direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.24D4, C4○D4⋊1C28, (C22×C8)⋊4C14, (C22×C56)⋊8C2, D4.5(C2×C28), C4.53(D4×C14), Q8.5(C2×C28), (C2×C28).518D4, C28.460(C2×D4), C4.3(C22×C28), D4⋊C4⋊20C14, C23.23(C7×D4), C42⋊C2⋊2C14, Q8⋊C4⋊20C14, C22.43(D4×C14), C14.114(C4○D8), (C2×C28).892C23, C28.148(C22×C4), (C2×C56).358C22, (C22×C14).127D4, C28.116(C22⋊C4), (D4×C14).287C22, (Q8×C14).251C22, (C22×C28).583C22, (C7×C4○D4)⋊7C4, C2.1(C7×C4○D8), C4⋊C4.38(C2×C14), (C2×C8).61(C2×C14), (C2×C4).49(C2×C28), (C2×C4○D4).4C14, (C7×D4).27(C2×C4), (C2×C4).122(C7×D4), C4.32(C7×C22⋊C4), (C7×Q8).29(C2×C4), (C7×D4⋊C4)⋊43C2, (C2×C28).270(C2×C4), (C7×Q8⋊C4)⋊43C2, (C14×C4○D4).18C2, (C2×D4).45(C2×C14), (C2×C14).619(C2×D4), C2.19(C14×C22⋊C4), (C2×Q8).36(C2×C14), C22.4(C7×C22⋊C4), (C7×C42⋊C2)⋊23C2, (C7×C4⋊C4).359C22, C14.107(C2×C22⋊C4), (C2×C4).67(C22×C14), (C2×C14).31(C22⋊C4), (C22×C4).112(C2×C14), SmallGroup(448,824)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.24D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=1, e4=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 258 in 158 conjugacy classes, 82 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C28, C28, C28, C2×C14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C23.24D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×D4⋊C4, C7×Q8⋊C4, C7×C42⋊C2, C22×C56, C14×C4○D4, C7×C23.24D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C2×C22⋊C4, C4○D8, C2×C28, C7×D4, C22×C14, C23.24D4, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4, C7×C4○D8, C7×C23.24D4
►On 224 points
Generators in S224
(1 170 219 51 211 43 203)(2 171 220 52 212 44 204)(3 172 221 53 213 45 205)(4 173 222 54 214 46 206)(5 174 223 55 215 47 207)(6 175 224 56 216 48 208)(7 176 217 49 209 41 201)(8 169 218 50 210 42 202)(9 161 193 25 185 17 177)(10 162 194 26 186 18 178)(11 163 195 27 187 19 179)(12 164 196 28 188 20 180)(13 165 197 29 189 21 181)(14 166 198 30 190 22 182)(15 167 199 31 191 23 183)(16 168 200 32 192 24 184)(33 131 75 123 67 115 60)(34 132 76 124 68 116 61)(35 133 77 125 69 117 62)(36 134 78 126 70 118 63)(37 135 79 127 71 119 64)(38 136 80 128 72 120 57)(39 129 73 121 65 113 58)(40 130 74 122 66 114 59)(81 112 160 104 152 96 143)(82 105 153 97 145 89 144)(83 106 154 98 146 90 137)(84 107 155 99 147 91 138)(85 108 156 100 148 92 139)(86 109 157 101 149 93 140)(87 110 158 102 150 94 141)(88 111 159 103 151 95 142)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)(193 197)(194 198)(195 199)(196 200)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 145)(16 146)(17 155)(18 156)(19 157)(20 158)(21 159)(22 160)(23 153)(24 154)(25 84)(26 85)(27 86)(28 87)(29 88)(30 81)(31 82)(32 83)(33 221)(34 222)(35 223)(36 224)(37 217)(38 218)(39 219)(40 220)(41 127)(42 128)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 135)(50 136)(51 129)(52 130)(53 131)(54 132)(55 133)(56 134)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 203)(66 204)(67 205)(68 206)(69 207)(70 208)(71 201)(72 202)(73 211)(74 212)(75 213)(76 214)(77 215)(78 216)(79 209)(80 210)(89 167)(90 168)(91 161)(92 162)(93 163)(94 164)(95 165)(96 166)(97 183)(98 184)(99 177)(100 178)(101 179)(102 180)(103 181)(104 182)(105 191)(106 192)(107 185)(108 186)(109 187)(110 188)(111 189)(112 190)(137 200)(138 193)(139 194)(140 195)(141 196)(142 197)(143 198)(144 199)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(145 149)(146 150)(147 151)(148 152)(153 157)(154 158)(155 159)(156 160)(161 165)(162 166)(163 167)(164 168)(169 173)(170 174)(171 175)(172 176)(177 181)(178 182)(179 183)(180 184)(185 189)(186 190)(187 191)(188 192)(193 197)(194 198)(195 199)(196 200)(201 205)(202 206)(203 207)(204 208)(209 213)(210 214)(211 215)(212 216)(217 221)(218 222)(219 223)(220 224)
(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 96 113 166)(2 165 114 95)(3 94 115 164)(4 163 116 93)(5 92 117 162)(6 161 118 91)(7 90 119 168)(8 167 120 89)(9 70 147 208)(10 207 148 69)(11 68 149 206)(12 205 150 67)(13 66 151 204)(14 203 152 65)(15 72 145 202)(16 201 146 71)(17 78 155 216)(18 215 156 77)(19 76 157 214)(20 213 158 75)(21 74 159 212)(22 211 160 73)(23 80 153 210)(24 209 154 79)(25 36 84 224)(26 223 85 35)(27 34 86 222)(28 221 87 33)(29 40 88 220)(30 219 81 39)(31 38 82 218)(32 217 83 37)(41 98 127 184)(42 183 128 97)(43 104 121 182)(44 181 122 103)(45 102 123 180)(46 179 124 101)(47 100 125 178)(48 177 126 99)(49 106 135 192)(50 191 136 105)(51 112 129 190)(52 189 130 111)(53 110 131 188)(54 187 132 109)(55 108 133 186)(56 185 134 107)(57 144 169 199)(58 198 170 143)(59 142 171 197)(60 196 172 141)(61 140 173 195)(62 194 174 139)(63 138 175 193)(64 200 176 137)
G:=sub<Sym(224)| (1,170,219,51,211,43,203)(2,171,220,52,212,44,204)(3,172,221,53,213,45,205)(4,173,222,54,214,46,206)(5,174,223,55,215,47,207)(6,175,224,56,216,48,208)(7,176,217,49,209,41,201)(8,169,218,50,210,42,202)(9,161,193,25,185,17,177)(10,162,194,26,186,18,178)(11,163,195,27,187,19,179)(12,164,196,28,188,20,180)(13,165,197,29,189,21,181)(14,166,198,30,190,22,182)(15,167,199,31,191,23,183)(16,168,200,32,192,24,184)(33,131,75,123,67,115,60)(34,132,76,124,68,116,61)(35,133,77,125,69,117,62)(36,134,78,126,70,118,63)(37,135,79,127,71,119,64)(38,136,80,128,72,120,57)(39,129,73,121,65,113,58)(40,130,74,122,66,114,59)(81,112,160,104,152,96,143)(82,105,153,97,145,89,144)(83,106,154,98,146,90,137)(84,107,155,99,147,91,138)(85,108,156,100,148,92,139)(86,109,157,101,149,93,140)(87,110,158,102,150,94,141)(88,111,159,103,151,95,142), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,145)(16,146)(17,155)(18,156)(19,157)(20,158)(21,159)(22,160)(23,153)(24,154)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(33,221)(34,222)(35,223)(36,224)(37,217)(38,218)(39,219)(40,220)(41,127)(42,128)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,135)(50,136)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,203)(66,204)(67,205)(68,206)(69,207)(70,208)(71,201)(72,202)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,209)(80,210)(89,167)(90,168)(91,161)(92,162)(93,163)(94,164)(95,165)(96,166)(97,183)(98,184)(99,177)(100,178)(101,179)(102,180)(103,181)(104,182)(105,191)(106,192)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(137,200)(138,193)(139,194)(140,195)(141,196)(142,197)(143,198)(144,199), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224), (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), (1,96,113,166)(2,165,114,95)(3,94,115,164)(4,163,116,93)(5,92,117,162)(6,161,118,91)(7,90,119,168)(8,167,120,89)(9,70,147,208)(10,207,148,69)(11,68,149,206)(12,205,150,67)(13,66,151,204)(14,203,152,65)(15,72,145,202)(16,201,146,71)(17,78,155,216)(18,215,156,77)(19,76,157,214)(20,213,158,75)(21,74,159,212)(22,211,160,73)(23,80,153,210)(24,209,154,79)(25,36,84,224)(26,223,85,35)(27,34,86,222)(28,221,87,33)(29,40,88,220)(30,219,81,39)(31,38,82,218)(32,217,83,37)(41,98,127,184)(42,183,128,97)(43,104,121,182)(44,181,122,103)(45,102,123,180)(46,179,124,101)(47,100,125,178)(48,177,126,99)(49,106,135,192)(50,191,136,105)(51,112,129,190)(52,189,130,111)(53,110,131,188)(54,187,132,109)(55,108,133,186)(56,185,134,107)(57,144,169,199)(58,198,170,143)(59,142,171,197)(60,196,172,141)(61,140,173,195)(62,194,174,139)(63,138,175,193)(64,200,176,137)>;
