95,176,41)(9,39,220,85)(10,40,221,86)(11,33,222,87)(12,34,223,88)(13,35,224,81)(14,36,217,82)(15,37,218,83)(16,38,219,84)(17,206,71,160)(18,207,72,153)(19,208,65,154)(20,201,66,155)(21,202,67,156)(22,203,68,157)(23,204,69,158)(24,205,70,159)(25,214,79,168)(26,215,80,161)(27,216,73,162)(28,209,74,163)(29,210,75,164)(30,211,76,165)(31,212,77,166)(32,213,78,167)(49,130,103,184)(50,131,104,177)(51,132,97,178)(52,133,98,179)(53,134,99,180)(54,135,100,181)(55,136,101,182)(56,129,102,183)(57,138,111,192)(58,139,112,185)(59,140,105,186)(60,141,106,187)(61,142,107,188)(62,143,108,189)(63,144,109,190)(64,137,110,191)(113,199,124,145)(114,200,125,146)(115,193,126,147)(116,194,127,148)(117,195,128,149)(118,196,121,150)(119,197,122,151)(120,198,123,152), (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), (2,176)(3,7)(4,174)(6,172)(8,170)(9,224)(10,12)(11,222)(13,220)(14,16)(15,218)(17,21)(18,66)(20,72)(22,70)(24,68)(25,29)(26,74)(28,80)(30,78)(32,76)(33,37)(34,82)(36,88)(38,86)(40,84)(41,47)(42,92)(43,45)(44,90)(46,96)(48,94)(49,55)(50,100)(51,53)(52,98)(54,104)(56,102)(57,63)(58,108)(59,61)(60,106)(62,112)(64,110)(67,71)(75,79)(83,87)(89,91)(93,95)(97,99)(101,103)(105,107)(109,111)(113,124)(114,120)(115,122)(116,118)(117,128)(119,126)(121,127)(123,125)(129,133)(130,178)(132,184)(134,182)(136,180)(137,141)(138,186)(140,192)(142,190)(144,188)(145,149)(146,194)(148,200)(150,198)(152,196)(153,159)(154,204)(155,157)(156,202)(158,208)(160,206)(161,167)(162,212)(163,165)(164,210)(166,216)(168,214)(171,175)(179,183)(187,191)(195,199)(201,203)(205,207)(209,211)(213,215)(217,219)(221,223), (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) );
G=PermutationGroup([[(1,147,35,139,27,131,19),(2,148,36,140,28,132,20),(3,149,37,141,29,133,21),(4,150,38,142,30,134,22),(5,151,39,143,31,135,23),(6,152,40,144,32,136,24),(7,145,33,137,25,129,17),(8,146,34,138,26,130,18),(9,62,166,54,158,46,122),(10,63,167,55,159,47,123),(11,64,168,56,160,48,124),(12,57,161,49,153,41,125),(13,58,162,50,154,42,126),(14,59,163,51,155,43,127),(15,60,164,52,156,44,128),(16,61,165,53,157,45,121),(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),(89,116,217,105,209,97,201),(90,117,218,106,210,98,202),(91,118,219,107,211,99,203),(92,119,220,108,212,100,204),(93,120,221,109,213,101,205),(94,113,222,110,214,102,206),(95,114,223,111,215,103,207),(96,115,224,112,216,104,208)], [(1,96,169,42),(2,89,170,43),(3,90,171,44),(4,91,172,45),(5,92,173,46),(6,93,174,47),(7,94,175,48),(8,95,176,41),(9,39,220,85),(10,40,221,86),(11,33,222,87),(12,34,223,88),(13,35,224,81),(14,36,217,82),(15,37,218,83),(16,38,219,84),(17,206,71,160),(18,207,72,153),(19,208,65,154),(20,201,66,155),(21,202,67,156),(22,203,68,157),(23,204,69,158),(24,205,70,159),(25,214,79,168),(26,215,80,161),(27,216,73,162),(28,209,74,163),(29,210,75,164),(30,211,76,165),(31,212,77,166),(32,213,78,167),(49,130,103,184),(50,131,104,177),(51,132,97,178),(52,133,98,179),(53,134,99,180),(54,135,100,181),(55,136,101,182),(56,129,102,183),(57,138,111,192),(58,139,112,185),(59,140,105,186),(60,141,106,187),(61,142,107,188),(62,143,108,189),(63,144,109,190),(64,137,110,191),(113,199,124,145),(114,200,125,146),(115,193,126,147),(116,194,127,148),(117,195,128,149),(118,196,121,150),(119,197,122,151),(120,198,123,152)], [(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)], [(2,176),(3,7),(4,174),(6,172),(8,170),(9,224),(10,12),(11,222),(13,220),(14,16),(15,218),(17,21),(18,66),(20,72),(22,70),(24,68),(25,29),(26,74),(28,80),(30,78),(32,76),(33,37),(34,82),(36,88),(38,86),(40,84),(41,47),(42,92),(43,45),(44,90),(46,96),(48,94),(49,55),(50,100),(51,53),(52,98),(54,104),(56,102),(57,63),(58,108),(59,61),(60,106),(62,112),(64,110),(67,71),(75,79),(83,87),(89,91),(93,95),(97,99),(101,103),(105,107),(109,111),(113,124),(114,120),(115,122),(116,118),(117,128),(119,126),(121,127),(123,125),(129,133),(130,178),(132,184),(134,182),(136,180),(137,141),(138,186),(140,192),(142,190),(144,188),(145,149),(146,194),(148,200),(150,198),(152,196),(153,159),(154,204),(155,157),(156,202),(158,208),(160,206),(161,167),(162,212),(163,165),(164,210),(166,216),(168,214),(171,175),(179,183),(187,191),(195,199),(201,203),(205,207),(209,211),(213,215),(217,219),(221,223)], [(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)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14X | 28A | ··· | 28AJ | 28AK | ··· | 28BB | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | ··· | 2 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
154 irreducible representations
| dim | 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 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C4○D8 | C7×D4 | C7×C4○D4 | C7×C4○D8 |
| kernel | C7×C42.78C22 | C4×C56 | C7×D4⋊C4 | C7×Q8⋊C4 | C7×C4.4D4 | C7×C42.C2 | C42.78C22 | C4×C8 | D4⋊C4 | Q8⋊C4 | C4.4D4 | C42.C2 | C2×C28 | C28 | C14 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 6 | 6 | 12 | 12 | 6 | 6 | 2 | 4 | 8 | 12 | 24 | 48 |
Matrix representation of C7×C42.78C22 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 0 | 15 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 31 | 82 | 0 | 0 |
| 31 | 31 | 0 | 0 |
| 0 | 0 | 13 | 13 |
| 0 | 0 | 100 | 13 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,30,0,0,0,0,30],[0,98,0,0,15,0,0,0,0,0,98,0,0,0,0,98],[0,1,0,0,112,0,0,0,0,0,0,1,0,0,112,0],[112,0,0,0,0,1,0,0,0,0,1,0,0,0,0,112],[31,31,0,0,82,31,0,0,0,0,13,100,0,0,13,13] >;
C7×C42.78C22 in GAP, Magma, Sage, TeX
C_7\times C_4^2._{78}C_2^2 % in TeX
G:=Group("C7xC4^2.78C2^2"); // GroupNames label
G:=SmallGroup(448,896);
// by ID
G=gap.SmallGroup(448,896);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,2360,2438,310,9804,172,14117,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^4=d^2=1,e^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=b^-1*c^2,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=b^2*c*d>;
// generators/relations