direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.36D4, C4○D4⋊2C28, D4⋊4(C2×C28), Q8⋊4(C2×C28), C4.54(D4×C14), (C2×C28).314D4, C28.461(C2×D4), C4.4(C22×C28), D4⋊C4⋊14C14, C23.36(C7×D4), Q8⋊C4⋊14C14, C22.44(D4×C14), (C2×M4(2))⋊10C14, (C14×M4(2))⋊28C2, (C2×C56).322C22, C28.149(C22×C4), (C2×C28).893C23, (C22×C14).158D4, C28.108(C22⋊C4), C14.126(C8⋊C22), (D4×C14).288C22, (Q8×C14).252C22, C14.126(C8.C22), (C22×C28).410C22, (C7×C4○D4)⋊8C4, (C14×C4⋊C4)⋊37C2, (C2×C4⋊C4)⋊10C14, (C7×D4)⋊24(C2×C4), (C7×Q8)⋊22(C2×C4), C2.1(C7×C8⋊C22), C4⋊C4.39(C2×C14), (C2×C8).47(C2×C14), (C2×C4).20(C2×C28), (C2×C4○D4).5C14, (C2×C4).123(C7×D4), C4.23(C7×C22⋊C4), (C7×D4⋊C4)⋊37C2, C2.1(C7×C8.C22), (C2×C28).193(C2×C4), (C7×Q8⋊C4)⋊37C2, (C14×C4○D4).19C2, (C2×D4).46(C2×C14), (C2×C14).620(C2×D4), C2.20(C14×C22⋊C4), (C2×Q8).37(C2×C14), C22.5(C7×C22⋊C4), (C7×C4⋊C4).360C22, C14.108(C2×C22⋊C4), (C22×C4).29(C2×C14), (C2×C4).68(C22×C14), (C2×C14).32(C22⋊C4), SmallGroup(448,825)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.36D4
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, ebe-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 274 in 162 conjugacy classes, 82 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C14, C14, C4⋊C4, C4⋊C4, C2×C8, M4(2), 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, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C56, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C23.36D4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×M4(2), C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C7×D4⋊C4, C7×Q8⋊C4, C14×C4⋊C4, C14×M4(2), C14×C4○D4, C7×C23.36D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C2×C22⋊C4, C8⋊C22, C8.C22, C2×C28, C7×D4, C22×C14, C23.36D4, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4, C7×C8⋊C22, C7×C8.C22, C7×C23.36D4
►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 63 135 79 127 71 119)(34 64 136 80 128 72 120)(35 57 129 73 121 65 113)(36 58 130 74 122 66 114)(37 59 131 75 123 67 115)(38 60 132 76 124 68 116)(39 61 133 77 125 69 117)(40 62 134 78 126 70 118)(81 107 155 99 147 91 138)(82 108 156 100 148 92 139)(83 109 157 101 149 93 140)(84 110 158 102 150 94 141)(85 111 159 103 151 95 142)(86 112 160 104 152 96 143)(87 105 153 97 145 89 144)(88 106 154 98 146 90 137)
(1 115)(2 120)(3 117)(4 114)(5 119)(6 116)(7 113)(8 118)(9 149)(10 146)(11 151)(12 148)(13 145)(14 150)(15 147)(16 152)(17 157)(18 154)(19 159)(20 156)(21 153)(22 158)(23 155)(24 160)(25 83)(26 88)(27 85)(28 82)(29 87)(30 84)(31 81)(32 86)(33 174)(34 171)(35 176)(36 173)(37 170)(38 175)(39 172)(40 169)(41 121)(42 126)(43 123)(44 128)(45 125)(46 122)(47 127)(48 124)(49 129)(50 134)(51 131)(52 136)(53 133)(54 130)(55 135)(56 132)(57 217)(58 222)(59 219)(60 224)(61 221)(62 218)(63 223)(64 220)(65 201)(66 206)(67 203)(68 208)(69 205)(70 202)(71 207)(72 204)(73 209)(74 214)(75 211)(76 216)(77 213)(78 210)(79 215)(80 212)(89 165)(90 162)(91 167)(92 164)(93 161)(94 166)(95 163)(96 168)(97 181)(98 178)(99 183)(100 180)(101 177)(102 182)(103 179)(104 184)(105 189)(106 186)(107 191)(108 188)(109 185)(110 190)(111 187)(112 192)(137 194)(138 199)(139 196)(140 193)(141 198)(142 195)(143 200)(144 197)
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 113)(8 114)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 151)(16 152)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 87)(26 88)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 174)(34 175)(35 176)(36 169)(37 170)(38 171)(39 172)(40 173)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 217)(58 218)(59 219)(60 220)(61 221)(62 222)(63 223)(64 224)(65 201)(66 202)(67 203)(68 204)(69 205)(70 206)(71 207)(72 208)(73 209)(74 210)(75 211)(76 212)(77 213)(78 214)(79 215)(80 216)(89 161)(90 162)(91 163)(92 164)(93 165)(94 166)(95 167)(96 168)(97 177)(98 178)(99 179)(100 180)(101 181)(102 182)(103 183)(104 184)(105 185)(106 186)(107 187)(108 188)(109 189)(110 190)(111 191)(112 192)(137 194)(138 195)(139 196)(140 197)(141 198)(142 199)(143 200)(144 193)
(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 94 115 166)(2 165 116 93)(3 92 117 164)(4 163 118 91)(5 90 119 162)(6 161 120 89)(7 96 113 168)(8 167 114 95)(9 72 145 208)(10 207 146 71)(11 70 147 206)(12 205 148 69)(13 68 149 204)(14 203 150 67)(15 66 151 202)(16 201 152 65)(17 80 153 216)(18 215 154 79)(19 78 155 214)(20 213 156 77)(21 76 157 212)(22 211 158 75)(23 74 159 210)(24 209 160 73)(25 64 87 224)(26 223 88 63)(27 62 81 222)(28 221 82 61)(29 60 83 220)(30 219 84 59)(31 58 85 218)(32 217 86 57)(33 194 174 137)(34 144 175 193)(35 200 176 143)(36 142 169 199)(37 198 170 141)(38 140 171 197)(39 196 172 139)(40 138 173 195)(41 104 121 184)(42 183 122 103)(43 102 123 182)(44 181 124 101)(45 100 125 180)(46 179 126 99)(47 98 127 178)(48 177 128 97)(49 112 129 192)(50 191 130 111)(51 110 131 190)(52 189 132 109)(53 108 133 188)(54 187 134 107)(55 106 135 186)(56 185 136 105)
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)(33,63,135,79,127,71,119)(34,64,136,80,128,72,120)(35,57,129,73,121,65,113)(36,58,130,74,122,66,114)(37,59,131,75,123,67,115)(38,60,132,76,124,68,116)(39,61,133,77,125,69,117)(40,62,134,78,126,70,118)(81,107,155,99,147,91,138)(82,108,156,100,148,92,139)(83,109,157,101,149,93,140)(84,110,158,102,150,94,141)(85,111,159,103,151,95,142)(86,112,160,104,152,96,143)(87,105,153,97,145,89,144)(88,106,154,98,146,90,137), (1,115)(2,120)(3,117)(4,114)(5,119)(6,116)(7,113)(8,118)(9,149)(10,146)(11,151)(12,148)(13,145)(14,150)(15,147)(16,152)(17,157)(18,154)(19,159)(20,156)(21,153)(22,158)(23,155)(24,160)(25,83)(26,88)(27,85)(28,82)(29,87)(30,84)(31,81)(32,86)(33,174)(34,171)(35,176)(36,173)(37,170)(38,175)(39,172)(40,169)(41,121)(42,126)(43,123)(44,128)(45,125)(46,122)(47,127)(48,124)(49,129)(50,134)(51,131)(52,136)(53,133)(54,130)(55,135)(56,132)(57,217)(58,222)(59,219)(60,224)(61,221)(62,218)(63,223)(64,220)(65,201)(66,206)(67,203)(68,208)(69,205)(70,202)(71,207)(72,204)(73,209)(74,214)(75,211)(76,216)(77,213)(78,210)(79,215)(80,212)(89,165)(90,162)(91,167)(92,164)(93,161)(94,166)(95,163)(96,168)(97,181)(98,178)(99,183)(100,180)(101,177)(102,182)(103,179)(104,184)(105,189)(106,186)(107,191)(108,188)(109,185)(110,190)(111,187)(112,192)(137,194)(138,199)(139,196)(140,193)(141,198)(142,195)(143,200)(144,197), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,113)(8,114)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,87)(26,88)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,174)(34,175)(35,176)(36,169)(37,170)(38,171)(39,172)(40,173)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,217)(58,218)(59,219)(60,220)(61,221)(62,222)(63,223)(64,224)(65,201)(66,202)(67,203)(68,204)(69,205)(70,206)(71,207)(72,208)(73,209)(74,210)(75,211)(76,212)(77,213)(78,214)(79,215)(80,216)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,177)(98,178)(99,179)(100,180)(101,181)(102,182)(103,183)(104,184)(105,185)(106,186)(107,187)(108,188)(109,189)(110,190)(111,191)(112,192)(137,194)(138,195)(139,196)(140,197)(141,198)(142,199)(143,200)(144,193), (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), (1,94,115,166)(2,165,116,93)(3,92,117,164)(4,163,118,91)(5,90,119,162)(6,161,120,89)(7,96,113,168)(8,167,114,95)(9,72,145,208)(10,207,146,71)(11,70,147,206)(12,205,148,69)(13,68,149,204)(14,203,150,67)(15,66,151,202)(16,201,152,65)(17,80,153,216)(18,215,154,79)(19,78,155,214)(20,213,156,77)(21,76,157,212)(22,211,158,75)(23,74,159,210)(24,209,160,73)(25,64,87,224)(26,223,88,63)(27,62,81,222)(28,221,82,61)(29,60,83,220)(30,219,84,59)(31,58,85,218)(32,217,86,57)(33,194,174,137)(34,144,175,193)(35,200,176,143)(36,142,169,199)(37,198,170,141)(38,140,171,197)(39,196,172,139)(40,138,173,195)(41,104,121,184)(42,183,122,103)(43,102,123,182)(44,181,124,101)(45,100,125,180)(46,179,126,99)(47,98,127,178)(48,177,128,97)(49,112,129,192)(50,191,130,111)(51,110,131,190)(52,189,132,109)(53,108,133,188)(54,187,134,107)(55,106,135,186)(56,185,136,105)>;
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)(33,63,135,79,127,71,119)(34,64,136,80,128,72,120)(35,57,129,73,121,65,113)(36,58,130,74,122,66,114)(37,59,131,75,123,67,115)(38,60,132,76,124,68,116)(39,61,133,77,125,69,117)(40,62,134,78,126,70,118)(81,107,155,99,147,91,138)(82,108,156,100,148,92,139)(83,109,157,101,149,93,140)(84,110,158,102,150,94,141)(85,111,159,103,151,95,142)(86,112,160,104,152,96,143)(87,105,153,97,145,89,144)(88,106,154,98,146,90,137), (1,115)(2,120)(3,117)(4,114)(5,119)(6,116)(7,113)(8,118)(9,149)(10,146)(11,151)(12,148)(13,145)(14,150)(15,147)(16,152)(17,157)(18,154)(19,159)(20,156)(21,153)(22,158)(23,155)(24,160)(25,83)(26,88)(27,85)(28,82)(29,87)(30,84)(31,81)(32,86)(33,174)(34,171)(35,176)(36,173)(37,170)(38,175)(39,172)(40,169)(41,121)(42,126)(43,123)(44,128)(45,125)(46,122)(47,127)(48,124)(49,129)(50,134)(51,131)(52,136)(53,133)(54,130)(55,135)(56,132)(57,217)(58,222)(59,219)(60,224)(61,221)(62,218)(63,223)(64,220)(65,201)(66,206)(67,203)(68,208)(69,205)(70,202)(71,207)(72,204)(73,209)(74,214)(75,211)(76,216)(77,213)(78,210)(79,215)(80,212)(89,165)(90,162)(91,167)(92,164)(93,161)(94,166)(95,163)(96,168)(97,181)(98,178)(99,183)(100,180)(101,177)(102,182)(103,179)(104,184)(105,189)(106,186)(107,191)(108,188)(109,185)(110,190)(111,187)(112,192)(137,194)(138,199)(139,196)(140,193)(141,198)(142,195)(143,200)(144,197), (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,113)(8,114)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,151)(16,152)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,87)(26,88)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,174)(34,175)(35,176)(36,169)(37,170)(38,171)(39,172)(40,173)(41,121)(42,122)(43,123)(44,124)(45,125)(46,126)(47,127)(48,128)(49,129)(50,130)(51,131)(52,132)(53,133)(54,134)(55,135)(56,136)(57,217)(58,218)(59,219)(60,220)(61,221)(62,222)(63,223)(64,224)(65,201)(66,202)(67,203)(68,204)(69,205)(70,206)(71,207)(72,208)(73,209)(74,210)(75,211)(76,212)(77,213)(78,214)(79,215)(80,216)(89,161)(90,162)(91,163)(92,164)(93,165)(94,166)(95,167)(96,168)(97,177)(98,178)(99,179)(100,180)(101,181)(102,182)(103,183)(104,184)(105,185)(106,186)(107,187)(108,188)(109,189)(110,190)(111,191)(112,192)(137,194)(138,195)(139,196)(140,197)(141,198)(142,199)(143,200)(144,193), (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), (1,94,115,166)(2,165,116,93)(3,92,117,164)(4,163,118,91)(5,90,119,162)(6,161,120,89)(7,96,113,168)(8,167,114,95)(9,72,145,208)(10,207,146,71)(11,70,147,206)(12,205,148,69)(13,68,149,204)(14,203,150,67)(15,66,151,202)(16,201,152,65)(17,80,153,216)(18,215,154,79)(19,78,155,214)(20,213,156,77)(21,76,157,212)(22,211,158,75)(23,74,159,210)(24,209,160,73)(25,64,87,224)(26,223,88,63)(27,62,81,222)(28,221,82,61)(29,60,83,220)(30,219,84,59)(31,58,85,218)(32,217,86,57)(33,194,174,137)(34,144,175,193)(35,200,176,143)(36,142,169,199)(37,198,170,141)(38,140,171,197)(39,196,172,139)(40,138,173,195)(41,104,121,184)(42,183,122,103)(43,102,123,182)(44,181,124,101)(45,100,125,180)(46,179,126,99)(47,98,127,178)(48,177,128,97)(49,112,129,192)(50,191,130,111)(51,110,131,190)(52,189,132,109)(53,108,133,188)(54,187,134,107)(55,106,135,186)(56,185,136,105) );
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),(33,63,135,79,127,71,119),(34,64,136,80,128,72,120),(35,57,129,73,121,65,113),(36,58,130,74,122,66,114),(37,59,131,75,123,67,115),(38,60,132,76,124,68,116),(39,61,133,77,125,69,117),(40,62,134,78,126,70,118),(81,107,155,99,147,91,138),(82,108,156,100,148,92,139),(83,109,157,101,149,93,140),(84,110,158,102,150,94,141),(85,111,159,103,151,95,142),(86,112,160,104,152,96,143),(87,105,153,97,145,89,144),(88,106,154,98,146,90,137)], [(1,115),(2,120),(3,117),(4,114),(5,119),(6,116),(7,113),(8,118),(9,149),(10,146),(11,151),(12,148),(13,145),(14,150),(15,147),(16,152),(17,157),(18,154),(19,159),(20,156),(21,153),(22,158),(23,155),(24,160),(25,83),(26,88),(27,85),(28,82),(29,87),(30,84),(31,81),(32,86),(33,174),(34,171),(35,176),(36,173),(37,170),(38,175),(39,172),(40,169),(41,121),(42,126),(43,123),(44,128),(45,125),(46,122),(47,127),(48,124),(49,129),(50,134),(51,131),(52,136),(53,133),(54,130),(55,135),(56,132),(57,217),(58,222),(59,219),(60,224),(61,221),(62,218),(63,223),(64,220),(65,201),(66,206),(67,203),(68,208),(69,205),(70,202),(71,207),(72,204),(73,209),(74,214),(75,211),(76,216),(77,213),(78,210),(79,215),(80,212),(89,165),(90,162),(91,167),(92,164),(93,161),(94,166),(95,163),(96,168),(97,181),(98,178),(99,183),(100,180),(101,177),(102,182),(103,179),(104,184),(105,189),(106,186),(107,191),(108,188),(109,185),(110,190),(111,187),(112,192),(137,194),(138,199),(139,196),(140,193),(141,198),(142,195),(143,200),(144,197)], [(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,113),(8,114),(9,145),(10,146),(11,147),(12,148),(13,149),(14,150),(15,151),(16,152),(17,153),(18,154),(19,155),(20,156),(21,157),(22,158),(23,159),(24,160),(25,87),(26,88),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,174),(34,175),(35,176),(36,169),(37,170),(38,171),(39,172),(40,173),(41,121),(42,122),(43,123),(44,124),(45,125),(46,126),(47,127),(48,128),(49,129),(50,130),(51,131),(52,132),(53,133),(54,134),(55,135),(56,136),(57,217),(58,218),(59,219),(60,220),(61,221),(62,222),(63,223),(64,224),(65,201),(66,202),(67,203),(68,204),(69,205),(70,206),(71,207),(72,208),(73,209),(74,210),(75,211),(76,212),(77,213),(78,214),(79,215),(80,216),(89,161),(90,162),(91,163),(92,164),(93,165),(94,166),(95,167),(96,168),(97,177),(98,178),(99,179),(100,180),(101,181),(102,182),(103,183),(104,184),(105,185),(106,186),(107,187),(108,188),(109,189),(110,190),(111,191),(112,192),(137,194),(138,195),(139,196),(140,197),(141,198),(142,199),(143,200),(144,193)], [(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)], [(1,94,115,166),(2,165,116,93),(3,92,117,164),(4,163,118,91),(5,90,119,162),(6,161,120,89),(7,96,113,168),(8,167,114,95),(9,72,145,208),(10,207,146,71),(11,70,147,206),(12,205,148,69),(13,68,149,204),(14,203,150,67),(15,66,151,202),(16,201,152,65),(17,80,153,216),(18,215,154,79),(19,78,155,214),(20,213,156,77),(21,76,157,212),(22,211,158,75),(23,74,159,210),(24,209,160,73),(25,64,87,224),(26,223,88,63),(27,62,81,222),(28,221,82,61),(29,60,83,220),(30,219,84,59),(31,58,85,218),(32,217,86,57),(33,194,174,137),(34,144,175,193),(35,200,176,143),(36,142,169,199),(37,198,170,141),(38,140,171,197),(39,196,172,139),(40,138,173,195),(41,104,121,184),(42,183,122,103),(43,102,123,182),(44,181,124,101),(45,100,125,180),(46,179,126,99),(47,98,127,178),(48,177,128,97),(49,112,129,192),(50,191,130,111),(51,110,131,190),(52,189,132,109),(53,108,133,188),(54,187,134,107),(55,106,135,186),(56,185,136,105)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28BH | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C14 | C28 | D4 | D4 | C7×D4 | C7×D4 | C8⋊C22 | C8.C22 | C7×C8⋊C22 | C7×C8.C22 |
| kernel | C7×C23.36D4 | C7×D4⋊C4 | C7×Q8⋊C4 | C14×C4⋊C4 | C14×M4(2) | C14×C4○D4 | C7×C4○D4 | C23.36D4 | D4⋊C4 | Q8⋊C4 | C2×C4⋊C4 | C2×M4(2) | C2×C4○D4 | C4○D4 | C2×C28 | C22×C14 | C2×C4 | C23 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 6 | 12 | 12 | 6 | 6 | 6 | 48 | 3 | 1 | 18 | 6 | 1 | 1 | 6 | 6 |
Matrix representation of C7×C23.36D4 ►in GL6(𝔽113)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 14 | 0 | 1 | 0 |
| 0 | 0 | 14 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 69 | 63 | 0 | 0 | 0 | 0 |
| 50 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 49 | 0 | 110 | 10 |
| 0 | 0 | 92 | 0 | 50 | 60 |
| 0 | 0 | 84 | 55 | 21 | 43 |
| 0 | 0 | 79 | 5 | 21 | 43 |
| 63 | 69 | 0 | 0 | 0 | 0 |
| 44 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 3 | 0 | 0 |
| 0 | 0 | 108 | 63 | 0 | 0 |
| 0 | 0 | 21 | 92 | 58 | 108 |
| 0 | 0 | 0 | 92 | 108 | 55 |
G:=sub<GL(6,GF(113))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,14,14,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[69,50,0,0,0,0,63,44,0,0,0,0,0,0,49,92,84,79,0,0,0,0,55,5,0,0,110,50,21,21,0,0,10,60,43,43],[63,44,0,0,0,0,69,50,0,0,0,0,0,0,50,108,21,0,0,0,3,63,92,92,0,0,0,0,58,108,0,0,0,0,108,55] >;
C7×C23.36D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{36}D_4 % in TeX
G:=Group("C7xC2^3.36D4"); // GroupNames label
G:=SmallGroup(448,825);
// by ID
G=gap.SmallGroup(448,825);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,2403,9804,4911,172]);
// 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,e*b*e^-1=b*d=d*b,b*f=f*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