direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4.5D4, C56.100D4, D4.5(C7×D4), C8.20(C7×D4), Q8.5(C7×D4), (C2×Q16)⋊8C14, C8○D4.1C14, (C7×D4).30D4, C4.39(D4×C14), (C7×Q8).30D4, C8.C22.C14, C8.C4⋊7C14, (C14×Q16)⋊22C2, C28.400(C2×D4), C4.10D4⋊4C14, (C2×C56).274C22, (C2×C28).615C23, M4(2).5(C2×C14), C14.156(C4⋊D4), (Q8×C14).168C22, (C7×M4(2)).49C22, (C7×C8○D4).4C2, (C2×C8).26(C2×C14), C2.25(C7×C4⋊D4), (C7×C8.C4)⋊16C2, C22.8(C7×C4○D4), C4○D4.12(C2×C14), (C2×Q8).12(C2×C14), (C7×C8.C22).2C2, (C7×C4.10D4)⋊10C2, (C2×C4).10(C22×C14), (C7×C4○D4).57C22, (C2×C14).117(C4○D4), SmallGroup(448,881)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4.5D4
G = < a,b,c,d,e | a7=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d3 >
Subgroups: 162 in 100 conjugacy classes, 50 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C14, C14, C2×C8, C2×C8, M4(2), M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C56, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, D4.5D4, C2×C56, C2×C56, C7×M4(2), C7×M4(2), C7×M4(2), C7×SD16, C7×Q16, Q8×C14, C7×C4○D4, C7×C4.10D4, C7×C8.C4, C7×C8○D4, C14×Q16, C7×C8.C22, C7×D4.5D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C7×D4, C22×C14, D4.5D4, D4×C14, C7×C4○D4, C7×C4⋊D4, C7×D4.5D4
►On 224 points
Generators in S224
(1 31 47 79 95 23 39)(2 32 48 80 96 24 40)(3 25 41 73 89 17 33)(4 26 42 74 90 18 34)(5 27 43 75 91 19 35)(6 28 44 76 92 20 36)(7 29 45 77 93 21 37)(8 30 46 78 94 22 38)(9 56 88 104 117 70 112)(10 49 81 97 118 71 105)(11 50 82 98 119 72 106)(12 51 83 99 120 65 107)(13 52 84 100 113 66 108)(14 53 85 101 114 67 109)(15 54 86 102 115 68 110)(16 55 87 103 116 69 111)(57 172 221 195 162 144 213)(58 173 222 196 163 137 214)(59 174 223 197 164 138 215)(60 175 224 198 165 139 216)(61 176 217 199 166 140 209)(62 169 218 200 167 141 210)(63 170 219 193 168 142 211)(64 171 220 194 161 143 212)(121 188 155 129 206 180 147)(122 189 156 130 207 181 148)(123 190 157 131 208 182 149)(124 191 158 132 201 183 150)(125 192 159 133 202 184 151)(126 185 160 134 203 177 152)(127 186 153 135 204 178 145)(128 187 154 136 205 179 146)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(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 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)(65 71 69 67)(66 72 70 68)(73 75 77 79)(74 76 78 80)(81 87 85 83)(82 88 86 84)(89 91 93 95)(90 92 94 96)(97 103 101 99)(98 104 102 100)(105 111 109 107)(106 112 110 108)(113 119 117 115)(114 120 118 116)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 143 141 139)(138 144 142 140)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)(161 167 165 163)(162 168 166 164)(169 175 173 171)(170 176 174 172)(177 179 181 183)(178 180 182 184)(185 187 189 191)(186 188 190 192)(193 199 197 195)(194 200 198 196)(201 203 205 207)(202 204 206 208)(209 215 213 211)(210 216 214 212)(217 223 221 219)(218 224 222 220)
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 94)(10 95)(11 96)(12 89)(13 90)(14 91)(15 92)(16 93)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 120)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 81)(40 82)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 71)(48 72)(57 191)(58 192)(59 185)(60 186)(61 187)(62 188)(63 189)(64 190)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 105)(80 106)(121 210)(122 211)(123 212)(124 213)(125 214)(126 215)(127 216)(128 209)(129 218)(130 219)(131 220)(132 221)(133 222)(134 223)(135 224)(136 217)(137 151)(138 152)(139 145)(140 146)(141 147)(142 148)(143 149)(144 150)(153 175)(154 176)(155 169)(156 170)(157 171)(158 172)(159 173)(160 174)(161 182)(162 183)(163 184)(164 177)(165 178)(166 179)(167 180)(168 181)(193 207)(194 208)(195 201)(196 202)(197 203)(198 204)(199 205)(200 206)
(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)
(1 127 5 123)(2 126 6 122)(3 125 7 121)(4 124 8 128)(9 193 13 197)(10 200 14 196)(11 199 15 195)(12 198 16 194)(17 184 21 180)(18 183 22 179)(19 182 23 178)(20 181 24 177)(25 192 29 188)(26 191 30 187)(27 190 31 186)(28 189 32 185)(33 151 37 147)(34 150 38 146)(35 149 39 145)(36 148 40 152)(41 159 45 155)(42 158 46 154)(43 157 47 153)(44 156 48 160)(49 167 53 163)(50 166 54 162)(51 165 55 161)(52 164 56 168)(57 119 61 115)(58 118 62 114)(59 117 63 113)(60 116 64 120)(65 175 69 171)(66 174 70 170)(67 173 71 169)(68 172 72 176)(73 133 77 129)(74 132 78 136)(75 131 79 135)(76 130 80 134)(81 141 85 137)(82 140 86 144)(83 139 87 143)(84 138 88 142)(89 202 93 206)(90 201 94 205)(91 208 95 204)(92 207 96 203)(97 210 101 214)(98 209 102 213)(99 216 103 212)(100 215 104 211)(105 218 109 222)(106 217 110 221)(107 224 111 220)(108 223 112 219)
G:=sub<Sym(224)| (1,31,47,79,95,23,39)(2,32,48,80,96,24,40)(3,25,41,73,89,17,33)(4,26,42,74,90,18,34)(5,27,43,75,91,19,35)(6,28,44,76,92,20,36)(7,29,45,77,93,21,37)(8,30,46,78,94,22,38)(9,56,88,104,117,70,112)(10,49,81,97,118,71,105)(11,50,82,98,119,72,106)(12,51,83,99,120,65,107)(13,52,84,100,113,66,108)(14,53,85,101,114,67,109)(15,54,86,102,115,68,110)(16,55,87,103,116,69,111)(57,172,221,195,162,144,213)(58,173,222,196,163,137,214)(59,174,223,197,164,138,215)(60,175,224,198,165,139,216)(61,176,217,199,166,140,209)(62,169,218,200,167,141,210)(63,170,219,193,168,142,211)(64,171,220,194,161,143,212)(121,188,155,129,206,180,147)(122,189,156,130,207,181,148)(123,190,157,131,208,182,149)(124,191,158,132,201,183,150)(125,192,159,133,202,184,151)(126,185,160,134,203,177,152)(127,186,153,135,204,178,145)(128,187,154,136,205,179,146), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(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)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,75,77,79)(74,76,78,80)(81,87,85,83)(82,88,86,84)(89,91,93,95)(90,92,94,96)(97,103,101,99)(98,104,102,100)(105,111,109,107)(106,112,110,108)(113,119,117,115)(114,120,118,116)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,143,141,139)(138,144,142,140)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,167,165,163)(162,168,166,164)(169,175,173,171)(170,176,174,172)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,199,197,195)(194,200,198,196)(201,203,205,207)(202,204,206,208)(209,215,213,211)(210,216,214,212)(217,223,221,219)(218,224,222,220), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,120)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,81)(40,82)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(57,191)(58,192)(59,185)(60,186)(61,187)(62,188)(63,189)(64,190)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,105)(80,106)(121,210)(122,211)(123,212)(124,213)(125,214)(126,215)(127,216)(128,209)(129,218)(130,219)(131,220)(132,221)(133,222)(134,223)(135,224)(136,217)(137,151)(138,152)(139,145)(140,146)(141,147)(142,148)(143,149)(144,150)(153,175)(154,176)(155,169)(156,170)(157,171)(158,172)(159,173)(160,174)(161,182)(162,183)(163,184)(164,177)(165,178)(166,179)(167,180)(168,181)(193,207)(194,208)(195,201)(196,202)(197,203)(198,204)(199,205)(200,206), (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), (1,127,5,123)(2,126,6,122)(3,125,7,121)(4,124,8,128)(9,193,13,197)(10,200,14,196)(11,199,15,195)(12,198,16,194)(17,184,21,180)(18,183,22,179)(19,182,23,178)(20,181,24,177)(25,192,29,188)(26,191,30,187)(27,190,31,186)(28,189,32,185)(33,151,37,147)(34,150,38,146)(35,149,39,145)(36,148,40,152)(41,159,45,155)(42,158,46,154)(43,157,47,153)(44,156,48,160)(49,167,53,163)(50,166,54,162)(51,165,55,161)(52,164,56,168)(57,119,61,115)(58,118,62,114)(59,117,63,113)(60,116,64,120)(65,175,69,171)(66,174,70,170)(67,173,71,169)(68,172,72,176)(73,133,77,129)(74,132,78,136)(75,131,79,135)(76,130,80,134)(81,141,85,137)(82,140,86,144)(83,139,87,143)(84,138,88,142)(89,202,93,206)(90,201,94,205)(91,208,95,204)(92,207,96,203)(97,210,101,214)(98,209,102,213)(99,216,103,212)(100,215,104,211)(105,218,109,222)(106,217,110,221)(107,224,111,220)(108,223,112,219)>;
G:=Group( (1,31,47,79,95,23,39)(2,32,48,80,96,24,40)(3,25,41,73,89,17,33)(4,26,42,74,90,18,34)(5,27,43,75,91,19,35)(6,28,44,76,92,20,36)(7,29,45,77,93,21,37)(8,30,46,78,94,22,38)(9,56,88,104,117,70,112)(10,49,81,97,118,71,105)(11,50,82,98,119,72,106)(12,51,83,99,120,65,107)(13,52,84,100,113,66,108)(14,53,85,101,114,67,109)(15,54,86,102,115,68,110)(16,55,87,103,116,69,111)(57,172,221,195,162,144,213)(58,173,222,196,163,137,214)(59,174,223,197,164,138,215)(60,175,224,198,165,139,216)(61,176,217,199,166,140,209)(62,169,218,200,167,141,210)(63,170,219,193,168,142,211)(64,171,220,194,161,143,212)(121,188,155,129,206,180,147)(122,189,156,130,207,181,148)(123,190,157,131,208,182,149)(124,191,158,132,201,183,150)(125,192,159,133,202,184,151)(126,185,160,134,203,177,152)(127,186,153,135,204,178,145)(128,187,154,136,205,179,146), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(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)(49,55,53,51)(50,56,54,52)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,75,77,79)(74,76,78,80)(81,87,85,83)(82,88,86,84)(89,91,93,95)(90,92,94,96)(97,103,101,99)(98,104,102,100)(105,111,109,107)(106,112,110,108)(113,119,117,115)(114,120,118,116)(121,123,125,127)(122,124,126,128)(129,131,133,135)(130,132,134,136)(137,143,141,139)(138,144,142,140)(145,147,149,151)(146,148,150,152)(153,155,157,159)(154,156,158,160)(161,167,165,163)(162,168,166,164)(169,175,173,171)(170,176,174,172)(177,179,181,183)(178,180,182,184)(185,187,189,191)(186,188,190,192)(193,199,197,195)(194,200,198,196)(201,203,205,207)(202,204,206,208)(209,215,213,211)(210,216,214,212)(217,223,221,219)(218,224,222,220), (1,97)(2,98)(3,99)(4,100)(5,101)(6,102)(7,103)(8,104)(9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,120)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,81)(40,82)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,71)(48,72)(57,191)(58,192)(59,185)(60,186)(61,187)(62,188)(63,189)(64,190)(73,107)(74,108)(75,109)(76,110)(77,111)(78,112)(79,105)(80,106)(121,210)(122,211)(123,212)(124,213)(125,214)(126,215)(127,216)(128,209)(129,218)(130,219)(131,220)(132,221)(133,222)(134,223)(135,224)(136,217)(137,151)(138,152)(139,145)(140,146)(141,147)(142,148)(143,149)(144,150)(153,175)(154,176)(155,169)(156,170)(157,171)(158,172)(159,173)(160,174)(161,182)(162,183)(163,184)(164,177)(165,178)(166,179)(167,180)(168,181)(193,207)(194,208)(195,201)(196,202)(197,203)(198,204)(199,205)(200,206), (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), (1,127,5,123)(2,126,6,122)(3,125,7,121)(4,124,8,128)(9,193,13,197)(10,200,14,196)(11,199,15,195)(12,198,16,194)(17,184,21,180)(18,183,22,179)(19,182,23,178)(20,181,24,177)(25,192,29,188)(26,191,30,187)(27,190,31,186)(28,189,32,185)(33,151,37,147)(34,150,38,146)(35,149,39,145)(36,148,40,152)(41,159,45,155)(42,158,46,154)(43,157,47,153)(44,156,48,160)(49,167,53,163)(50,166,54,162)(51,165,55,161)(52,164,56,168)(57,119,61,115)(58,118,62,114)(59,117,63,113)(60,116,64,120)(65,175,69,171)(66,174,70,170)(67,173,71,169)(68,172,72,176)(73,133,77,129)(74,132,78,136)(75,131,79,135)(76,130,80,134)(81,141,85,137)(82,140,86,144)(83,139,87,143)(84,138,88,142)(89,202,93,206)(90,201,94,205)(91,208,95,204)(92,207,96,203)(97,210,101,214)(98,209,102,213)(99,216,103,212)(100,215,104,211)(105,218,109,222)(106,217,110,221)(107,224,111,220)(108,223,112,219) );
G=PermutationGroup([[(1,31,47,79,95,23,39),(2,32,48,80,96,24,40),(3,25,41,73,89,17,33),(4,26,42,74,90,18,34),(5,27,43,75,91,19,35),(6,28,44,76,92,20,36),(7,29,45,77,93,21,37),(8,30,46,78,94,22,38),(9,56,88,104,117,70,112),(10,49,81,97,118,71,105),(11,50,82,98,119,72,106),(12,51,83,99,120,65,107),(13,52,84,100,113,66,108),(14,53,85,101,114,67,109),(15,54,86,102,115,68,110),(16,55,87,103,116,69,111),(57,172,221,195,162,144,213),(58,173,222,196,163,137,214),(59,174,223,197,164,138,215),(60,175,224,198,165,139,216),(61,176,217,199,166,140,209),(62,169,218,200,167,141,210),(63,170,219,193,168,142,211),(64,171,220,194,161,143,212),(121,188,155,129,206,180,147),(122,189,156,130,207,181,148),(123,190,157,131,208,182,149),(124,191,158,132,201,183,150),(125,192,159,133,202,184,151),(126,185,160,134,203,177,152),(127,186,153,135,204,178,145),(128,187,154,136,205,179,146)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(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),(49,55,53,51),(50,56,54,52),(57,63,61,59),(58,64,62,60),(65,71,69,67),(66,72,70,68),(73,75,77,79),(74,76,78,80),(81,87,85,83),(82,88,86,84),(89,91,93,95),(90,92,94,96),(97,103,101,99),(98,104,102,100),(105,111,109,107),(106,112,110,108),(113,119,117,115),(114,120,118,116),(121,123,125,127),(122,124,126,128),(129,131,133,135),(130,132,134,136),(137,143,141,139),(138,144,142,140),(145,147,149,151),(146,148,150,152),(153,155,157,159),(154,156,158,160),(161,167,165,163),(162,168,166,164),(169,175,173,171),(170,176,174,172),(177,179,181,183),(178,180,182,184),(185,187,189,191),(186,188,190,192),(193,199,197,195),(194,200,198,196),(201,203,205,207),(202,204,206,208),(209,215,213,211),(210,216,214,212),(217,223,221,219),(218,224,222,220)], [(1,97),(2,98),(3,99),(4,100),(5,101),(6,102),(7,103),(8,104),(9,94),(10,95),(11,96),(12,89),(13,90),(14,91),(15,92),(16,93),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,120),(26,113),(27,114),(28,115),(29,116),(30,117),(31,118),(32,119),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,81),(40,82),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,71),(48,72),(57,191),(58,192),(59,185),(60,186),(61,187),(62,188),(63,189),(64,190),(73,107),(74,108),(75,109),(76,110),(77,111),(78,112),(79,105),(80,106),(121,210),(122,211),(123,212),(124,213),(125,214),(126,215),(127,216),(128,209),(129,218),(130,219),(131,220),(132,221),(133,222),(134,223),(135,224),(136,217),(137,151),(138,152),(139,145),(140,146),(141,147),(142,148),(143,149),(144,150),(153,175),(154,176),(155,169),(156,170),(157,171),(158,172),(159,173),(160,174),(161,182),(162,183),(163,184),(164,177),(165,178),(166,179),(167,180),(168,181),(193,207),(194,208),(195,201),(196,202),(197,203),(198,204),(199,205),(200,206)], [(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)], [(1,127,5,123),(2,126,6,122),(3,125,7,121),(4,124,8,128),(9,193,13,197),(10,200,14,196),(11,199,15,195),(12,198,16,194),(17,184,21,180),(18,183,22,179),(19,182,23,178),(20,181,24,177),(25,192,29,188),(26,191,30,187),(27,190,31,186),(28,189,32,185),(33,151,37,147),(34,150,38,146),(35,149,39,145),(36,148,40,152),(41,159,45,155),(42,158,46,154),(43,157,47,153),(44,156,48,160),(49,167,53,163),(50,166,54,162),(51,165,55,161),(52,164,56,168),(57,119,61,115),(58,118,62,114),(59,117,63,113),(60,116,64,120),(65,175,69,171),(66,174,70,170),(67,173,71,169),(68,172,72,176),(73,133,77,129),(74,132,78,136),(75,131,79,135),(76,130,80,134),(81,141,85,137),(82,140,86,144),(83,139,87,143),(84,138,88,142),(89,202,93,206),(90,201,94,205),(91,208,95,204),(92,207,96,203),(97,210,101,214),(98,209,102,213),(99,216,103,212),(100,215,104,211),(105,218,109,222),(106,217,110,221),(107,224,111,220),(108,223,112,219)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14R | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28AD | 56A | ··· | 56L | 56M | ··· | 56AD | 56AE | ··· | 56AP |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D4 | C4○D4 | C7×D4 | C7×D4 | C7×D4 | C7×C4○D4 | D4.5D4 | C7×D4.5D4 |
| kernel | C7×D4.5D4 | C7×C4.10D4 | C7×C8.C4 | C7×C8○D4 | C14×Q16 | C7×C8.C22 | D4.5D4 | C4.10D4 | C8.C4 | C8○D4 | C2×Q16 | C8.C22 | C56 | C7×D4 | C7×Q8 | C2×C14 | C8 | D4 | Q8 | C22 | C7 | C1 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 6 | 12 | 6 | 6 | 6 | 12 | 2 | 1 | 1 | 2 | 12 | 6 | 6 | 12 | 2 | 12 |
Matrix representation of C7×D4.5D4 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 49 |
| 1 | 111 | 0 | 0 |
| 1 | 112 | 0 | 0 |
| 1 | 112 | 0 | 112 |
| 0 | 112 | 1 | 0 |
| 112 | 0 | 0 | 2 |
| 112 | 0 | 1 | 1 |
| 112 | 1 | 0 | 1 |
| 0 | 0 | 0 | 1 |
| 51 | 62 | 0 | 0 |
| 82 | 0 | 0 | 0 |
| 0 | 31 | 82 | 82 |
| 82 | 31 | 31 | 82 |
| 63 | 110 | 0 | 0 |
| 5 | 50 | 0 | 0 |
| 58 | 55 | 55 | 5 |
| 0 | 55 | 5 | 58 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,49,0,0,0,0,49],[1,1,1,0,111,112,112,112,0,0,0,1,0,0,112,0],[112,112,112,0,0,0,1,0,0,1,0,0,2,1,1,1],[51,82,0,82,62,0,31,31,0,0,82,31,0,0,82,82],[63,5,58,0,110,50,55,55,0,0,55,5,0,0,5,58] >;
C7×D4.5D4 in GAP, Magma, Sage, TeX
C_7\times D_4._5D_4
% in TeX
G:=Group("C7xD4.5D4"); // GroupNames label
G:=SmallGroup(448,881);
// by ID
G=gap.SmallGroup(448,881);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1968,2438,9804,172,14117,3547,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=1,d^4=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^3>;
// generators/relations