G:=Group( (1,170,219,51,211,43,203)(2,171,220,52,212,44,204)(3,172,221,53,213,45,205)(4,173,222,54,214,46,206)(5,174,223,55,215,47,207)(6,175,224,56,216,48,208)(7,176,217,49,209,41,201)(8,169,218,50,210,42,202)(9,161,193,25,185,17,177)(10,162,194,26,186,18,178)(11,163,195,27,187,19,179)(12,164,196,28,188,20,180)(13,165,197,29,189,21,181)(14,166,198,30,190,22,182)(15,167,199,31,191,23,183)(16,168,200,32,192,24,184)(33,131,75,123,67,115,60)(34,132,76,124,68,116,61)(35,133,77,125,69,117,62)(36,134,78,126,70,118,63)(37,135,79,127,71,119,64)(38,136,80,128,72,120,57)(39,129,73,121,65,113,58)(40,130,74,122,66,114,59)(81,112,160,104,152,96,143)(82,105,153,97,145,89,144)(83,106,154,98,146,90,137)(84,107,155,99,147,91,138)(85,108,156,100,148,92,139)(86,109,157,101,149,93,140)(87,110,158,102,150,94,141)(88,111,159,103,151,95,142), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,145)(16,146)(17,155)(18,156)(19,157)(20,158)(21,159)(22,160)(23,153)(24,154)(25,84)(26,85)(27,86)(28,87)(29,88)(30,81)(31,82)(32,83)(33,221)(34,222)(35,223)(36,224)(37,217)(38,218)(39,219)(40,220)(41,127)(42,128)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,135)(50,136)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,203)(66,204)(67,205)(68,206)(69,207)(70,208)(71,201)(72,202)(73,211)(74,212)(75,213)(76,214)(77,215)(78,216)(79,209)(80,210)(89,167)(90,168)(91,161)(92,162)(93,163)(94,164)(95,165)(96,166)(97,183)(98,184)(99,177)(100,178)(101,179)(102,180)(103,181)(104,182)(105,191)(106,192)(107,185)(108,186)(109,187)(110,188)(111,189)(112,190)(137,200)(138,193)(139,194)(140,195)(141,196)(142,197)(143,198)(144,199), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224), (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), (1,96,113,166)(2,165,114,95)(3,94,115,164)(4,163,116,93)(5,92,117,162)(6,161,118,91)(7,90,119,168)(8,167,120,89)(9,70,147,208)(10,207,148,69)(11,68,149,206)(12,205,150,67)(13,66,151,204)(14,203,152,65)(15,72,145,202)(16,201,146,71)(17,78,155,216)(18,215,156,77)(19,76,157,214)(20,213,158,75)(21,74,159,212)(22,211,160,73)(23,80,153,210)(24,209,154,79)(25,36,84,224)(26,223,85,35)(27,34,86,222)(28,221,87,33)(29,40,88,220)(30,219,81,39)(31,38,82,218)(32,217,83,37)(41,98,127,184)(42,183,128,97)(43,104,121,182)(44,181,122,103)(45,102,123,180)(46,179,124,101)(47,100,125,178)(48,177,126,99)(49,106,135,192)(50,191,136,105)(51,112,129,190)(52,189,130,111)(53,110,131,188)(54,187,132,109)(55,108,133,186)(56,185,134,107)(57,144,169,199)(58,198,170,143)(59,142,171,197)(60,196,172,141)(61,140,173,195)(62,194,174,139)(63,138,175,193)(64,200,176,137) );
G=PermutationGroup([[(1,170,219,51,211,43,203),(2,171,220,52,212,44,204),(3,172,221,53,213,45,205),(4,173,222,54,214,46,206),(5,174,223,55,215,47,207),(6,175,224,56,216,48,208),(7,176,217,49,209,41,201),(8,169,218,50,210,42,202),(9,161,193,25,185,17,177),(10,162,194,26,186,18,178),(11,163,195,27,187,19,179),(12,164,196,28,188,20,180),(13,165,197,29,189,21,181),(14,166,198,30,190,22,182),(15,167,199,31,191,23,183),(16,168,200,32,192,24,184),(33,131,75,123,67,115,60),(34,132,76,124,68,116,61),(35,133,77,125,69,117,62),(36,134,78,126,70,118,63),(37,135,79,127,71,119,64),(38,136,80,128,72,120,57),(39,129,73,121,65,113,58),(40,130,74,122,66,114,59),(81,112,160,104,152,96,143),(82,105,153,97,145,89,144),(83,106,154,98,146,90,137),(84,107,155,99,147,91,138),(85,108,156,100,148,92,139),(86,109,157,101,149,93,140),(87,110,158,102,150,94,141),(88,111,159,103,151,95,142)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160),(161,165),(162,166),(163,167),(164,168),(177,181),(178,182),(179,183),(180,184),(185,189),(186,190),(187,191),(188,192),(193,197),(194,198),(195,199),(196,200)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,147),(10,148),(11,149),(12,150),(13,151),(14,152),(15,145),(16,146),(17,155),(18,156),(19,157),(20,158),(21,159),(22,160),(23,153),(24,154),(25,84),(26,85),(27,86),(28,87),(29,88),(30,81),(31,82),(32,83),(33,221),(34,222),(35,223),(36,224),(37,217),(38,218),(39,219),(40,220),(41,127),(42,128),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,135),(50,136),(51,129),(52,130),(53,131),(54,132),(55,133),(56,134),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,203),(66,204),(67,205),(68,206),(69,207),(70,208),(71,201),(72,202),(73,211),(74,212),(75,213),(76,214),(77,215),(78,216),(79,209),(80,210),(89,167),(90,168),(91,161),(92,162),(93,163),(94,164),(95,165),(96,166),(97,183),(98,184),(99,177),(100,178),(101,179),(102,180),(103,181),(104,182),(105,191),(106,192),(107,185),(108,186),(109,187),(110,188),(111,189),(112,190),(137,200),(138,193),(139,194),(140,195),(141,196),(142,197),(143,198),(144,199)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(145,149),(146,150),(147,151),(148,152),(153,157),(154,158),(155,159),(156,160),(161,165),(162,166),(163,167),(164,168),(169,173),(170,174),(171,175),(172,176),(177,181),(178,182),(179,183),(180,184),(185,189),(186,190),(187,191),(188,192),(193,197),(194,198),(195,199),(196,200),(201,205),(202,206),(203,207),(204,208),(209,213),(210,214),(211,215),(212,216),(217,221),(218,222),(219,223),(220,224)], [(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)], [(1,96,113,166),(2,165,114,95),(3,94,115,164),(4,163,116,93),(5,92,117,162),(6,161,118,91),(7,90,119,168),(8,167,120,89),(9,70,147,208),(10,207,148,69),(11,68,149,206),(12,205,150,67),(13,66,151,204),(14,203,152,65),(15,72,145,202),(16,201,146,71),(17,78,155,216),(18,215,156,77),(19,76,157,214),(20,213,158,75),(21,74,159,212),(22,211,160,73),(23,80,153,210),(24,209,154,79),(25,36,84,224),(26,223,85,35),(27,34,86,222),(28,221,87,33),(29,40,88,220),(30,219,81,39),(31,38,82,218),(32,217,83,37),(41,98,127,184),(42,183,128,97),(43,104,121,182),(44,181,122,103),(45,102,123,180),(46,179,124,101),(47,100,125,178),(48,177,126,99),(49,106,135,192),(50,191,136,105),(51,112,129,190),(52,189,130,111),(53,110,131,188),(54,187,132,109),(55,108,133,186),(56,185,134,107),(57,144,169,199),(58,198,170,143),(59,142,171,197),(60,196,172,141),(61,140,173,195),(62,194,174,139),(63,138,175,193),(64,200,176,137)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28AJ | 28AK | ··· | 28BT | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
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 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | D4 | C4○D8 | C7×D4 | C7×D4 | C7×C4○D8 |
| kernel | C7×C23.24D4 | C7×D4⋊C4 | C7×Q8⋊C4 | C7×C42⋊C2 | C22×C56 | C14×C4○D4 | C7×C4○D4 | C23.24D4 | D4⋊C4 | Q8⋊C4 | C42⋊C2 | C22×C8 | C2×C4○D4 | C4○D4 | C2×C28 | C22×C14 | C14 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 6 | 12 | 12 | 6 | 6 | 6 | 48 | 3 | 1 | 8 | 18 | 6 | 48 |
Matrix representation of C7×C23.24D4 ►in GL3(𝔽113) generated by
| 16 | 0 | 0 |
| 0 | 49 | 0 |
| 0 | 0 | 49 |
| 112 | 0 | 0 |
| 0 | 1 | 29 |
| 0 | 0 | 112 |
| 112 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 1 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 98 | 0 | 0 |
| 0 | 18 | 108 |
| 0 | 0 | 69 |
| 98 | 0 | 0 |
| 0 | 78 | 70 |
| 0 | 18 | 35 |
G:=sub<GL(3,GF(113))| [16,0,0,0,49,0,0,0,49],[112,0,0,0,1,0,0,29,112],[112,0,0,0,112,0,0,0,112],[1,0,0,0,112,0,0,0,112],[98,0,0,0,18,0,0,108,69],[98,0,0,0,78,18,0,70,35] >;
C7×C23.24D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{24}D_4 % in TeX
G:=Group("C7xC2^3.24D4"); // GroupNames label
G:=SmallGroup(448,824);
// by ID
G=gap.SmallGroup(448,824);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1192,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
// generators/relations