G = C14.752- 1+4 order 448 = 26·7
30th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.752- 1+4, C14.502+ 1+4, C28⋊Q8⋊23C2, C4⋊C4.94D14, (C2×Dic7)⋊4Q8, C22⋊Q8.8D7, C22.6(Q8×D7), (C2×Q8).74D14, Dic7.4(C2×Q8), Dic7.Q8⋊16C2, (C2×C28).49C23, C22⋊C4.53D14, Dic7⋊Q8⋊12C2, C14.33(C22×Q8), (C2×C14).167C24, (C22×C4).232D14, C4⋊Dic7.47C22, C2.52(D4⋊6D14), C28.48D4.19C2, C22⋊Dic14.3C2, (Q8×C14).102C22, C23.185(C22×D7), C22.188(C23×D7), C23.D7.31C22, Dic7⋊C4.161C22, (C22×C14).195C23, (C22×C28).314C22, C7⋊3(C23.41C23), (C4×Dic7).101C22, (C2×Dic7).231C23, C23.11D14.2C2, C2.33(D4.10D14), (C2×Dic14).158C22, (C22×Dic7).117C22, C2.16(C2×Q8×D7), (C2×C14).6(C2×Q8), (C7×C22⋊Q8).8C2, (C7×C4⋊C4).153C22, (C2×Dic7⋊C4).23C2, (C2×C4).181(C22×D7), (C7×C22⋊C4).22C22, SmallGroup(448,1076)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.752- 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=a7b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a7c, ce=ec, ede=b2d >
Subgroups: 796 in 206 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×Q8, C22×C14, C23.41C23, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, C22×Dic7, C22×C28, Q8×C14, C23.11D14, C22⋊Dic14, C28⋊Q8, Dic7.Q8, C2×Dic7⋊C4, C28.48D4, Dic7⋊Q8, C7×C22⋊Q8, C14.752- 1+4
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C24, D14, C22×Q8, 2+ 1+4, 2- 1+4, C22×D7, C23.41C23, Q8×D7, C23×D7, D4⋊6D14, C2×Q8×D7, D4.10D14, C14.752- 1+4
►On 224 points
Generators in S224
(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 90 39 62)(2 89 40 61)(3 88 41 60)(4 87 42 59)(5 86 29 58)(6 85 30 57)(7 98 31 70)(8 97 32 69)(9 96 33 68)(10 95 34 67)(11 94 35 66)(12 93 36 65)(13 92 37 64)(14 91 38 63)(15 214 112 159)(16 213 99 158)(17 212 100 157)(18 211 101 156)(19 224 102 155)(20 223 103 168)(21 222 104 167)(22 221 105 166)(23 220 106 165)(24 219 107 164)(25 218 108 163)(26 217 109 162)(27 216 110 161)(28 215 111 160)(43 176 145 183)(44 175 146 196)(45 174 147 195)(46 173 148 194)(47 172 149 193)(48 171 150 192)(49 170 151 191)(50 169 152 190)(51 182 153 189)(52 181 154 188)(53 180 141 187)(54 179 142 186)(55 178 143 185)(56 177 144 184)(71 124 205 137)(72 123 206 136)(73 122 207 135)(74 121 208 134)(75 120 209 133)(76 119 210 132)(77 118 197 131)(78 117 198 130)(79 116 199 129)(80 115 200 128)(81 114 201 127)(82 113 202 140)(83 126 203 139)(84 125 204 138)
(1 114 8 121)(2 115 9 122)(3 116 10 123)(4 117 11 124)(5 118 12 125)(6 119 13 126)(7 120 14 113)(15 47 22 54)(16 48 23 55)(17 49 24 56)(18 50 25 43)(19 51 26 44)(20 52 27 45)(21 53 28 46)(29 131 36 138)(30 132 37 139)(31 133 38 140)(32 134 39 127)(33 135 40 128)(34 136 41 129)(35 137 42 130)(57 210 64 203)(58 197 65 204)(59 198 66 205)(60 199 67 206)(61 200 68 207)(62 201 69 208)(63 202 70 209)(71 87 78 94)(72 88 79 95)(73 89 80 96)(74 90 81 97)(75 91 82 98)(76 92 83 85)(77 93 84 86)(99 150 106 143)(100 151 107 144)(101 152 108 145)(102 153 109 146)(103 154 110 147)(104 141 111 148)(105 142 112 149)(155 182 162 175)(156 169 163 176)(157 170 164 177)(158 171 165 178)(159 172 166 179)(160 173 167 180)(161 174 168 181)(183 211 190 218)(184 212 191 219)(185 213 192 220)(186 214 193 221)(187 215 194 222)(188 216 195 223)(189 217 196 224)
(1 175 32 189)(2 176 33 190)(3 177 34 191)(4 178 35 192)(5 179 36 193)(6 180 37 194)(7 181 38 195)(8 182 39 196)(9 169 40 183)(10 170 41 184)(11 171 42 185)(12 172 29 186)(13 173 30 187)(14 174 31 188)(15 77 105 204)(16 78 106 205)(17 79 107 206)(18 80 108 207)(19 81 109 208)(20 82 110 209)(21 83 111 210)(22 84 112 197)(23 71 99 198)(24 72 100 199)(25 73 101 200)(26 74 102 201)(27 75 103 202)(28 76 104 203)(43 96 152 61)(44 97 153 62)(45 98 154 63)(46 85 141 64)(47 86 142 65)(48 87 143 66)(49 88 144 67)(50 89 145 68)(51 90 146 69)(52 91 147 70)(53 92 148 57)(54 93 149 58)(55 94 150 59)(56 95 151 60)(113 161 133 223)(114 162 134 224)(115 163 135 211)(116 164 136 212)(117 165 137 213)(118 166 138 214)(119 167 139 215)(120 168 140 216)(121 155 127 217)(122 156 128 218)(123 157 129 219)(124 158 130 220)(125 159 131 221)(126 160 132 222)
(15 112)(16 99)(17 100)(18 101)(19 102)(20 103)(21 104)(22 105)(23 106)(24 107)(25 108)(26 109)(27 110)(28 111)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 141)(54 142)(55 143)(56 144)(155 224)(156 211)(157 212)(158 213)(159 214)(160 215)(161 216)(162 217)(163 218)(164 219)(165 220)(166 221)(167 222)(168 223)(169 190)(170 191)(171 192)(172 193)(173 194)(174 195)(175 196)(176 183)(177 184)(178 185)(179 186)(180 187)(181 188)(182 189)
G:=sub<Sym(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), (1,90,39,62)(2,89,40,61)(3,88,41,60)(4,87,42,59)(5,86,29,58)(6,85,30,57)(7,98,31,70)(8,97,32,69)(9,96,33,68)(10,95,34,67)(11,94,35,66)(12,93,36,65)(13,92,37,64)(14,91,38,63)(15,214,112,159)(16,213,99,158)(17,212,100,157)(18,211,101,156)(19,224,102,155)(20,223,103,168)(21,222,104,167)(22,221,105,166)(23,220,106,165)(24,219,107,164)(25,218,108,163)(26,217,109,162)(27,216,110,161)(28,215,111,160)(43,176,145,183)(44,175,146,196)(45,174,147,195)(46,173,148,194)(47,172,149,193)(48,171,150,192)(49,170,151,191)(50,169,152,190)(51,182,153,189)(52,181,154,188)(53,180,141,187)(54,179,142,186)(55,178,143,185)(56,177,144,184)(71,124,205,137)(72,123,206,136)(73,122,207,135)(74,121,208,134)(75,120,209,133)(76,119,210,132)(77,118,197,131)(78,117,198,130)(79,116,199,129)(80,115,200,128)(81,114,201,127)(82,113,202,140)(83,126,203,139)(84,125,204,138), (1,114,8,121)(2,115,9,122)(3,116,10,123)(4,117,11,124)(5,118,12,125)(6,119,13,126)(7,120,14,113)(15,47,22,54)(16,48,23,55)(17,49,24,56)(18,50,25,43)(19,51,26,44)(20,52,27,45)(21,53,28,46)(29,131,36,138)(30,132,37,139)(31,133,38,140)(32,134,39,127)(33,135,40,128)(34,136,41,129)(35,137,42,130)(57,210,64,203)(58,197,65,204)(59,198,66,205)(60,199,67,206)(61,200,68,207)(62,201,69,208)(63,202,70,209)(71,87,78,94)(72,88,79,95)(73,89,80,96)(74,90,81,97)(75,91,82,98)(76,92,83,85)(77,93,84,86)(99,150,106,143)(100,151,107,144)(101,152,108,145)(102,153,109,146)(103,154,110,147)(104,141,111,148)(105,142,112,149)(155,182,162,175)(156,169,163,176)(157,170,164,177)(158,171,165,178)(159,172,166,179)(160,173,167,180)(161,174,168,181)(183,211,190,218)(184,212,191,219)(185,213,192,220)(186,214,193,221)(187,215,194,222)(188,216,195,223)(189,217,196,224), (1,175,32,189)(2,176,33,190)(3,177,34,191)(4,178,35,192)(5,179,36,193)(6,180,37,194)(7,181,38,195)(8,182,39,196)(9,169,40,183)(10,170,41,184)(11,171,42,185)(12,172,29,186)(13,173,30,187)(14,174,31,188)(15,77,105,204)(16,78,106,205)(17,79,107,206)(18,80,108,207)(19,81,109,208)(20,82,110,209)(21,83,111,210)(22,84,112,197)(23,71,99,198)(24,72,100,199)(25,73,101,200)(26,74,102,201)(27,75,103,202)(28,76,104,203)(43,96,152,61)(44,97,153,62)(45,98,154,63)(46,85,141,64)(47,86,142,65)(48,87,143,66)(49,88,144,67)(50,89,145,68)(51,90,146,69)(52,91,147,70)(53,92,148,57)(54,93,149,58)(55,94,150,59)(56,95,151,60)(113,161,133,223)(114,162,134,224)(115,163,135,211)(116,164,136,212)(117,165,137,213)(118,166,138,214)(119,167,139,215)(120,168,140,216)(121,155,127,217)(122,156,128,218)(123,157,129,219)(124,158,130,220)(125,159,131,221)(126,160,132,222), (15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,141)(54,142)(55,143)(56,144)(155,224)(156,211)(157,212)(158,213)(159,214)(160,215)(161,216)(162,217)(163,218)(164,219)(165,220)(166,221)(167,222)(168,223)(169,190)(170,191)(171,192)(172,193)(173,194)(174,195)(175,196)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)>;
G:=Group( (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), (1,90,39,62)(2,89,40,61)(3,88,41,60)(4,87,42,59)(5,86,29,58)(6,85,30,57)(7,98,31,70)(8,97,32,69)(9,96,33,68)(10,95,34,67)(11,94,35,66)(12,93,36,65)(13,92,37,64)(14,91,38,63)(15,214,112,159)(16,213,99,158)(17,212,100,157)(18,211,101,156)(19,224,102,155)(20,223,103,168)(21,222,104,167)(22,221,105,166)(23,220,106,165)(24,219,107,164)(25,218,108,163)(26,217,109,162)(27,216,110,161)(28,215,111,160)(43,176,145,183)(44,175,146,196)(45,174,147,195)(46,173,148,194)(47,172,149,193)(48,171,150,192)(49,170,151,191)(50,169,152,190)(51,182,153,189)(52,181,154,188)(53,180,141,187)(54,179,142,186)(55,178,143,185)(56,177,144,184)(71,124,205,137)(72,123,206,136)(73,122,207,135)(74,121,208,134)(75,120,209,133)(76,119,210,132)(77,118,197,131)(78,117,198,130)(79,116,199,129)(80,115,200,128)(81,114,201,127)(82,113,202,140)(83,126,203,139)(84,125,204,138), (1,114,8,121)(2,115,9,122)(3,116,10,123)(4,117,11,124)(5,118,12,125)(6,119,13,126)(7,120,14,113)(15,47,22,54)(16,48,23,55)(17,49,24,56)(18,50,25,43)(19,51,26,44)(20,52,27,45)(21,53,28,46)(29,131,36,138)(30,132,37,139)(31,133,38,140)(32,134,39,127)(33,135,40,128)(34,136,41,129)(35,137,42,130)(57,210,64,203)(58,197,65,204)(59,198,66,205)(60,199,67,206)(61,200,68,207)(62,201,69,208)(63,202,70,209)(71,87,78,94)(72,88,79,95)(73,89,80,96)(74,90,81,97)(75,91,82,98)(76,92,83,85)(77,93,84,86)(99,150,106,143)(100,151,107,144)(101,152,108,145)(102,153,109,146)(103,154,110,147)(104,141,111,148)(105,142,112,149)(155,182,162,175)(156,169,163,176)(157,170,164,177)(158,171,165,178)(159,172,166,179)(160,173,167,180)(161,174,168,181)(183,211,190,218)(184,212,191,219)(185,213,192,220)(186,214,193,221)(187,215,194,222)(188,216,195,223)(189,217,196,224), (1,175,32,189)(2,176,33,190)(3,177,34,191)(4,178,35,192)(5,179,36,193)(6,180,37,194)(7,181,38,195)(8,182,39,196)(9,169,40,183)(10,170,41,184)(11,171,42,185)(12,172,29,186)(13,173,30,187)(14,174,31,188)(15,77,105,204)(16,78,106,205)(17,79,107,206)(18,80,108,207)(19,81,109,208)(20,82,110,209)(21,83,111,210)(22,84,112,197)(23,71,99,198)(24,72,100,199)(25,73,101,200)(26,74,102,201)(27,75,103,202)(28,76,104,203)(43,96,152,61)(44,97,153,62)(45,98,154,63)(46,85,141,64)(47,86,142,65)(48,87,143,66)(49,88,144,67)(50,89,145,68)(51,90,146,69)(52,91,147,70)(53,92,148,57)(54,93,149,58)(55,94,150,59)(56,95,151,60)(113,161,133,223)(114,162,134,224)(115,163,135,211)(116,164,136,212)(117,165,137,213)(118,166,138,214)(119,167,139,215)(120,168,140,216)(121,155,127,217)(122,156,128,218)(123,157,129,219)(124,158,130,220)(125,159,131,221)(126,160,132,222), (15,112)(16,99)(17,100)(18,101)(19,102)(20,103)(21,104)(22,105)(23,106)(24,107)(25,108)(26,109)(27,110)(28,111)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,141)(54,142)(55,143)(56,144)(155,224)(156,211)(157,212)(158,213)(159,214)(160,215)(161,216)(162,217)(163,218)(164,219)(165,220)(166,221)(167,222)(168,223)(169,190)(170,191)(171,192)(172,193)(173,194)(174,195)(175,196)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189) );
G=PermutationGroup([[(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)], [(1,90,39,62),(2,89,40,61),(3,88,41,60),(4,87,42,59),(5,86,29,58),(6,85,30,57),(7,98,31,70),(8,97,32,69),(9,96,33,68),(10,95,34,67),(11,94,35,66),(12,93,36,65),(13,92,37,64),(14,91,38,63),(15,214,112,159),(16,213,99,158),(17,212,100,157),(18,211,101,156),(19,224,102,155),(20,223,103,168),(21,222,104,167),(22,221,105,166),(23,220,106,165),(24,219,107,164),(25,218,108,163),(26,217,109,162),(27,216,110,161),(28,215,111,160),(43,176,145,183),(44,175,146,196),(45,174,147,195),(46,173,148,194),(47,172,149,193),(48,171,150,192),(49,170,151,191),(50,169,152,190),(51,182,153,189),(52,181,154,188),(53,180,141,187),(54,179,142,186),(55,178,143,185),(56,177,144,184),(71,124,205,137),(72,123,206,136),(73,122,207,135),(74,121,208,134),(75,120,209,133),(76,119,210,132),(77,118,197,131),(78,117,198,130),(79,116,199,129),(80,115,200,128),(81,114,201,127),(82,113,202,140),(83,126,203,139),(84,125,204,138)], [(1,114,8,121),(2,115,9,122),(3,116,10,123),(4,117,11,124),(5,118,12,125),(6,119,13,126),(7,120,14,113),(15,47,22,54),(16,48,23,55),(17,49,24,56),(18,50,25,43),(19,51,26,44),(20,52,27,45),(21,53,28,46),(29,131,36,138),(30,132,37,139),(31,133,38,140),(32,134,39,127),(33,135,40,128),(34,136,41,129),(35,137,42,130),(57,210,64,203),(58,197,65,204),(59,198,66,205),(60,199,67,206),(61,200,68,207),(62,201,69,208),(63,202,70,209),(71,87,78,94),(72,88,79,95),(73,89,80,96),(74,90,81,97),(75,91,82,98),(76,92,83,85),(77,93,84,86),(99,150,106,143),(100,151,107,144),(101,152,108,145),(102,153,109,146),(103,154,110,147),(104,141,111,148),(105,142,112,149),(155,182,162,175),(156,169,163,176),(157,170,164,177),(158,171,165,178),(159,172,166,179),(160,173,167,180),(161,174,168,181),(183,211,190,218),(184,212,191,219),(185,213,192,220),(186,214,193,221),(187,215,194,222),(188,216,195,223),(189,217,196,224)], [(1,175,32,189),(2,176,33,190),(3,177,34,191),(4,178,35,192),(5,179,36,193),(6,180,37,194),(7,181,38,195),(8,182,39,196),(9,169,40,183),(10,170,41,184),(11,171,42,185),(12,172,29,186),(13,173,30,187),(14,174,31,188),(15,77,105,204),(16,78,106,205),(17,79,107,206),(18,80,108,207),(19,81,109,208),(20,82,110,209),(21,83,111,210),(22,84,112,197),(23,71,99,198),(24,72,100,199),(25,73,101,200),(26,74,102,201),(27,75,103,202),(28,76,104,203),(43,96,152,61),(44,97,153,62),(45,98,154,63),(46,85,141,64),(47,86,142,65),(48,87,143,66),(49,88,144,67),(50,89,145,68),(51,90,146,69),(52,91,147,70),(53,92,148,57),(54,93,149,58),(55,94,150,59),(56,95,151,60),(113,161,133,223),(114,162,134,224),(115,163,135,211),(116,164,136,212),(117,165,137,213),(118,166,138,214),(119,167,139,215),(120,168,140,216),(121,155,127,217),(122,156,128,218),(123,157,129,219),(124,158,130,220),(125,159,131,221),(126,160,132,222)], [(15,112),(16,99),(17,100),(18,101),(19,102),(20,103),(21,104),(22,105),(23,106),(24,107),(25,108),(26,109),(27,110),(28,111),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,141),(54,142),(55,143),(56,144),(155,224),(156,211),(157,212),(158,213),(159,214),(160,215),(161,216),(162,217),(163,218),(164,219),(165,220),(166,221),(167,222),(168,223),(169,190),(170,191),(171,192),(172,193),(173,194),(174,195),(175,196),(176,183),(177,184),(178,185),(179,186),(180,187),(181,188),(182,189)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | Q8×D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.752- 1+4 | C23.11D14 | C22⋊Dic14 | C28⋊Q8 | Dic7.Q8 | C2×Dic7⋊C4 | C28.48D4 | Dic7⋊Q8 | C7×C22⋊Q8 | C2×Dic7 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 4 | 3 | 6 | 9 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
Matrix representation of C14.752- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 21 | 0 | 0 |
| 0 | 0 | 8 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 21 |
| 0 | 0 | 0 | 0 | 8 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 5 | 0 | 0 |
| 0 | 0 | 28 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 5 |
| 0 | 0 | 0 | 0 | 28 | 27 |
| 8 | 5 | 0 | 0 | 0 | 0 |
| 16 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 16 | 0 | 0 |
| 0 | 0 | 13 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 16 |
| 0 | 0 | 0 | 0 | 13 | 5 |
| 20 | 27 | 0 | 0 | 0 | 0 |
| 12 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,26,8,0,0,0,0,21,21,0,0,0,0,0,0,26,8,0,0,0,0,21,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,28,0,0,0,0,5,27,0,0,0,0,0,0,2,28,0,0,0,0,5,27],[8,16,0,0,0,0,5,21,0,0,0,0,0,0,24,13,0,0,0,0,16,5,0,0,0,0,0,0,24,13,0,0,0,0,16,5],[20,12,0,0,0,0,27,9,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,1,0,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;
C14.752- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{75}2_-^{1+4} % in TeX
G:=Group("C14.75ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1076);
// by ID
G=gap.SmallGroup(448,1076);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,387,100,1123,185,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=a^7*b^2,b*a*b^-1=a^-1,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,d*c*d^-1=a^7*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations