direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.9D14, C28.36C24, Dic14.31C23, C7⋊C8.15C23, C4○D4.41D14, C28.428(C2×D4), (C2×C28).219D4, C4.36(C23×D7), (C2×D4).233D14, C14⋊5(C8.C22), D4.D7⋊18C22, (C2×Q8).191D14, (C7×D4).24C23, D4.24(C22×D7), C7⋊Q16⋊17C22, (C7×Q8).24C23, Q8.24(C22×D7), (C2×C28).558C23, (C22×C4).283D14, C14.161(C22×D4), (C22×C14).125D4, C23.69(C7⋊D4), C4.Dic7⋊38C22, (C22×Dic14)⋊21C2, (C2×Dic14)⋊70C22, (D4×C14).273C22, (Q8×C14).238C22, (C22×C28).293C22, C7⋊6(C2×C8.C22), C4.31(C2×C7⋊D4), (C2×D4.D7)⋊31C2, (C2×C4○D4).10D7, (C2×C7⋊Q16)⋊31C2, (C2×C14).77(C2×D4), (C14×C4○D4).11C2, (C2×C4).96(C7⋊D4), (C2×C7⋊C8).184C22, (C2×C4.Dic7)⋊32C2, C2.34(C22×C7⋊D4), (C7×C4○D4).50C22, (C2×C4).247(C22×D7), C22.120(C2×C7⋊D4), SmallGroup(448,1276)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4.9D14
G = < a,b,c,d,e | a2=b4=c2=d14=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >
Subgroups: 916 in 258 conjugacy classes, 111 normal (27 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, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic7, C28, C28, C28, C2×C14, C2×C14, C2×C14, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C7⋊C8, Dic14, Dic14, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, C2×C8.C22, C2×C7⋊C8, C4.Dic7, D4.D7, C7⋊Q16, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C7×C4○D4, C2×C4.Dic7, C2×D4.D7, C2×C7⋊Q16, D4.9D14, C22×Dic14, C14×C4○D4, C2×D4.9D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8.C22, C22×D4, C7⋊D4, C22×D7, C2×C8.C22, C2×C7⋊D4, C23×D7, D4.9D14, C22×C7⋊D4, C2×D4.9D14
►On 224 points
Generators in S224
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 200)(30 201)(31 202)(32 203)(33 204)(34 205)(35 206)(36 207)(37 208)(38 209)(39 210)(40 197)(41 198)(42 199)(43 190)(44 191)(45 192)(46 193)(47 194)(48 195)(49 196)(50 183)(51 184)(52 185)(53 186)(54 187)(55 188)(56 189)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 128)(72 129)(73 130)(74 131)(75 132)(76 133)(77 134)(78 135)(79 136)(80 137)(81 138)(82 139)(83 140)(84 127)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 155)(122 156)(123 157)(124 158)(125 159)(126 160)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(1 175 144 90)(2 176 145 91)(3 177 146 92)(4 178 147 93)(5 179 148 94)(6 180 149 95)(7 181 150 96)(8 182 151 97)(9 169 152 98)(10 170 153 85)(11 171 154 86)(12 172 141 87)(13 173 142 88)(14 174 143 89)(15 222 110 61)(16 223 111 62)(17 224 112 63)(18 211 99 64)(19 212 100 65)(20 213 101 66)(21 214 102 67)(22 215 103 68)(23 216 104 69)(24 217 105 70)(25 218 106 57)(26 219 107 58)(27 220 108 59)(28 221 109 60)(29 121 207 162)(30 122 208 163)(31 123 209 164)(32 124 210 165)(33 125 197 166)(34 126 198 167)(35 113 199 168)(36 114 200 155)(37 115 201 156)(38 116 202 157)(39 117 203 158)(40 118 204 159)(41 119 205 160)(42 120 206 161)(43 84 183 134)(44 71 184 135)(45 72 185 136)(46 73 186 137)(47 74 187 138)(48 75 188 139)(49 76 189 140)(50 77 190 127)(51 78 191 128)(52 79 192 129)(53 80 193 130)(54 81 194 131)(55 82 195 132)(56 83 196 133)
(1 198)(2 35)(3 200)(4 37)(5 202)(6 39)(7 204)(8 41)(9 206)(10 29)(11 208)(12 31)(13 210)(14 33)(15 193)(16 54)(17 195)(18 56)(19 183)(20 44)(21 185)(22 46)(23 187)(24 48)(25 189)(26 50)(27 191)(28 52)(30 154)(32 142)(34 144)(36 146)(38 148)(40 150)(42 152)(43 100)(45 102)(47 104)(49 106)(51 108)(53 110)(55 112)(57 140)(58 77)(59 128)(60 79)(61 130)(62 81)(63 132)(64 83)(65 134)(66 71)(67 136)(68 73)(69 138)(70 75)(72 214)(74 216)(76 218)(78 220)(80 222)(82 224)(84 212)(85 121)(86 163)(87 123)(88 165)(89 125)(90 167)(91 113)(92 155)(93 115)(94 157)(95 117)(96 159)(97 119)(98 161)(99 196)(101 184)(103 186)(105 188)(107 190)(109 192)(111 194)(114 177)(116 179)(118 181)(120 169)(122 171)(124 173)(126 175)(127 219)(129 221)(131 223)(133 211)(135 213)(137 215)(139 217)(141 209)(143 197)(145 199)(147 201)(149 203)(151 205)(153 207)(156 178)(158 180)(160 182)(162 170)(164 172)(166 174)(168 176)
(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 66 144 213)(2 65 145 212)(3 64 146 211)(4 63 147 224)(5 62 148 223)(6 61 149 222)(7 60 150 221)(8 59 151 220)(9 58 152 219)(10 57 153 218)(11 70 154 217)(12 69 141 216)(13 68 142 215)(14 67 143 214)(15 180 110 95)(16 179 111 94)(17 178 112 93)(18 177 99 92)(19 176 100 91)(20 175 101 90)(21 174 102 89)(22 173 103 88)(23 172 104 87)(24 171 105 86)(25 170 106 85)(26 169 107 98)(27 182 108 97)(28 181 109 96)(29 49 207 189)(30 48 208 188)(31 47 209 187)(32 46 210 186)(33 45 197 185)(34 44 198 184)(35 43 199 183)(36 56 200 196)(37 55 201 195)(38 54 202 194)(39 53 203 193)(40 52 204 192)(41 51 205 191)(42 50 206 190)(71 126 135 167)(72 125 136 166)(73 124 137 165)(74 123 138 164)(75 122 139 163)(76 121 140 162)(77 120 127 161)(78 119 128 160)(79 118 129 159)(80 117 130 158)(81 116 131 157)(82 115 132 156)(83 114 133 155)(84 113 134 168)
G:=sub<Sym(224)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,200)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,197)(41,198)(42,199)(43,190)(44,191)(45,192)(46,193)(47,194)(48,195)(49,196)(50,183)(51,184)(52,185)(53,186)(54,187)(55,188)(56,189)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,137)(81,138)(82,139)(83,140)(84,127)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,175,144,90)(2,176,145,91)(3,177,146,92)(4,178,147,93)(5,179,148,94)(6,180,149,95)(7,181,150,96)(8,182,151,97)(9,169,152,98)(10,170,153,85)(11,171,154,86)(12,172,141,87)(13,173,142,88)(14,174,143,89)(15,222,110,61)(16,223,111,62)(17,224,112,63)(18,211,99,64)(19,212,100,65)(20,213,101,66)(21,214,102,67)(22,215,103,68)(23,216,104,69)(24,217,105,70)(25,218,106,57)(26,219,107,58)(27,220,108,59)(28,221,109,60)(29,121,207,162)(30,122,208,163)(31,123,209,164)(32,124,210,165)(33,125,197,166)(34,126,198,167)(35,113,199,168)(36,114,200,155)(37,115,201,156)(38,116,202,157)(39,117,203,158)(40,118,204,159)(41,119,205,160)(42,120,206,161)(43,84,183,134)(44,71,184,135)(45,72,185,136)(46,73,186,137)(47,74,187,138)(48,75,188,139)(49,76,189,140)(50,77,190,127)(51,78,191,128)(52,79,192,129)(53,80,193,130)(54,81,194,131)(55,82,195,132)(56,83,196,133), (1,198)(2,35)(3,200)(4,37)(5,202)(6,39)(7,204)(8,41)(9,206)(10,29)(11,208)(12,31)(13,210)(14,33)(15,193)(16,54)(17,195)(18,56)(19,183)(20,44)(21,185)(22,46)(23,187)(24,48)(25,189)(26,50)(27,191)(28,52)(30,154)(32,142)(34,144)(36,146)(38,148)(40,150)(42,152)(43,100)(45,102)(47,104)(49,106)(51,108)(53,110)(55,112)(57,140)(58,77)(59,128)(60,79)(61,130)(62,81)(63,132)(64,83)(65,134)(66,71)(67,136)(68,73)(69,138)(70,75)(72,214)(74,216)(76,218)(78,220)(80,222)(82,224)(84,212)(85,121)(86,163)(87,123)(88,165)(89,125)(90,167)(91,113)(92,155)(93,115)(94,157)(95,117)(96,159)(97,119)(98,161)(99,196)(101,184)(103,186)(105,188)(107,190)(109,192)(111,194)(114,177)(116,179)(118,181)(120,169)(122,171)(124,173)(126,175)(127,219)(129,221)(131,223)(133,211)(135,213)(137,215)(139,217)(141,209)(143,197)(145,199)(147,201)(149,203)(151,205)(153,207)(156,178)(158,180)(160,182)(162,170)(164,172)(166,174)(168,176), (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), (1,66,144,213)(2,65,145,212)(3,64,146,211)(4,63,147,224)(5,62,148,223)(6,61,149,222)(7,60,150,221)(8,59,151,220)(9,58,152,219)(10,57,153,218)(11,70,154,217)(12,69,141,216)(13,68,142,215)(14,67,143,214)(15,180,110,95)(16,179,111,94)(17,178,112,93)(18,177,99,92)(19,176,100,91)(20,175,101,90)(21,174,102,89)(22,173,103,88)(23,172,104,87)(24,171,105,86)(25,170,106,85)(26,169,107,98)(27,182,108,97)(28,181,109,96)(29,49,207,189)(30,48,208,188)(31,47,209,187)(32,46,210,186)(33,45,197,185)(34,44,198,184)(35,43,199,183)(36,56,200,196)(37,55,201,195)(38,54,202,194)(39,53,203,193)(40,52,204,192)(41,51,205,191)(42,50,206,190)(71,126,135,167)(72,125,136,166)(73,124,137,165)(74,123,138,164)(75,122,139,163)(76,121,140,162)(77,120,127,161)(78,119,128,160)(79,118,129,159)(80,117,130,158)(81,116,131,157)(82,115,132,156)(83,114,133,155)(84,113,134,168)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,200)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,197)(41,198)(42,199)(43,190)(44,191)(45,192)(46,193)(47,194)(48,195)(49,196)(50,183)(51,184)(52,185)(53,186)(54,187)(55,188)(56,189)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,128)(72,129)(73,130)(74,131)(75,132)(76,133)(77,134)(78,135)(79,136)(80,137)(81,138)(82,139)(83,140)(84,127)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,155)(122,156)(123,157)(124,158)(125,159)(126,160)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,175,144,90)(2,176,145,91)(3,177,146,92)(4,178,147,93)(5,179,148,94)(6,180,149,95)(7,181,150,96)(8,182,151,97)(9,169,152,98)(10,170,153,85)(11,171,154,86)(12,172,141,87)(13,173,142,88)(14,174,143,89)(15,222,110,61)(16,223,111,62)(17,224,112,63)(18,211,99,64)(19,212,100,65)(20,213,101,66)(21,214,102,67)(22,215,103,68)(23,216,104,69)(24,217,105,70)(25,218,106,57)(26,219,107,58)(27,220,108,59)(28,221,109,60)(29,121,207,162)(30,122,208,163)(31,123,209,164)(32,124,210,165)(33,125,197,166)(34,126,198,167)(35,113,199,168)(36,114,200,155)(37,115,201,156)(38,116,202,157)(39,117,203,158)(40,118,204,159)(41,119,205,160)(42,120,206,161)(43,84,183,134)(44,71,184,135)(45,72,185,136)(46,73,186,137)(47,74,187,138)(48,75,188,139)(49,76,189,140)(50,77,190,127)(51,78,191,128)(52,79,192,129)(53,80,193,130)(54,81,194,131)(55,82,195,132)(56,83,196,133), (1,198)(2,35)(3,200)(4,37)(5,202)(6,39)(7,204)(8,41)(9,206)(10,29)(11,208)(12,31)(13,210)(14,33)(15,193)(16,54)(17,195)(18,56)(19,183)(20,44)(21,185)(22,46)(23,187)(24,48)(25,189)(26,50)(27,191)(28,52)(30,154)(32,142)(34,144)(36,146)(38,148)(40,150)(42,152)(43,100)(45,102)(47,104)(49,106)(51,108)(53,110)(55,112)(57,140)(58,77)(59,128)(60,79)(61,130)(62,81)(63,132)(64,83)(65,134)(66,71)(67,136)(68,73)(69,138)(70,75)(72,214)(74,216)(76,218)(78,220)(80,222)(82,224)(84,212)(85,121)(86,163)(87,123)(88,165)(89,125)(90,167)(91,113)(92,155)(93,115)(94,157)(95,117)(96,159)(97,119)(98,161)(99,196)(101,184)(103,186)(105,188)(107,190)(109,192)(111,194)(114,177)(116,179)(118,181)(120,169)(122,171)(124,173)(126,175)(127,219)(129,221)(131,223)(133,211)(135,213)(137,215)(139,217)(141,209)(143,197)(145,199)(147,201)(149,203)(151,205)(153,207)(156,178)(158,180)(160,182)(162,170)(164,172)(166,174)(168,176), (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), (1,66,144,213)(2,65,145,212)(3,64,146,211)(4,63,147,224)(5,62,148,223)(6,61,149,222)(7,60,150,221)(8,59,151,220)(9,58,152,219)(10,57,153,218)(11,70,154,217)(12,69,141,216)(13,68,142,215)(14,67,143,214)(15,180,110,95)(16,179,111,94)(17,178,112,93)(18,177,99,92)(19,176,100,91)(20,175,101,90)(21,174,102,89)(22,173,103,88)(23,172,104,87)(24,171,105,86)(25,170,106,85)(26,169,107,98)(27,182,108,97)(28,181,109,96)(29,49,207,189)(30,48,208,188)(31,47,209,187)(32,46,210,186)(33,45,197,185)(34,44,198,184)(35,43,199,183)(36,56,200,196)(37,55,201,195)(38,54,202,194)(39,53,203,193)(40,52,204,192)(41,51,205,191)(42,50,206,190)(71,126,135,167)(72,125,136,166)(73,124,137,165)(74,123,138,164)(75,122,139,163)(76,121,140,162)(77,120,127,161)(78,119,128,160)(79,118,129,159)(80,117,130,158)(81,116,131,157)(82,115,132,156)(83,114,133,155)(84,113,134,168) );
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,200),(30,201),(31,202),(32,203),(33,204),(34,205),(35,206),(36,207),(37,208),(38,209),(39,210),(40,197),(41,198),(42,199),(43,190),(44,191),(45,192),(46,193),(47,194),(48,195),(49,196),(50,183),(51,184),(52,185),(53,186),(54,187),(55,188),(56,189),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,128),(72,129),(73,130),(74,131),(75,132),(76,133),(77,134),(78,135),(79,136),(80,137),(81,138),(82,139),(83,140),(84,127),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,161),(114,162),(115,163),(116,164),(117,165),(118,166),(119,167),(120,168),(121,155),(122,156),(123,157),(124,158),(125,159),(126,160),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)], [(1,175,144,90),(2,176,145,91),(3,177,146,92),(4,178,147,93),(5,179,148,94),(6,180,149,95),(7,181,150,96),(8,182,151,97),(9,169,152,98),(10,170,153,85),(11,171,154,86),(12,172,141,87),(13,173,142,88),(14,174,143,89),(15,222,110,61),(16,223,111,62),(17,224,112,63),(18,211,99,64),(19,212,100,65),(20,213,101,66),(21,214,102,67),(22,215,103,68),(23,216,104,69),(24,217,105,70),(25,218,106,57),(26,219,107,58),(27,220,108,59),(28,221,109,60),(29,121,207,162),(30,122,208,163),(31,123,209,164),(32,124,210,165),(33,125,197,166),(34,126,198,167),(35,113,199,168),(36,114,200,155),(37,115,201,156),(38,116,202,157),(39,117,203,158),(40,118,204,159),(41,119,205,160),(42,120,206,161),(43,84,183,134),(44,71,184,135),(45,72,185,136),(46,73,186,137),(47,74,187,138),(48,75,188,139),(49,76,189,140),(50,77,190,127),(51,78,191,128),(52,79,192,129),(53,80,193,130),(54,81,194,131),(55,82,195,132),(56,83,196,133)], [(1,198),(2,35),(3,200),(4,37),(5,202),(6,39),(7,204),(8,41),(9,206),(10,29),(11,208),(12,31),(13,210),(14,33),(15,193),(16,54),(17,195),(18,56),(19,183),(20,44),(21,185),(22,46),(23,187),(24,48),(25,189),(26,50),(27,191),(28,52),(30,154),(32,142),(34,144),(36,146),(38,148),(40,150),(42,152),(43,100),(45,102),(47,104),(49,106),(51,108),(53,110),(55,112),(57,140),(58,77),(59,128),(60,79),(61,130),(62,81),(63,132),(64,83),(65,134),(66,71),(67,136),(68,73),(69,138),(70,75),(72,214),(74,216),(76,218),(78,220),(80,222),(82,224),(84,212),(85,121),(86,163),(87,123),(88,165),(89,125),(90,167),(91,113),(92,155),(93,115),(94,157),(95,117),(96,159),(97,119),(98,161),(99,196),(101,184),(103,186),(105,188),(107,190),(109,192),(111,194),(114,177),(116,179),(118,181),(120,169),(122,171),(124,173),(126,175),(127,219),(129,221),(131,223),(133,211),(135,213),(137,215),(139,217),(141,209),(143,197),(145,199),(147,201),(149,203),(151,205),(153,207),(156,178),(158,180),(160,182),(162,170),(164,172),(166,174),(168,176)], [(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)], [(1,66,144,213),(2,65,145,212),(3,64,146,211),(4,63,147,224),(5,62,148,223),(6,61,149,222),(7,60,150,221),(8,59,151,220),(9,58,152,219),(10,57,153,218),(11,70,154,217),(12,69,141,216),(13,68,142,215),(14,67,143,214),(15,180,110,95),(16,179,111,94),(17,178,112,93),(18,177,99,92),(19,176,100,91),(20,175,101,90),(21,174,102,89),(22,173,103,88),(23,172,104,87),(24,171,105,86),(25,170,106,85),(26,169,107,98),(27,182,108,97),(28,181,109,96),(29,49,207,189),(30,48,208,188),(31,47,209,187),(32,46,210,186),(33,45,197,185),(34,44,198,184),(35,43,199,183),(36,56,200,196),(37,55,201,195),(38,54,202,194),(39,53,203,193),(40,52,204,192),(41,51,205,191),(42,50,206,190),(71,126,135,167),(72,125,136,166),(73,124,137,165),(74,123,138,164),(75,122,139,163),(76,121,140,162),(77,120,127,161),(78,119,128,160),(79,118,129,159),(80,117,130,158),(81,116,131,157),(82,115,132,156),(83,114,133,155),(84,113,134,168)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8.C22 | D4.9D14 |
| kernel | C2×D4.9D14 | C2×C4.Dic7 | C2×D4.D7 | C2×C7⋊Q16 | D4.9D14 | C22×Dic14 | C14×C4○D4 | C2×C28 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 1 | 3 | 3 | 3 | 3 | 12 | 18 | 6 | 2 | 12 |
Matrix representation of C2×D4.9D14 ►in GL6(𝔽113)
| 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 | 17 | 8 | 0 | 0 |
| 0 | 0 | 105 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 105 |
| 0 | 0 | 0 | 0 | 8 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 105 |
| 0 | 0 | 0 | 0 | 8 | 17 |
| 0 | 0 | 17 | 8 | 0 | 0 |
| 0 | 0 | 105 | 96 | 0 | 0 |
| 106 | 0 | 0 | 0 | 0 | 0 |
| 59 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 103 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 89 | 10 |
| 0 | 0 | 0 | 0 | 103 | 103 |
| 86 | 68 | 0 | 0 | 0 | 0 |
| 89 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 18 | 0 | 0 |
| 0 | 0 | 56 | 102 | 0 | 0 |
| 0 | 0 | 0 | 0 | 70 | 105 |
| 0 | 0 | 0 | 0 | 90 | 43 |
G:=sub<GL(6,GF(113))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,105,0,0,0,0,8,96,0,0,0,0,0,0,96,8,0,0,0,0,105,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,105,0,0,0,0,8,96,0,0,96,8,0,0,0,0,105,17,0,0],[106,59,0,0,0,0,0,16,0,0,0,0,0,0,24,10,0,0,0,0,103,10,0,0,0,0,0,0,89,103,0,0,0,0,10,103],[86,89,0,0,0,0,68,27,0,0,0,0,0,0,11,56,0,0,0,0,18,102,0,0,0,0,0,0,70,90,0,0,0,0,105,43] >;
C2×D4.9D14 in GAP, Magma, Sage, TeX
C_2\times D_4._9D_{14} % in TeX
G:=Group("C2xD4.9D14"); // GroupNames label
G:=SmallGroup(448,1276);
// by ID
G=gap.SmallGroup(448,1276);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,675,297,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^14=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations