direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.38D4, (C2×Q8)⋊6C28, (Q8×C14)⋊16C4, C4.56(D4×C14), Q8.6(C2×C28), C28.463(C2×D4), (C2×C28).316D4, C4.6(C22×C28), C23.38(C7×D4), Q8⋊C4⋊15C14, C22.46(D4×C14), (C22×Q8).5C14, C28.83(C22⋊C4), (C2×C56).323C22, C42⋊C2.4C14, (C2×C28).895C23, C28.151(C22×C4), (C22×C14).160D4, (C14×M4(2)).30C2, (C2×M4(2)).12C14, (Q8×C14).253C22, C14.127(C8.C22), (C22×C28).412C22, (C2×C4).24(C7×D4), (Q8×C2×C14).15C2, C4⋊C4.40(C2×C14), (C2×C8).48(C2×C14), (C2×C4).22(C2×C28), C4.15(C7×C22⋊C4), (C7×Q8).30(C2×C4), C2.2(C7×C8.C22), (C2×C28).195(C2×C4), (C7×Q8⋊C4)⋊38C2, (C2×C14).622(C2×D4), C2.22(C14×C22⋊C4), (C2×Q8).38(C2×C14), (C7×C4⋊C4).361C22, C14.110(C2×C22⋊C4), (C2×C4).70(C22×C14), (C22×C4).31(C2×C14), C22.21(C7×C22⋊C4), (C2×C14).82(C22⋊C4), (C7×C42⋊C2).18C2, SmallGroup(448,827)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.38D4
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=fbf-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce3 >
Subgroups: 226 in 150 conjugacy classes, 82 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C28, C28, C28, C2×C14, C2×C14, C2×C14, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C56, C2×C28, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×C14, C23.38D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×M4(2), C22×C28, C22×C28, Q8×C14, Q8×C14, C7×Q8⋊C4, C7×C42⋊C2, C14×M4(2), Q8×C2×C14, C7×C23.38D4
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, C2×C28, C7×D4, C22×C14, C23.38D4, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4, C7×C8.C22, C7×C23.38D4
►On 224 points
Generators in S224
(1 175 223 55 215 47 207)(2 176 224 56 216 48 208)(3 169 217 49 209 41 201)(4 170 218 50 210 42 202)(5 171 219 51 211 43 203)(6 172 220 52 212 44 204)(7 173 221 53 213 45 205)(8 174 222 54 214 46 206)(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)(33 129 80 121 72 113 57)(34 130 73 122 65 114 58)(35 131 74 123 66 115 59)(36 132 75 124 67 116 60)(37 133 76 125 68 117 61)(38 134 77 126 69 118 62)(39 135 78 127 70 119 63)(40 136 79 128 71 120 64)(81 107 156 99 148 91 143)(82 108 157 100 149 92 144)(83 109 158 101 150 93 137)(84 110 159 102 151 94 138)(85 111 160 103 152 95 139)(86 112 153 104 145 96 140)(87 105 154 97 146 89 141)(88 106 155 98 147 90 142)
(1 113)(2 118)(3 115)(4 120)(5 117)(6 114)(7 119)(8 116)(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 82)(26 87)(27 84)(28 81)(29 86)(30 83)(31 88)(32 85)(33 223)(34 220)(35 217)(36 222)(37 219)(38 224)(39 221)(40 218)(41 123)(42 128)(43 125)(44 122)(45 127)(46 124)(47 121)(48 126)(49 131)(50 136)(51 133)(52 130)(53 135)(54 132)(55 129)(56 134)(57 175)(58 172)(59 169)(60 174)(61 171)(62 176)(63 173)(64 170)(65 204)(66 201)(67 206)(68 203)(69 208)(70 205)(71 202)(72 207)(73 212)(74 209)(75 214)(76 211)(77 216)(78 213)(79 210)(80 215)(89 162)(90 167)(91 164)(92 161)(93 166)(94 163)(95 168)(96 165)(97 178)(98 183)(99 180)(100 177)(101 182)(102 179)(103 184)(104 181)(105 186)(106 191)(107 188)(108 185)(109 190)(110 187)(111 192)(112 189)(137 198)(138 195)(139 200)(140 197)(141 194)(142 199)(143 196)(144 193)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 149)(10 150)(11 151)(12 152)(13 145)(14 146)(15 147)(16 148)(17 157)(18 158)(19 159)(20 160)(21 153)(22 154)(23 155)(24 156)(25 82)(26 83)(27 84)(28 85)(29 86)(30 87)(31 88)(32 81)(33 223)(34 224)(35 217)(36 218)(37 219)(38 220)(39 221)(40 222)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 121)(48 122)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 129)(56 130)(57 175)(58 176)(59 169)(60 170)(61 171)(62 172)(63 173)(64 174)(65 208)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 216)(74 209)(75 210)(76 211)(77 212)(78 213)(79 214)(80 215)(89 166)(90 167)(91 168)(92 161)(93 162)(94 163)(95 164)(96 165)(97 182)(98 183)(99 184)(100 177)(101 178)(102 179)(103 180)(104 181)(105 190)(106 191)(107 192)(108 185)(109 186)(110 187)(111 188)(112 189)(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 93 113 162)(2 165 114 96)(3 91 115 168)(4 163 116 94)(5 89 117 166)(6 161 118 92)(7 95 119 164)(8 167 120 90)(9 69 149 204)(10 207 150 72)(11 67 151 202)(12 205 152 70)(13 65 145 208)(14 203 146 68)(15 71 147 206)(16 201 148 66)(17 77 157 212)(18 215 158 80)(19 75 159 210)(20 213 160 78)(21 73 153 216)(22 211 154 76)(23 79 155 214)(24 209 156 74)(25 38 82 220)(26 223 83 33)(27 36 84 218)(28 221 85 39)(29 34 86 224)(30 219 87 37)(31 40 88 222)(32 217 81 35)(41 99 123 184)(42 179 124 102)(43 97 125 182)(44 177 126 100)(45 103 127 180)(46 183 128 98)(47 101 121 178)(48 181 122 104)(49 107 131 192)(50 187 132 110)(51 105 133 190)(52 185 134 108)(53 111 135 188)(54 191 136 106)(55 109 129 186)(56 189 130 112)(57 194 175 137)(58 140 176 197)(59 200 169 143)(60 138 170 195)(61 198 171 141)(62 144 172 193)(63 196 173 139)(64 142 174 199)
G:=sub<Sym(224)| (1,175,223,55,215,47,207)(2,176,224,56,216,48,208)(3,169,217,49,209,41,201)(4,170,218,50,210,42,202)(5,171,219,51,211,43,203)(6,172,220,52,212,44,204)(7,173,221,53,213,45,205)(8,174,222,54,214,46,206)(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)(33,129,80,121,72,113,57)(34,130,73,122,65,114,58)(35,131,74,123,66,115,59)(36,132,75,124,67,116,60)(37,133,76,125,68,117,61)(38,134,77,126,69,118,62)(39,135,78,127,70,119,63)(40,136,79,128,71,120,64)(81,107,156,99,148,91,143)(82,108,157,100,149,92,144)(83,109,158,101,150,93,137)(84,110,159,102,151,94,138)(85,111,160,103,152,95,139)(86,112,153,104,145,96,140)(87,105,154,97,146,89,141)(88,106,155,98,147,90,142), (1,113)(2,118)(3,115)(4,120)(5,117)(6,114)(7,119)(8,116)(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)(25,82)(26,87)(27,84)(28,81)(29,86)(30,83)(31,88)(32,85)(33,223)(34,220)(35,217)(36,222)(37,219)(38,224)(39,221)(40,218)(41,123)(42,128)(43,125)(44,122)(45,127)(46,124)(47,121)(48,126)(49,131)(50,136)(51,133)(52,130)(53,135)(54,132)(55,129)(56,134)(57,175)(58,172)(59,169)(60,174)(61,171)(62,176)(63,173)(64,170)(65,204)(66,201)(67,206)(68,203)(69,208)(70,205)(71,202)(72,207)(73,212)(74,209)(75,214)(76,211)(77,216)(78,213)(79,210)(80,215)(89,162)(90,167)(91,164)(92,161)(93,166)(94,163)(95,168)(96,165)(97,178)(98,183)(99,180)(100,177)(101,182)(102,179)(103,184)(104,181)(105,186)(106,191)(107,188)(108,185)(109,190)(110,187)(111,192)(112,189)(137,198)(138,195)(139,200)(140,197)(141,194)(142,199)(143,196)(144,193), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,149)(10,150)(11,151)(12,152)(13,145)(14,146)(15,147)(16,148)(17,157)(18,158)(19,159)(20,160)(21,153)(22,154)(23,155)(24,156)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,81)(33,223)(34,224)(35,217)(36,218)(37,219)(38,220)(39,221)(40,222)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,121)(48,122)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,129)(56,130)(57,175)(58,176)(59,169)(60,170)(61,171)(62,172)(63,173)(64,174)(65,208)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,216)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(89,166)(90,167)(91,168)(92,161)(93,162)(94,163)(95,164)(96,165)(97,182)(98,183)(99,184)(100,177)(101,178)(102,179)(103,180)(104,181)(105,190)(106,191)(107,192)(108,185)(109,186)(110,187)(111,188)(112,189)(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), (1,93,113,162)(2,165,114,96)(3,91,115,168)(4,163,116,94)(5,89,117,166)(6,161,118,92)(7,95,119,164)(8,167,120,90)(9,69,149,204)(10,207,150,72)(11,67,151,202)(12,205,152,70)(13,65,145,208)(14,203,146,68)(15,71,147,206)(16,201,148,66)(17,77,157,212)(18,215,158,80)(19,75,159,210)(20,213,160,78)(21,73,153,216)(22,211,154,76)(23,79,155,214)(24,209,156,74)(25,38,82,220)(26,223,83,33)(27,36,84,218)(28,221,85,39)(29,34,86,224)(30,219,87,37)(31,40,88,222)(32,217,81,35)(41,99,123,184)(42,179,124,102)(43,97,125,182)(44,177,126,100)(45,103,127,180)(46,183,128,98)(47,101,121,178)(48,181,122,104)(49,107,131,192)(50,187,132,110)(51,105,133,190)(52,185,134,108)(53,111,135,188)(54,191,136,106)(55,109,129,186)(56,189,130,112)(57,194,175,137)(58,140,176,197)(59,200,169,143)(60,138,170,195)(61,198,171,141)(62,144,172,193)(63,196,173,139)(64,142,174,199)>;
G:=Group( (1,175,223,55,215,47,207)(2,176,224,56,216,48,208)(3,169,217,49,209,41,201)(4,170,218,50,210,42,202)(5,171,219,51,211,43,203)(6,172,220,52,212,44,204)(7,173,221,53,213,45,205)(8,174,222,54,214,46,206)(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)(33,129,80,121,72,113,57)(34,130,73,122,65,114,58)(35,131,74,123,66,115,59)(36,132,75,124,67,116,60)(37,133,76,125,68,117,61)(38,134,77,126,69,118,62)(39,135,78,127,70,119,63)(40,136,79,128,71,120,64)(81,107,156,99,148,91,143)(82,108,157,100,149,92,144)(83,109,158,101,150,93,137)(84,110,159,102,151,94,138)(85,111,160,103,152,95,139)(86,112,153,104,145,96,140)(87,105,154,97,146,89,141)(88,106,155,98,147,90,142), (1,113)(2,118)(3,115)(4,120)(5,117)(6,114)(7,119)(8,116)(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)(25,82)(26,87)(27,84)(28,81)(29,86)(30,83)(31,88)(32,85)(33,223)(34,220)(35,217)(36,222)(37,219)(38,224)(39,221)(40,218)(41,123)(42,128)(43,125)(44,122)(45,127)(46,124)(47,121)(48,126)(49,131)(50,136)(51,133)(52,130)(53,135)(54,132)(55,129)(56,134)(57,175)(58,172)(59,169)(60,174)(61,171)(62,176)(63,173)(64,170)(65,204)(66,201)(67,206)(68,203)(69,208)(70,205)(71,202)(72,207)(73,212)(74,209)(75,214)(76,211)(77,216)(78,213)(79,210)(80,215)(89,162)(90,167)(91,164)(92,161)(93,166)(94,163)(95,168)(96,165)(97,178)(98,183)(99,180)(100,177)(101,182)(102,179)(103,184)(104,181)(105,186)(106,191)(107,188)(108,185)(109,190)(110,187)(111,192)(112,189)(137,198)(138,195)(139,200)(140,197)(141,194)(142,199)(143,196)(144,193), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,149)(10,150)(11,151)(12,152)(13,145)(14,146)(15,147)(16,148)(17,157)(18,158)(19,159)(20,160)(21,153)(22,154)(23,155)(24,156)(25,82)(26,83)(27,84)(28,85)(29,86)(30,87)(31,88)(32,81)(33,223)(34,224)(35,217)(36,218)(37,219)(38,220)(39,221)(40,222)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,121)(48,122)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,129)(56,130)(57,175)(58,176)(59,169)(60,170)(61,171)(62,172)(63,173)(64,174)(65,208)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,216)(74,209)(75,210)(76,211)(77,212)(78,213)(79,214)(80,215)(89,166)(90,167)(91,168)(92,161)(93,162)(94,163)(95,164)(96,165)(97,182)(98,183)(99,184)(100,177)(101,178)(102,179)(103,180)(104,181)(105,190)(106,191)(107,192)(108,185)(109,186)(110,187)(111,188)(112,189)(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), (1,93,113,162)(2,165,114,96)(3,91,115,168)(4,163,116,94)(5,89,117,166)(6,161,118,92)(7,95,119,164)(8,167,120,90)(9,69,149,204)(10,207,150,72)(11,67,151,202)(12,205,152,70)(13,65,145,208)(14,203,146,68)(15,71,147,206)(16,201,148,66)(17,77,157,212)(18,215,158,80)(19,75,159,210)(20,213,160,78)(21,73,153,216)(22,211,154,76)(23,79,155,214)(24,209,156,74)(25,38,82,220)(26,223,83,33)(27,36,84,218)(28,221,85,39)(29,34,86,224)(30,219,87,37)(31,40,88,222)(32,217,81,35)(41,99,123,184)(42,179,124,102)(43,97,125,182)(44,177,126,100)(45,103,127,180)(46,183,128,98)(47,101,121,178)(48,181,122,104)(49,107,131,192)(50,187,132,110)(51,105,133,190)(52,185,134,108)(53,111,135,188)(54,191,136,106)(55,109,129,186)(56,189,130,112)(57,194,175,137)(58,140,176,197)(59,200,169,143)(60,138,170,195)(61,198,171,141)(62,144,172,193)(63,196,173,139)(64,142,174,199) );
G=PermutationGroup([[(1,175,223,55,215,47,207),(2,176,224,56,216,48,208),(3,169,217,49,209,41,201),(4,170,218,50,210,42,202),(5,171,219,51,211,43,203),(6,172,220,52,212,44,204),(7,173,221,53,213,45,205),(8,174,222,54,214,46,206),(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),(33,129,80,121,72,113,57),(34,130,73,122,65,114,58),(35,131,74,123,66,115,59),(36,132,75,124,67,116,60),(37,133,76,125,68,117,61),(38,134,77,126,69,118,62),(39,135,78,127,70,119,63),(40,136,79,128,71,120,64),(81,107,156,99,148,91,143),(82,108,157,100,149,92,144),(83,109,158,101,150,93,137),(84,110,159,102,151,94,138),(85,111,160,103,152,95,139),(86,112,153,104,145,96,140),(87,105,154,97,146,89,141),(88,106,155,98,147,90,142)], [(1,113),(2,118),(3,115),(4,120),(5,117),(6,114),(7,119),(8,116),(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),(25,82),(26,87),(27,84),(28,81),(29,86),(30,83),(31,88),(32,85),(33,223),(34,220),(35,217),(36,222),(37,219),(38,224),(39,221),(40,218),(41,123),(42,128),(43,125),(44,122),(45,127),(46,124),(47,121),(48,126),(49,131),(50,136),(51,133),(52,130),(53,135),(54,132),(55,129),(56,134),(57,175),(58,172),(59,169),(60,174),(61,171),(62,176),(63,173),(64,170),(65,204),(66,201),(67,206),(68,203),(69,208),(70,205),(71,202),(72,207),(73,212),(74,209),(75,214),(76,211),(77,216),(78,213),(79,210),(80,215),(89,162),(90,167),(91,164),(92,161),(93,166),(94,163),(95,168),(96,165),(97,178),(98,183),(99,180),(100,177),(101,182),(102,179),(103,184),(104,181),(105,186),(106,191),(107,188),(108,185),(109,190),(110,187),(111,192),(112,189),(137,198),(138,195),(139,200),(140,197),(141,194),(142,199),(143,196),(144,193)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,149),(10,150),(11,151),(12,152),(13,145),(14,146),(15,147),(16,148),(17,157),(18,158),(19,159),(20,160),(21,153),(22,154),(23,155),(24,156),(25,82),(26,83),(27,84),(28,85),(29,86),(30,87),(31,88),(32,81),(33,223),(34,224),(35,217),(36,218),(37,219),(38,220),(39,221),(40,222),(41,123),(42,124),(43,125),(44,126),(45,127),(46,128),(47,121),(48,122),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,129),(56,130),(57,175),(58,176),(59,169),(60,170),(61,171),(62,172),(63,173),(64,174),(65,208),(66,201),(67,202),(68,203),(69,204),(70,205),(71,206),(72,207),(73,216),(74,209),(75,210),(76,211),(77,212),(78,213),(79,214),(80,215),(89,166),(90,167),(91,168),(92,161),(93,162),(94,163),(95,164),(96,165),(97,182),(98,183),(99,184),(100,177),(101,178),(102,179),(103,180),(104,181),(105,190),(106,191),(107,192),(108,185),(109,186),(110,187),(111,188),(112,189),(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)], [(1,93,113,162),(2,165,114,96),(3,91,115,168),(4,163,116,94),(5,89,117,166),(6,161,118,92),(7,95,119,164),(8,167,120,90),(9,69,149,204),(10,207,150,72),(11,67,151,202),(12,205,152,70),(13,65,145,208),(14,203,146,68),(15,71,147,206),(16,201,148,66),(17,77,157,212),(18,215,158,80),(19,75,159,210),(20,213,160,78),(21,73,153,216),(22,211,154,76),(23,79,155,214),(24,209,156,74),(25,38,82,220),(26,223,83,33),(27,36,84,218),(28,221,85,39),(29,34,86,224),(30,219,87,37),(31,40,88,222),(32,217,81,35),(41,99,123,184),(42,179,124,102),(43,97,125,182),(44,177,126,100),(45,103,127,180),(46,183,128,98),(47,101,121,178),(48,181,122,104),(49,107,131,192),(50,187,132,110),(51,105,133,190),(52,185,134,108),(53,111,135,188),(54,191,136,106),(55,109,129,186),(56,189,130,112),(57,194,175,137),(58,140,176,197),(59,200,169,143),(60,138,170,195),(61,198,171,141),(62,144,172,193),(63,196,173,139),(64,142,174,199)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28BT | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C14 | C28 | D4 | D4 | C7×D4 | C7×D4 | C8.C22 | C7×C8.C22 |
| kernel | C7×C23.38D4 | C7×Q8⋊C4 | C7×C42⋊C2 | C14×M4(2) | Q8×C2×C14 | Q8×C14 | C23.38D4 | Q8⋊C4 | C42⋊C2 | C2×M4(2) | C22×Q8 | C2×Q8 | C2×C28 | C22×C14 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 6 | 24 | 6 | 6 | 6 | 48 | 3 | 1 | 18 | 6 | 2 | 12 |
Matrix representation of C7×C23.38D4 ►in GL6(𝔽113)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 101 | 112 | 0 |
| 0 | 0 | 85 | 12 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 18 | 54 | 0 | 0 | 0 | 0 |
| 84 | 95 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 44 | 23 | 91 |
| 0 | 0 | 96 | 88 | 56 | 79 |
| 0 | 0 | 68 | 43 | 99 | 9 |
| 0 | 0 | 34 | 14 | 14 | 104 |
| 72 | 78 | 0 | 0 | 0 | 0 |
| 19 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 101 | 111 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 109 | 58 | 79 | 107 |
| 0 | 0 | 4 | 56 | 34 | 6 |
G:=sub<GL(6,GF(113))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,28,85,0,0,0,1,101,12,0,0,0,0,112,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[18,84,0,0,0,0,54,95,0,0,0,0,0,0,48,96,68,34,0,0,44,88,43,14,0,0,23,56,99,14,0,0,91,79,9,104],[72,19,0,0,0,0,78,41,0,0,0,0,0,0,28,0,109,4,0,0,101,0,58,56,0,0,111,1,79,34,0,0,0,1,107,6] >;
C7×C23.38D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._{38}D_4 % in TeX
G:=Group("C7xC2^3.38D4"); // GroupNames label
G:=SmallGroup(448,827);
// by ID
G=gap.SmallGroup(448,827);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,1576,2403,1192,9804,4911,172]);
// 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,e*b*e^-1=f*b*f^-1=b*d=d*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*e^3>;
// generators/relations