direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×Q8○D8, C28.87C24, C56.54C23, 2- (1+4)⋊3C14, C4○D8⋊6C14, C8○D4⋊5C14, (C7×D4).47D4, D8.4(C2×C14), D4.13(C7×D4), C4.47(D4×C14), SD16.(C2×C14), (C7×Q8).47D4, Q8.13(C7×D4), (C14×Q16)⋊26C2, (C2×Q16)⋊12C14, C28.408(C2×D4), C8.C22⋊5C14, Q16.4(C2×C14), C22.9(D4×C14), C4.10(C23×C14), C8.11(C22×C14), (C7×D4).40C23, (C7×D8).14C22, D4.7(C22×C14), (C7×Q8).41C23, Q8.7(C22×C14), (C2×C56).281C22, (C2×C28).689C23, C14.208(C22×D4), M4(2).6(C2×C14), (C7×2- (1+4))⋊8C2, (C7×Q16).16C22, (C7×SD16).3C22, (Q8×C14).189C22, (C7×M4(2)).31C22, C2.32(D4×C2×C14), (C7×C8○D4)⋊14C2, (C7×C4○D8)⋊13C2, (C2×C8).33(C2×C14), C4○D4.5(C2×C14), (C2×C14).186(C2×D4), (C7×C8.C22)⋊12C2, (C2×Q8).32(C2×C14), (C2×C4).50(C22×C14), (C7×C4○D4).35C22, SmallGroup(448,1361)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 346 in 248 conjugacy classes, 158 normal (18 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C7, C8, C8 [×3], C2×C4 [×3], C2×C4 [×12], D4 [×5], D4 [×6], Q8, Q8 [×6], Q8 [×6], C14, C14 [×5], C2×C8 [×3], M4(2) [×3], D8, SD16 [×6], Q16 [×9], C2×Q8 [×6], C2×Q8 [×2], C4○D4, C4○D4 [×6], C4○D4 [×6], C28, C28 [×3], C28 [×6], C2×C14 [×3], C2×C14 [×2], C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22 [×6], 2- (1+4) [×2], C56, C56 [×3], C2×C28 [×3], C2×C28 [×12], C7×D4 [×5], C7×D4 [×6], C7×Q8, C7×Q8 [×6], C7×Q8 [×6], Q8○D8, C2×C56 [×3], C7×M4(2) [×3], C7×D8, C7×SD16 [×6], C7×Q16 [×9], Q8×C14 [×6], Q8×C14 [×2], C7×C4○D4, C7×C4○D4 [×6], C7×C4○D4 [×6], C7×C8○D4, C14×Q16 [×3], C7×C4○D8 [×3], C7×C8.C22 [×6], C7×2- (1+4) [×2], C7×Q8○D8
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C24, C2×C14 [×35], C22×D4, C7×D4 [×4], C22×C14 [×15], Q8○D8, D4×C14 [×6], C23×C14, D4×C2×C14, C7×Q8○D8
Generators and relations
G = < a,b,c,d,e | a7=b4=e2=1, c2=d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d3 >
►On 224 points
Generators in S224
(1 127 46 68 201 119 38)(2 128 47 69 202 120 39)(3 121 48 70 203 113 40)(4 122 41 71 204 114 33)(5 123 42 72 205 115 34)(6 124 43 65 206 116 35)(7 125 44 66 207 117 36)(8 126 45 67 208 118 37)(9 89 164 140 97 180 148)(10 90 165 141 98 181 149)(11 91 166 142 99 182 150)(12 92 167 143 100 183 151)(13 93 168 144 101 184 152)(14 94 161 137 102 177 145)(15 95 162 138 103 178 146)(16 96 163 139 104 179 147)(17 196 82 25 105 188 156)(18 197 83 26 106 189 157)(19 198 84 27 107 190 158)(20 199 85 28 108 191 159)(21 200 86 29 109 192 160)(22 193 87 30 110 185 153)(23 194 88 31 111 186 154)(24 195 81 32 112 187 155)(49 79 212 174 64 220 130)(50 80 213 175 57 221 131)(51 73 214 176 58 222 132)(52 74 215 169 59 223 133)(53 75 216 170 60 224 134)(54 76 209 171 61 217 135)(55 77 210 172 62 218 136)(56 78 211 173 63 219 129)
(1 211 5 215)(2 212 6 216)(3 213 7 209)(4 214 8 210)(9 107 13 111)(10 108 14 112)(11 109 15 105)(12 110 16 106)(17 142 21 138)(18 143 22 139)(19 144 23 140)(20 137 24 141)(25 150 29 146)(26 151 30 147)(27 152 31 148)(28 145 32 149)(33 73 37 77)(34 74 38 78)(35 75 39 79)(36 76 40 80)(41 58 45 62)(42 59 46 63)(43 60 47 64)(44 61 48 57)(49 116 53 120)(50 117 54 113)(51 118 55 114)(52 119 56 115)(65 224 69 220)(66 217 70 221)(67 218 71 222)(68 219 72 223)(81 181 85 177)(82 182 86 178)(83 183 87 179)(84 184 88 180)(89 190 93 186)(90 191 94 187)(91 192 95 188)(92 185 96 189)(97 198 101 194)(98 199 102 195)(99 200 103 196)(100 193 104 197)(121 175 125 171)(122 176 126 172)(123 169 127 173)(124 170 128 174)(129 205 133 201)(130 206 134 202)(131 207 135 203)(132 208 136 204)(153 163 157 167)(154 164 158 168)(155 165 159 161)(156 166 160 162)
(1 209 5 213)(2 210 6 214)(3 211 7 215)(4 212 8 216)(9 105 13 109)(10 106 14 110)(11 107 15 111)(12 108 16 112)(17 144 21 140)(18 137 22 141)(19 138 23 142)(20 139 24 143)(25 152 29 148)(26 145 30 149)(27 146 31 150)(28 147 32 151)(33 79 37 75)(34 80 38 76)(35 73 39 77)(36 74 40 78)(41 64 45 60)(42 57 46 61)(43 58 47 62)(44 59 48 63)(49 118 53 114)(50 119 54 115)(51 120 55 116)(52 113 56 117)(65 222 69 218)(66 223 70 219)(67 224 71 220)(68 217 72 221)(81 183 85 179)(82 184 86 180)(83 177 87 181)(84 178 88 182)(89 188 93 192)(90 189 94 185)(91 190 95 186)(92 191 96 187)(97 196 101 200)(98 197 102 193)(99 198 103 194)(100 199 104 195)(121 173 125 169)(122 174 126 170)(123 175 127 171)(124 176 128 172)(129 207 133 203)(130 208 134 204)(131 201 135 205)(132 202 136 206)(153 165 157 161)(154 166 158 162)(155 167 159 163)(156 168 160 164)
(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 18)(2 17)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 132)(10 131)(11 130)(12 129)(13 136)(14 135)(15 134)(16 133)(25 69)(26 68)(27 67)(28 66)(29 65)(30 72)(31 71)(32 70)(33 154)(34 153)(35 160)(36 159)(37 158)(38 157)(39 156)(40 155)(41 88)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)(49 91)(50 90)(51 89)(52 96)(53 95)(54 94)(55 93)(56 92)(57 181)(58 180)(59 179)(60 178)(61 177)(62 184)(63 183)(64 182)(73 164)(74 163)(75 162)(76 161)(77 168)(78 167)(79 166)(80 165)(97 176)(98 175)(99 174)(100 173)(101 172)(102 171)(103 170)(104 169)(105 202)(106 201)(107 208)(108 207)(109 206)(110 205)(111 204)(112 203)(113 187)(114 186)(115 185)(116 192)(117 191)(118 190)(119 189)(120 188)(121 195)(122 194)(123 193)(124 200)(125 199)(126 198)(127 197)(128 196)(137 209)(138 216)(139 215)(140 214)(141 213)(142 212)(143 211)(144 210)(145 217)(146 224)(147 223)(148 222)(149 221)(150 220)(151 219)(152 218)
G:=sub<Sym(224)| (1,127,46,68,201,119,38)(2,128,47,69,202,120,39)(3,121,48,70,203,113,40)(4,122,41,71,204,114,33)(5,123,42,72,205,115,34)(6,124,43,65,206,116,35)(7,125,44,66,207,117,36)(8,126,45,67,208,118,37)(9,89,164,140,97,180,148)(10,90,165,141,98,181,149)(11,91,166,142,99,182,150)(12,92,167,143,100,183,151)(13,93,168,144,101,184,152)(14,94,161,137,102,177,145)(15,95,162,138,103,178,146)(16,96,163,139,104,179,147)(17,196,82,25,105,188,156)(18,197,83,26,106,189,157)(19,198,84,27,107,190,158)(20,199,85,28,108,191,159)(21,200,86,29,109,192,160)(22,193,87,30,110,185,153)(23,194,88,31,111,186,154)(24,195,81,32,112,187,155)(49,79,212,174,64,220,130)(50,80,213,175,57,221,131)(51,73,214,176,58,222,132)(52,74,215,169,59,223,133)(53,75,216,170,60,224,134)(54,76,209,171,61,217,135)(55,77,210,172,62,218,136)(56,78,211,173,63,219,129), (1,211,5,215)(2,212,6,216)(3,213,7,209)(4,214,8,210)(9,107,13,111)(10,108,14,112)(11,109,15,105)(12,110,16,106)(17,142,21,138)(18,143,22,139)(19,144,23,140)(20,137,24,141)(25,150,29,146)(26,151,30,147)(27,152,31,148)(28,145,32,149)(33,73,37,77)(34,74,38,78)(35,75,39,79)(36,76,40,80)(41,58,45,62)(42,59,46,63)(43,60,47,64)(44,61,48,57)(49,116,53,120)(50,117,54,113)(51,118,55,114)(52,119,56,115)(65,224,69,220)(66,217,70,221)(67,218,71,222)(68,219,72,223)(81,181,85,177)(82,182,86,178)(83,183,87,179)(84,184,88,180)(89,190,93,186)(90,191,94,187)(91,192,95,188)(92,185,96,189)(97,198,101,194)(98,199,102,195)(99,200,103,196)(100,193,104,197)(121,175,125,171)(122,176,126,172)(123,169,127,173)(124,170,128,174)(129,205,133,201)(130,206,134,202)(131,207,135,203)(132,208,136,204)(153,163,157,167)(154,164,158,168)(155,165,159,161)(156,166,160,162), (1,209,5,213)(2,210,6,214)(3,211,7,215)(4,212,8,216)(9,105,13,109)(10,106,14,110)(11,107,15,111)(12,108,16,112)(17,144,21,140)(18,137,22,141)(19,138,23,142)(20,139,24,143)(25,152,29,148)(26,145,30,149)(27,146,31,150)(28,147,32,151)(33,79,37,75)(34,80,38,76)(35,73,39,77)(36,74,40,78)(41,64,45,60)(42,57,46,61)(43,58,47,62)(44,59,48,63)(49,118,53,114)(50,119,54,115)(51,120,55,116)(52,113,56,117)(65,222,69,218)(66,223,70,219)(67,224,71,220)(68,217,72,221)(81,183,85,179)(82,184,86,180)(83,177,87,181)(84,178,88,182)(89,188,93,192)(90,189,94,185)(91,190,95,186)(92,191,96,187)(97,196,101,200)(98,197,102,193)(99,198,103,194)(100,199,104,195)(121,173,125,169)(122,174,126,170)(123,175,127,171)(124,176,128,172)(129,207,133,203)(130,208,134,204)(131,201,135,205)(132,202,136,206)(153,165,157,161)(154,166,158,162)(155,167,159,163)(156,168,160,164), (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), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,132)(10,131)(11,130)(12,129)(13,136)(14,135)(15,134)(16,133)(25,69)(26,68)(27,67)(28,66)(29,65)(30,72)(31,71)(32,70)(33,154)(34,153)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,91)(50,90)(51,89)(52,96)(53,95)(54,94)(55,93)(56,92)(57,181)(58,180)(59,179)(60,178)(61,177)(62,184)(63,183)(64,182)(73,164)(74,163)(75,162)(76,161)(77,168)(78,167)(79,166)(80,165)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,202)(106,201)(107,208)(108,207)(109,206)(110,205)(111,204)(112,203)(113,187)(114,186)(115,185)(116,192)(117,191)(118,190)(119,189)(120,188)(121,195)(122,194)(123,193)(124,200)(125,199)(126,198)(127,197)(128,196)(137,209)(138,216)(139,215)(140,214)(141,213)(142,212)(143,211)(144,210)(145,217)(146,224)(147,223)(148,222)(149,221)(150,220)(151,219)(152,218)>;
G:=Group( (1,127,46,68,201,119,38)(2,128,47,69,202,120,39)(3,121,48,70,203,113,40)(4,122,41,71,204,114,33)(5,123,42,72,205,115,34)(6,124,43,65,206,116,35)(7,125,44,66,207,117,36)(8,126,45,67,208,118,37)(9,89,164,140,97,180,148)(10,90,165,141,98,181,149)(11,91,166,142,99,182,150)(12,92,167,143,100,183,151)(13,93,168,144,101,184,152)(14,94,161,137,102,177,145)(15,95,162,138,103,178,146)(16,96,163,139,104,179,147)(17,196,82,25,105,188,156)(18,197,83,26,106,189,157)(19,198,84,27,107,190,158)(20,199,85,28,108,191,159)(21,200,86,29,109,192,160)(22,193,87,30,110,185,153)(23,194,88,31,111,186,154)(24,195,81,32,112,187,155)(49,79,212,174,64,220,130)(50,80,213,175,57,221,131)(51,73,214,176,58,222,132)(52,74,215,169,59,223,133)(53,75,216,170,60,224,134)(54,76,209,171,61,217,135)(55,77,210,172,62,218,136)(56,78,211,173,63,219,129), (1,211,5,215)(2,212,6,216)(3,213,7,209)(4,214,8,210)(9,107,13,111)(10,108,14,112)(11,109,15,105)(12,110,16,106)(17,142,21,138)(18,143,22,139)(19,144,23,140)(20,137,24,141)(25,150,29,146)(26,151,30,147)(27,152,31,148)(28,145,32,149)(33,73,37,77)(34,74,38,78)(35,75,39,79)(36,76,40,80)(41,58,45,62)(42,59,46,63)(43,60,47,64)(44,61,48,57)(49,116,53,120)(50,117,54,113)(51,118,55,114)(52,119,56,115)(65,224,69,220)(66,217,70,221)(67,218,71,222)(68,219,72,223)(81,181,85,177)(82,182,86,178)(83,183,87,179)(84,184,88,180)(89,190,93,186)(90,191,94,187)(91,192,95,188)(92,185,96,189)(97,198,101,194)(98,199,102,195)(99,200,103,196)(100,193,104,197)(121,175,125,171)(122,176,126,172)(123,169,127,173)(124,170,128,174)(129,205,133,201)(130,206,134,202)(131,207,135,203)(132,208,136,204)(153,163,157,167)(154,164,158,168)(155,165,159,161)(156,166,160,162), (1,209,5,213)(2,210,6,214)(3,211,7,215)(4,212,8,216)(9,105,13,109)(10,106,14,110)(11,107,15,111)(12,108,16,112)(17,144,21,140)(18,137,22,141)(19,138,23,142)(20,139,24,143)(25,152,29,148)(26,145,30,149)(27,146,31,150)(28,147,32,151)(33,79,37,75)(34,80,38,76)(35,73,39,77)(36,74,40,78)(41,64,45,60)(42,57,46,61)(43,58,47,62)(44,59,48,63)(49,118,53,114)(50,119,54,115)(51,120,55,116)(52,113,56,117)(65,222,69,218)(66,223,70,219)(67,224,71,220)(68,217,72,221)(81,183,85,179)(82,184,86,180)(83,177,87,181)(84,178,88,182)(89,188,93,192)(90,189,94,185)(91,190,95,186)(92,191,96,187)(97,196,101,200)(98,197,102,193)(99,198,103,194)(100,199,104,195)(121,173,125,169)(122,174,126,170)(123,175,127,171)(124,176,128,172)(129,207,133,203)(130,208,134,204)(131,201,135,205)(132,202,136,206)(153,165,157,161)(154,166,158,162)(155,167,159,163)(156,168,160,164), (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), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,132)(10,131)(11,130)(12,129)(13,136)(14,135)(15,134)(16,133)(25,69)(26,68)(27,67)(28,66)(29,65)(30,72)(31,71)(32,70)(33,154)(34,153)(35,160)(36,159)(37,158)(38,157)(39,156)(40,155)(41,88)(42,87)(43,86)(44,85)(45,84)(46,83)(47,82)(48,81)(49,91)(50,90)(51,89)(52,96)(53,95)(54,94)(55,93)(56,92)(57,181)(58,180)(59,179)(60,178)(61,177)(62,184)(63,183)(64,182)(73,164)(74,163)(75,162)(76,161)(77,168)(78,167)(79,166)(80,165)(97,176)(98,175)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,202)(106,201)(107,208)(108,207)(109,206)(110,205)(111,204)(112,203)(113,187)(114,186)(115,185)(116,192)(117,191)(118,190)(119,189)(120,188)(121,195)(122,194)(123,193)(124,200)(125,199)(126,198)(127,197)(128,196)(137,209)(138,216)(139,215)(140,214)(141,213)(142,212)(143,211)(144,210)(145,217)(146,224)(147,223)(148,222)(149,221)(150,220)(151,219)(152,218) );
G=PermutationGroup([(1,127,46,68,201,119,38),(2,128,47,69,202,120,39),(3,121,48,70,203,113,40),(4,122,41,71,204,114,33),(5,123,42,72,205,115,34),(6,124,43,65,206,116,35),(7,125,44,66,207,117,36),(8,126,45,67,208,118,37),(9,89,164,140,97,180,148),(10,90,165,141,98,181,149),(11,91,166,142,99,182,150),(12,92,167,143,100,183,151),(13,93,168,144,101,184,152),(14,94,161,137,102,177,145),(15,95,162,138,103,178,146),(16,96,163,139,104,179,147),(17,196,82,25,105,188,156),(18,197,83,26,106,189,157),(19,198,84,27,107,190,158),(20,199,85,28,108,191,159),(21,200,86,29,109,192,160),(22,193,87,30,110,185,153),(23,194,88,31,111,186,154),(24,195,81,32,112,187,155),(49,79,212,174,64,220,130),(50,80,213,175,57,221,131),(51,73,214,176,58,222,132),(52,74,215,169,59,223,133),(53,75,216,170,60,224,134),(54,76,209,171,61,217,135),(55,77,210,172,62,218,136),(56,78,211,173,63,219,129)], [(1,211,5,215),(2,212,6,216),(3,213,7,209),(4,214,8,210),(9,107,13,111),(10,108,14,112),(11,109,15,105),(12,110,16,106),(17,142,21,138),(18,143,22,139),(19,144,23,140),(20,137,24,141),(25,150,29,146),(26,151,30,147),(27,152,31,148),(28,145,32,149),(33,73,37,77),(34,74,38,78),(35,75,39,79),(36,76,40,80),(41,58,45,62),(42,59,46,63),(43,60,47,64),(44,61,48,57),(49,116,53,120),(50,117,54,113),(51,118,55,114),(52,119,56,115),(65,224,69,220),(66,217,70,221),(67,218,71,222),(68,219,72,223),(81,181,85,177),(82,182,86,178),(83,183,87,179),(84,184,88,180),(89,190,93,186),(90,191,94,187),(91,192,95,188),(92,185,96,189),(97,198,101,194),(98,199,102,195),(99,200,103,196),(100,193,104,197),(121,175,125,171),(122,176,126,172),(123,169,127,173),(124,170,128,174),(129,205,133,201),(130,206,134,202),(131,207,135,203),(132,208,136,204),(153,163,157,167),(154,164,158,168),(155,165,159,161),(156,166,160,162)], [(1,209,5,213),(2,210,6,214),(3,211,7,215),(4,212,8,216),(9,105,13,109),(10,106,14,110),(11,107,15,111),(12,108,16,112),(17,144,21,140),(18,137,22,141),(19,138,23,142),(20,139,24,143),(25,152,29,148),(26,145,30,149),(27,146,31,150),(28,147,32,151),(33,79,37,75),(34,80,38,76),(35,73,39,77),(36,74,40,78),(41,64,45,60),(42,57,46,61),(43,58,47,62),(44,59,48,63),(49,118,53,114),(50,119,54,115),(51,120,55,116),(52,113,56,117),(65,222,69,218),(66,223,70,219),(67,224,71,220),(68,217,72,221),(81,183,85,179),(82,184,86,180),(83,177,87,181),(84,178,88,182),(89,188,93,192),(90,189,94,185),(91,190,95,186),(92,191,96,187),(97,196,101,200),(98,197,102,193),(99,198,103,194),(100,199,104,195),(121,173,125,169),(122,174,126,170),(123,175,127,171),(124,176,128,172),(129,207,133,203),(130,208,134,204),(131,201,135,205),(132,202,136,206),(153,165,157,161),(154,166,158,162),(155,167,159,163),(156,168,160,164)], [(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)], [(1,18),(2,17),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,132),(10,131),(11,130),(12,129),(13,136),(14,135),(15,134),(16,133),(25,69),(26,68),(27,67),(28,66),(29,65),(30,72),(31,71),(32,70),(33,154),(34,153),(35,160),(36,159),(37,158),(38,157),(39,156),(40,155),(41,88),(42,87),(43,86),(44,85),(45,84),(46,83),(47,82),(48,81),(49,91),(50,90),(51,89),(52,96),(53,95),(54,94),(55,93),(56,92),(57,181),(58,180),(59,179),(60,178),(61,177),(62,184),(63,183),(64,182),(73,164),(74,163),(75,162),(76,161),(77,168),(78,167),(79,166),(80,165),(97,176),(98,175),(99,174),(100,173),(101,172),(102,171),(103,170),(104,169),(105,202),(106,201),(107,208),(108,207),(109,206),(110,205),(111,204),(112,203),(113,187),(114,186),(115,185),(116,192),(117,191),(118,190),(119,189),(120,188),(121,195),(122,194),(123,193),(124,200),(125,199),(126,198),(127,197),(128,196),(137,209),(138,216),(139,215),(140,214),(141,213),(142,212),(143,211),(144,210),(145,217),(146,224),(147,223),(148,222),(149,221),(150,220),(151,219),(152,218)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 109 | 0 | 0 | 0 |
| 0 | 109 | 0 | 0 |
| 0 | 0 | 109 | 0 |
| 0 | 0 | 0 | 109 |
| 21 | 21 | 6 | 5 |
| 0 | 0 | 112 | 0 |
| 0 | 1 | 0 | 0 |
| 2 | 1 | 92 | 92 |
| 92 | 0 | 108 | 108 |
| 0 | 0 | 0 | 112 |
| 111 | 112 | 21 | 21 |
| 0 | 1 | 0 | 0 |
| 0 | 31 | 0 | 27 |
| 51 | 51 | 86 | 86 |
| 0 | 0 | 82 | 82 |
| 0 | 0 | 31 | 82 |
| 99 | 30 | 72 | 25 |
| 0 | 0 | 84 | 47 |
| 19 | 95 | 83 | 83 |
| 55 | 37 | 44 | 44 |
G:=sub<GL(4,GF(113))| [109,0,0,0,0,109,0,0,0,0,109,0,0,0,0,109],[21,0,0,2,21,0,1,1,6,112,0,92,5,0,0,92],[92,0,111,0,0,0,112,1,108,0,21,0,108,112,21,0],[0,51,0,0,31,51,0,0,0,86,82,31,27,86,82,82],[99,0,19,55,30,0,95,37,72,84,83,44,25,47,83,44] >;
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 8E | 14A | ··· | 14F | 14G | ··· | 14X | 14Y | ··· | 14AJ | 28A | ··· | 28X | 28Y | ··· | 28BH | 56A | ··· | 56L | 56M | ··· | 56AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 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 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | C7×D4 | C7×D4 | Q8○D8 | C7×Q8○D8 |
| kernel | C7×Q8○D8 | C7×C8○D4 | C14×Q16 | C7×C4○D8 | C7×C8.C22 | C7×2- (1+4) | Q8○D8 | C8○D4 | C2×Q16 | C4○D8 | C8.C22 | 2- (1+4) | C7×D4 | C7×Q8 | D4 | Q8 | C7 | C1 |
| # reps | 1 | 1 | 3 | 3 | 6 | 2 | 6 | 6 | 18 | 18 | 36 | 12 | 3 | 1 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_7\times Q_8\circ D_8
% in TeX
G:=Group("C7xQ8oD8"); // GroupNames label
G:=SmallGroup(448,1361);
// by ID
G=gap.SmallGroup(448,1361);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,1576,1641,14117,7068,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=e^2=1,c^2=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