G = C14.252- 1+4 order 448 = 26·7
25th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.252- 1+4, C14.582+ 1+4, C28⋊Q8⋊29C2, C22⋊Q8⋊23D7, C4⋊C4.101D14, (C2×Q8).79D14, D14⋊3Q8⋊25C2, D14⋊Q8⋊24C2, Dic7.Q8⋊22C2, D14⋊C4.8C22, C22⋊C4.23D14, C4.Dic14⋊27C2, Dic7⋊Q8⋊18C2, C28.48D4⋊47C2, (C2×C28).176C23, (C2×C14).190C24, D14.D4.3C2, (C22×C4).252D14, C2.60(D4⋊6D14), C23.D14⋊26C2, Dic7⋊C4.81C22, C4⋊Dic7.222C22, Dic7.D4.3C2, (Q8×C14).119C22, (C2×Dic7).96C23, (C22×D7).81C23, C22.211(C23×D7), C23.126(C22×D7), C23.D7.36C22, (C22×C28).318C22, (C22×C14).218C23, C7⋊1(C22.57C24), (C4×Dic7).117C22, (C2×Dic14).35C22, C23.23D14.2C2, C2.39(D4.10D14), C2.26(Q8.10D14), C4⋊C4⋊D7⋊25C2, (C7×C22⋊Q8)⋊26C2, (C2×C4×D7).106C22, (C7×C4⋊C4).170C22, (C2×C4).187(C22×D7), (C2×C7⋊D4).42C22, (C7×C22⋊C4).45C22, SmallGroup(448,1099)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.252- 1+4
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=a7b-1, bd=db, ebe-1=a7b, dcd-1=a7c, ce=ec, ede-1=b2d >
Subgroups: 828 in 196 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4 [×6], C2×C4 [×9], D4, Q8 [×3], C23, C23, D7, C14 [×3], C14, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8 [×2], Dic7 [×7], C28 [×6], D14 [×3], C2×C14, C2×C14 [×3], C22⋊Q8, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], Dic14 [×2], C4×D7, C2×Dic7 [×7], C7⋊D4, C2×C28 [×6], C2×C28, C7×Q8, C22×D7, C22×C14, C22.57C24, C4×Dic7 [×3], Dic7⋊C4 [×9], C4⋊Dic7 [×4], D14⋊C4 [×5], C23.D7 [×3], C7×C22⋊C4 [×2], C7×C4⋊C4 [×3], C2×Dic14 [×2], C2×C4×D7, C2×C7⋊D4, C22×C28, Q8×C14, C23.D14 [×2], D14.D4, Dic7.D4, C28⋊Q8, Dic7.Q8, C4.Dic14, D14⋊Q8, C4⋊C4⋊D7 [×2], C28.48D4, C23.23D14, Dic7⋊Q8, D14⋊3Q8, C7×C22⋊Q8, C14.252- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ 1+4, 2- 1+4 [×2], C22×D7 [×7], C22.57C24, C23×D7, D4⋊6D14, Q8.10D14, D4.10D14, C14.252- 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)
(1 155 96 202)(2 156 97 203)(3 157 98 204)(4 158 85 205)(5 159 86 206)(6 160 87 207)(7 161 88 208)(8 162 89 209)(9 163 90 210)(10 164 91 197)(11 165 92 198)(12 166 93 199)(13 167 94 200)(14 168 95 201)(15 129 213 43)(16 130 214 44)(17 131 215 45)(18 132 216 46)(19 133 217 47)(20 134 218 48)(21 135 219 49)(22 136 220 50)(23 137 221 51)(24 138 222 52)(25 139 223 53)(26 140 224 54)(27 127 211 55)(28 128 212 56)(29 67 180 126)(30 68 181 113)(31 69 182 114)(32 70 169 115)(33 57 170 116)(34 58 171 117)(35 59 172 118)(36 60 173 119)(37 61 174 120)(38 62 175 121)(39 63 176 122)(40 64 177 123)(41 65 178 124)(42 66 179 125)(71 195 111 149)(72 196 112 150)(73 183 99 151)(74 184 100 152)(75 185 101 153)(76 186 102 154)(77 187 103 141)(78 188 104 142)(79 189 105 143)(80 190 106 144)(81 191 107 145)(82 192 108 146)(83 193 109 147)(84 194 110 148)
(1 171 8 178)(2 170 9 177)(3 169 10 176)(4 182 11 175)(5 181 12 174)(6 180 13 173)(7 179 14 172)(15 186 22 193)(16 185 23 192)(17 184 24 191)(18 183 25 190)(19 196 26 189)(20 195 27 188)(21 194 28 187)(29 94 36 87)(30 93 37 86)(31 92 38 85)(32 91 39 98)(33 90 40 97)(34 89 41 96)(35 88 42 95)(43 109 50 102)(44 108 51 101)(45 107 52 100)(46 106 53 99)(47 105 54 112)(48 104 55 111)(49 103 56 110)(57 156 64 163)(58 155 65 162)(59 168 66 161)(60 167 67 160)(61 166 68 159)(62 165 69 158)(63 164 70 157)(71 134 78 127)(72 133 79 140)(73 132 80 139)(74 131 81 138)(75 130 82 137)(76 129 83 136)(77 128 84 135)(113 206 120 199)(114 205 121 198)(115 204 122 197)(116 203 123 210)(117 202 124 209)(118 201 125 208)(119 200 126 207)(141 219 148 212)(142 218 149 211)(143 217 150 224)(144 216 151 223)(145 215 152 222)(146 214 153 221)(147 213 154 220)
(1 155 96 202)(2 156 97 203)(3 157 98 204)(4 158 85 205)(5 159 86 206)(6 160 87 207)(7 161 88 208)(8 162 89 209)(9 163 90 210)(10 164 91 197)(11 165 92 198)(12 166 93 199)(13 167 94 200)(14 168 95 201)(15 136 213 50)(16 137 214 51)(17 138 215 52)(18 139 216 53)(19 140 217 54)(20 127 218 55)(21 128 219 56)(22 129 220 43)(23 130 221 44)(24 131 222 45)(25 132 223 46)(26 133 224 47)(27 134 211 48)(28 135 212 49)(29 126 180 67)(30 113 181 68)(31 114 182 69)(32 115 169 70)(33 116 170 57)(34 117 171 58)(35 118 172 59)(36 119 173 60)(37 120 174 61)(38 121 175 62)(39 122 176 63)(40 123 177 64)(41 124 178 65)(42 125 179 66)(71 142 111 188)(72 143 112 189)(73 144 99 190)(74 145 100 191)(75 146 101 192)(76 147 102 193)(77 148 103 194)(78 149 104 195)(79 150 105 196)(80 151 106 183)(81 152 107 184)(82 153 108 185)(83 154 109 186)(84 141 110 187)
(1 194 96 148)(2 193 97 147)(3 192 98 146)(4 191 85 145)(5 190 86 144)(6 189 87 143)(7 188 88 142)(8 187 89 141)(9 186 90 154)(10 185 91 153)(11 184 92 152)(12 183 93 151)(13 196 94 150)(14 195 95 149)(15 33 213 170)(16 32 214 169)(17 31 215 182)(18 30 216 181)(19 29 217 180)(20 42 218 179)(21 41 219 178)(22 40 220 177)(23 39 221 176)(24 38 222 175)(25 37 223 174)(26 36 224 173)(27 35 211 172)(28 34 212 171)(43 123 129 64)(44 122 130 63)(45 121 131 62)(46 120 132 61)(47 119 133 60)(48 118 134 59)(49 117 135 58)(50 116 136 57)(51 115 137 70)(52 114 138 69)(53 113 139 68)(54 126 140 67)(55 125 127 66)(56 124 128 65)(71 161 111 208)(72 160 112 207)(73 159 99 206)(74 158 100 205)(75 157 101 204)(76 156 102 203)(77 155 103 202)(78 168 104 201)(79 167 105 200)(80 166 106 199)(81 165 107 198)(82 164 108 197)(83 163 109 210)(84 162 110 209)
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), (1,155,96,202)(2,156,97,203)(3,157,98,204)(4,158,85,205)(5,159,86,206)(6,160,87,207)(7,161,88,208)(8,162,89,209)(9,163,90,210)(10,164,91,197)(11,165,92,198)(12,166,93,199)(13,167,94,200)(14,168,95,201)(15,129,213,43)(16,130,214,44)(17,131,215,45)(18,132,216,46)(19,133,217,47)(20,134,218,48)(21,135,219,49)(22,136,220,50)(23,137,221,51)(24,138,222,52)(25,139,223,53)(26,140,224,54)(27,127,211,55)(28,128,212,56)(29,67,180,126)(30,68,181,113)(31,69,182,114)(32,70,169,115)(33,57,170,116)(34,58,171,117)(35,59,172,118)(36,60,173,119)(37,61,174,120)(38,62,175,121)(39,63,176,122)(40,64,177,123)(41,65,178,124)(42,66,179,125)(71,195,111,149)(72,196,112,150)(73,183,99,151)(74,184,100,152)(75,185,101,153)(76,186,102,154)(77,187,103,141)(78,188,104,142)(79,189,105,143)(80,190,106,144)(81,191,107,145)(82,192,108,146)(83,193,109,147)(84,194,110,148), (1,171,8,178)(2,170,9,177)(3,169,10,176)(4,182,11,175)(5,181,12,174)(6,180,13,173)(7,179,14,172)(15,186,22,193)(16,185,23,192)(17,184,24,191)(18,183,25,190)(19,196,26,189)(20,195,27,188)(21,194,28,187)(29,94,36,87)(30,93,37,86)(31,92,38,85)(32,91,39,98)(33,90,40,97)(34,89,41,96)(35,88,42,95)(43,109,50,102)(44,108,51,101)(45,107,52,100)(46,106,53,99)(47,105,54,112)(48,104,55,111)(49,103,56,110)(57,156,64,163)(58,155,65,162)(59,168,66,161)(60,167,67,160)(61,166,68,159)(62,165,69,158)(63,164,70,157)(71,134,78,127)(72,133,79,140)(73,132,80,139)(74,131,81,138)(75,130,82,137)(76,129,83,136)(77,128,84,135)(113,206,120,199)(114,205,121,198)(115,204,122,197)(116,203,123,210)(117,202,124,209)(118,201,125,208)(119,200,126,207)(141,219,148,212)(142,218,149,211)(143,217,150,224)(144,216,151,223)(145,215,152,222)(146,214,153,221)(147,213,154,220), (1,155,96,202)(2,156,97,203)(3,157,98,204)(4,158,85,205)(5,159,86,206)(6,160,87,207)(7,161,88,208)(8,162,89,209)(9,163,90,210)(10,164,91,197)(11,165,92,198)(12,166,93,199)(13,167,94,200)(14,168,95,201)(15,136,213,50)(16,137,214,51)(17,138,215,52)(18,139,216,53)(19,140,217,54)(20,127,218,55)(21,128,219,56)(22,129,220,43)(23,130,221,44)(24,131,222,45)(25,132,223,46)(26,133,224,47)(27,134,211,48)(28,135,212,49)(29,126,180,67)(30,113,181,68)(31,114,182,69)(32,115,169,70)(33,116,170,57)(34,117,171,58)(35,118,172,59)(36,119,173,60)(37,120,174,61)(38,121,175,62)(39,122,176,63)(40,123,177,64)(41,124,178,65)(42,125,179,66)(71,142,111,188)(72,143,112,189)(73,144,99,190)(74,145,100,191)(75,146,101,192)(76,147,102,193)(77,148,103,194)(78,149,104,195)(79,150,105,196)(80,151,106,183)(81,152,107,184)(82,153,108,185)(83,154,109,186)(84,141,110,187), (1,194,96,148)(2,193,97,147)(3,192,98,146)(4,191,85,145)(5,190,86,144)(6,189,87,143)(7,188,88,142)(8,187,89,141)(9,186,90,154)(10,185,91,153)(11,184,92,152)(12,183,93,151)(13,196,94,150)(14,195,95,149)(15,33,213,170)(16,32,214,169)(17,31,215,182)(18,30,216,181)(19,29,217,180)(20,42,218,179)(21,41,219,178)(22,40,220,177)(23,39,221,176)(24,38,222,175)(25,37,223,174)(26,36,224,173)(27,35,211,172)(28,34,212,171)(43,123,129,64)(44,122,130,63)(45,121,131,62)(46,120,132,61)(47,119,133,60)(48,118,134,59)(49,117,135,58)(50,116,136,57)(51,115,137,70)(52,114,138,69)(53,113,139,68)(54,126,140,67)(55,125,127,66)(56,124,128,65)(71,161,111,208)(72,160,112,207)(73,159,99,206)(74,158,100,205)(75,157,101,204)(76,156,102,203)(77,155,103,202)(78,168,104,201)(79,167,105,200)(80,166,106,199)(81,165,107,198)(82,164,108,197)(83,163,109,210)(84,162,110,209)>;
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), (1,155,96,202)(2,156,97,203)(3,157,98,204)(4,158,85,205)(5,159,86,206)(6,160,87,207)(7,161,88,208)(8,162,89,209)(9,163,90,210)(10,164,91,197)(11,165,92,198)(12,166,93,199)(13,167,94,200)(14,168,95,201)(15,129,213,43)(16,130,214,44)(17,131,215,45)(18,132,216,46)(19,133,217,47)(20,134,218,48)(21,135,219,49)(22,136,220,50)(23,137,221,51)(24,138,222,52)(25,139,223,53)(26,140,224,54)(27,127,211,55)(28,128,212,56)(29,67,180,126)(30,68,181,113)(31,69,182,114)(32,70,169,115)(33,57,170,116)(34,58,171,117)(35,59,172,118)(36,60,173,119)(37,61,174,120)(38,62,175,121)(39,63,176,122)(40,64,177,123)(41,65,178,124)(42,66,179,125)(71,195,111,149)(72,196,112,150)(73,183,99,151)(74,184,100,152)(75,185,101,153)(76,186,102,154)(77,187,103,141)(78,188,104,142)(79,189,105,143)(80,190,106,144)(81,191,107,145)(82,192,108,146)(83,193,109,147)(84,194,110,148), (1,171,8,178)(2,170,9,177)(3,169,10,176)(4,182,11,175)(5,181,12,174)(6,180,13,173)(7,179,14,172)(15,186,22,193)(16,185,23,192)(17,184,24,191)(18,183,25,190)(19,196,26,189)(20,195,27,188)(21,194,28,187)(29,94,36,87)(30,93,37,86)(31,92,38,85)(32,91,39,98)(33,90,40,97)(34,89,41,96)(35,88,42,95)(43,109,50,102)(44,108,51,101)(45,107,52,100)(46,106,53,99)(47,105,54,112)(48,104,55,111)(49,103,56,110)(57,156,64,163)(58,155,65,162)(59,168,66,161)(60,167,67,160)(61,166,68,159)(62,165,69,158)(63,164,70,157)(71,134,78,127)(72,133,79,140)(73,132,80,139)(74,131,81,138)(75,130,82,137)(76,129,83,136)(77,128,84,135)(113,206,120,199)(114,205,121,198)(115,204,122,197)(116,203,123,210)(117,202,124,209)(118,201,125,208)(119,200,126,207)(141,219,148,212)(142,218,149,211)(143,217,150,224)(144,216,151,223)(145,215,152,222)(146,214,153,221)(147,213,154,220), (1,155,96,202)(2,156,97,203)(3,157,98,204)(4,158,85,205)(5,159,86,206)(6,160,87,207)(7,161,88,208)(8,162,89,209)(9,163,90,210)(10,164,91,197)(11,165,92,198)(12,166,93,199)(13,167,94,200)(14,168,95,201)(15,136,213,50)(16,137,214,51)(17,138,215,52)(18,139,216,53)(19,140,217,54)(20,127,218,55)(21,128,219,56)(22,129,220,43)(23,130,221,44)(24,131,222,45)(25,132,223,46)(26,133,224,47)(27,134,211,48)(28,135,212,49)(29,126,180,67)(30,113,181,68)(31,114,182,69)(32,115,169,70)(33,116,170,57)(34,117,171,58)(35,118,172,59)(36,119,173,60)(37,120,174,61)(38,121,175,62)(39,122,176,63)(40,123,177,64)(41,124,178,65)(42,125,179,66)(71,142,111,188)(72,143,112,189)(73,144,99,190)(74,145,100,191)(75,146,101,192)(76,147,102,193)(77,148,103,194)(78,149,104,195)(79,150,105,196)(80,151,106,183)(81,152,107,184)(82,153,108,185)(83,154,109,186)(84,141,110,187), (1,194,96,148)(2,193,97,147)(3,192,98,146)(4,191,85,145)(5,190,86,144)(6,189,87,143)(7,188,88,142)(8,187,89,141)(9,186,90,154)(10,185,91,153)(11,184,92,152)(12,183,93,151)(13,196,94,150)(14,195,95,149)(15,33,213,170)(16,32,214,169)(17,31,215,182)(18,30,216,181)(19,29,217,180)(20,42,218,179)(21,41,219,178)(22,40,220,177)(23,39,221,176)(24,38,222,175)(25,37,223,174)(26,36,224,173)(27,35,211,172)(28,34,212,171)(43,123,129,64)(44,122,130,63)(45,121,131,62)(46,120,132,61)(47,119,133,60)(48,118,134,59)(49,117,135,58)(50,116,136,57)(51,115,137,70)(52,114,138,69)(53,113,139,68)(54,126,140,67)(55,125,127,66)(56,124,128,65)(71,161,111,208)(72,160,112,207)(73,159,99,206)(74,158,100,205)(75,157,101,204)(76,156,102,203)(77,155,103,202)(78,168,104,201)(79,167,105,200)(80,166,106,199)(81,165,107,198)(82,164,108,197)(83,163,109,210)(84,162,110,209) );
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)], [(1,155,96,202),(2,156,97,203),(3,157,98,204),(4,158,85,205),(5,159,86,206),(6,160,87,207),(7,161,88,208),(8,162,89,209),(9,163,90,210),(10,164,91,197),(11,165,92,198),(12,166,93,199),(13,167,94,200),(14,168,95,201),(15,129,213,43),(16,130,214,44),(17,131,215,45),(18,132,216,46),(19,133,217,47),(20,134,218,48),(21,135,219,49),(22,136,220,50),(23,137,221,51),(24,138,222,52),(25,139,223,53),(26,140,224,54),(27,127,211,55),(28,128,212,56),(29,67,180,126),(30,68,181,113),(31,69,182,114),(32,70,169,115),(33,57,170,116),(34,58,171,117),(35,59,172,118),(36,60,173,119),(37,61,174,120),(38,62,175,121),(39,63,176,122),(40,64,177,123),(41,65,178,124),(42,66,179,125),(71,195,111,149),(72,196,112,150),(73,183,99,151),(74,184,100,152),(75,185,101,153),(76,186,102,154),(77,187,103,141),(78,188,104,142),(79,189,105,143),(80,190,106,144),(81,191,107,145),(82,192,108,146),(83,193,109,147),(84,194,110,148)], [(1,171,8,178),(2,170,9,177),(3,169,10,176),(4,182,11,175),(5,181,12,174),(6,180,13,173),(7,179,14,172),(15,186,22,193),(16,185,23,192),(17,184,24,191),(18,183,25,190),(19,196,26,189),(20,195,27,188),(21,194,28,187),(29,94,36,87),(30,93,37,86),(31,92,38,85),(32,91,39,98),(33,90,40,97),(34,89,41,96),(35,88,42,95),(43,109,50,102),(44,108,51,101),(45,107,52,100),(46,106,53,99),(47,105,54,112),(48,104,55,111),(49,103,56,110),(57,156,64,163),(58,155,65,162),(59,168,66,161),(60,167,67,160),(61,166,68,159),(62,165,69,158),(63,164,70,157),(71,134,78,127),(72,133,79,140),(73,132,80,139),(74,131,81,138),(75,130,82,137),(76,129,83,136),(77,128,84,135),(113,206,120,199),(114,205,121,198),(115,204,122,197),(116,203,123,210),(117,202,124,209),(118,201,125,208),(119,200,126,207),(141,219,148,212),(142,218,149,211),(143,217,150,224),(144,216,151,223),(145,215,152,222),(146,214,153,221),(147,213,154,220)], [(1,155,96,202),(2,156,97,203),(3,157,98,204),(4,158,85,205),(5,159,86,206),(6,160,87,207),(7,161,88,208),(8,162,89,209),(9,163,90,210),(10,164,91,197),(11,165,92,198),(12,166,93,199),(13,167,94,200),(14,168,95,201),(15,136,213,50),(16,137,214,51),(17,138,215,52),(18,139,216,53),(19,140,217,54),(20,127,218,55),(21,128,219,56),(22,129,220,43),(23,130,221,44),(24,131,222,45),(25,132,223,46),(26,133,224,47),(27,134,211,48),(28,135,212,49),(29,126,180,67),(30,113,181,68),(31,114,182,69),(32,115,169,70),(33,116,170,57),(34,117,171,58),(35,118,172,59),(36,119,173,60),(37,120,174,61),(38,121,175,62),(39,122,176,63),(40,123,177,64),(41,124,178,65),(42,125,179,66),(71,142,111,188),(72,143,112,189),(73,144,99,190),(74,145,100,191),(75,146,101,192),(76,147,102,193),(77,148,103,194),(78,149,104,195),(79,150,105,196),(80,151,106,183),(81,152,107,184),(82,153,108,185),(83,154,109,186),(84,141,110,187)], [(1,194,96,148),(2,193,97,147),(3,192,98,146),(4,191,85,145),(5,190,86,144),(6,189,87,143),(7,188,88,142),(8,187,89,141),(9,186,90,154),(10,185,91,153),(11,184,92,152),(12,183,93,151),(13,196,94,150),(14,195,95,149),(15,33,213,170),(16,32,214,169),(17,31,215,182),(18,30,216,181),(19,29,217,180),(20,42,218,179),(21,41,219,178),(22,40,220,177),(23,39,221,176),(24,38,222,175),(25,37,223,174),(26,36,224,173),(27,35,211,172),(28,34,212,171),(43,123,129,64),(44,122,130,63),(45,121,131,62),(46,120,132,61),(47,119,133,60),(48,118,134,59),(49,117,135,58),(50,116,136,57),(51,115,137,70),(52,114,138,69),(53,113,139,68),(54,126,140,67),(55,125,127,66),(56,124,128,65),(71,161,111,208),(72,160,112,207),(73,159,99,206),(74,158,100,205),(75,157,101,204),(76,156,102,203),(77,155,103,202),(78,168,104,201),(79,167,105,200),(80,166,106,199),(81,165,107,198),(82,164,108,197),(83,163,109,210),(84,162,110,209)])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ 1+4 | 2- 1+4 | D4⋊6D14 | Q8.10D14 | D4.10D14 |
| kernel | C14.252- 1+4 | C23.D14 | D14.D4 | Dic7.D4 | C28⋊Q8 | Dic7.Q8 | C4.Dic14 | D14⋊Q8 | C4⋊C4⋊D7 | C28.48D4 | C23.23D14 | Dic7⋊Q8 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 6 | 9 | 3 | 3 | 1 | 2 | 6 | 6 | 6 |
Matrix representation of C14.252- 1+4 ►in GL8(𝔽29)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 0 | 22 | 0 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 13 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 26 | 0 | 9 | 17 |
| 1 | 0 | 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 8 | 28 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 0 | 27 | 15 |
| 0 | 0 | 0 | 0 | 19 | 0 | 1 | 20 |
| 0 | 0 | 0 | 0 | 0 | 5 | 9 | 5 |
| 0 | 0 | 0 | 0 | 10 | 20 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 20 | 12 |
| 28 | 0 | 0 | 22 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 1 | 0 | 0 | 0 | 0 |
| 8 | 28 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 9 | 5 |
| 0 | 0 | 0 | 0 | 19 | 0 | 1 | 20 |
G:=sub<GL(8,GF(29))| [4,0,14,15,0,0,0,0,0,4,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,9,14,0,15,0,0,0,0,0,13,0,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,13],[1,0,21,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,17,26,21,26,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,9,0,0,0,0,0,0,0,17],[1,0,21,8,0,0,0,0,0,0,0,28,0,0,0,0,22,1,28,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,20,19,0,10,0,0,0,0,0,0,5,20,0,0,0,0,27,1,9,0,0,0,0,0,15,20,5,0],[1,8,21,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,26,0,3,0,0,0,0,0,12,0,0,0,0,0,0,0,0,17,20,0,0,0,0,0,0,0,12],[28,21,8,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,22,1,28,1,0,0,0,0,0,0,0,0,9,10,0,19,0,0,0,0,15,20,5,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,5,20] >;
C14.252- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{25}2_-^{1+4} % in TeX
G:=Group("C14.25ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1099);
// by ID
G=gap.SmallGroup(448,1099);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=a^7*b^-1,b*d=d*b,e*b*e^-1=a^7*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations