G = C14.162- 1+4 order 448 = 26·7
16th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.162- 1+4, C28⋊Q8⋊24C2, C22⋊Q8⋊5D7, C4⋊C4.95D14, (C4×D7).11D4, C4.187(D4×D7), D14.43(C2×D4), C28.232(C2×D4), Dic7.7(C2×D4), D14⋊3Q8⋊14C2, D14⋊2Q8⋊24C2, (C2×C28).52C23, (C2×Q8).124D14, C22⋊C4.14D14, C14.74(C22×D4), D14.D4⋊24C2, D14.5D4⋊16C2, C28.48D4⋊36C2, (C2×C14).172C24, D14⋊C4.21C22, (C22×C4).234D14, Dic7.D4⋊24C2, (C2×D28).221C22, Dic7⋊C4.25C22, C4⋊Dic7.213C22, (Q8×C14).105C22, (C2×Dic7).87C23, C22.193(C23×D7), C23.117(C22×D7), C23.D7.33C22, (C22×C28).252C22, (C22×C14).200C23, C7⋊2(C23.38C23), (C4×Dic7).104C22, (C22×D7).194C23, C2.35(D4.10D14), C2.17(Q8.10D14), (C2×Dic14).246C22, (C2×Q8×D7)⋊6C2, C2.47(C2×D4×D7), C4⋊C4⋊7D7⋊25C2, (C7×C22⋊Q8)⋊8C2, (C2×C4×D7).93C22, (C2×C4○D28).20C2, (C7×C4⋊C4).156C22, (C2×C4).590(C22×D7), (C2×C7⋊D4).120C22, (C7×C22⋊C4).27C22, SmallGroup(448,1081)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.162- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=b2, ab=ba, cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=a7b, be=eb, dcd-1=a7c, ce=ec, ede-1=b2d >
Subgroups: 1244 in 270 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C42⋊C2, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, Dic14, C4×D7, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×C14, C23.38C23, C4×Dic7, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C4○D28, Q8×D7, C2×C7⋊D4, C22×C28, Q8×C14, D14.D4, Dic7.D4, C28⋊Q8, C4⋊C4⋊7D7, D14.5D4, D14⋊2Q8, C28.48D4, D14⋊3Q8, C7×C22⋊Q8, C2×C4○D28, C2×Q8×D7, C14.162- 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22×D4, 2- 1+4, C22×D7, C23.38C23, D4×D7, C23×D7, C2×D4×D7, Q8.10D14, D4.10D14, C14.162- 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)
(1 159 19 153)(2 160 20 154)(3 161 21 141)(4 162 22 142)(5 163 23 143)(6 164 24 144)(7 165 25 145)(8 166 26 146)(9 167 27 147)(10 168 28 148)(11 155 15 149)(12 156 16 150)(13 157 17 151)(14 158 18 152)(29 133 49 117)(30 134 50 118)(31 135 51 119)(32 136 52 120)(33 137 53 121)(34 138 54 122)(35 139 55 123)(36 140 56 124)(37 127 43 125)(38 128 44 126)(39 129 45 113)(40 130 46 114)(41 131 47 115)(42 132 48 116)(57 218 80 207)(58 219 81 208)(59 220 82 209)(60 221 83 210)(61 222 84 197)(62 223 71 198)(63 224 72 199)(64 211 73 200)(65 212 74 201)(66 213 75 202)(67 214 76 203)(68 215 77 204)(69 216 78 205)(70 217 79 206)(85 193 104 171)(86 194 105 172)(87 195 106 173)(88 196 107 174)(89 183 108 175)(90 184 109 176)(91 185 110 177)(92 186 111 178)(93 187 112 179)(94 188 99 180)(95 189 100 181)(96 190 101 182)(97 191 102 169)(98 192 103 170)
(1 153)(2 152)(3 151)(4 150)(5 149)(6 148)(7 147)(8 146)(9 145)(10 144)(11 143)(12 142)(13 141)(14 154)(15 163)(16 162)(17 161)(18 160)(19 159)(20 158)(21 157)(22 156)(23 155)(24 168)(25 167)(26 166)(27 165)(28 164)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 140)(37 139)(38 138)(39 137)(40 136)(41 135)(42 134)(43 123)(44 122)(45 121)(46 120)(47 119)(48 118)(49 117)(50 116)(51 115)(52 114)(53 113)(54 126)(55 125)(56 124)(57 199)(58 198)(59 197)(60 210)(61 209)(62 208)(63 207)(64 206)(65 205)(66 204)(67 203)(68 202)(69 201)(70 200)(71 219)(72 218)(73 217)(74 216)(75 215)(76 214)(77 213)(78 212)(79 211)(80 224)(81 223)(82 222)(83 221)(84 220)(85 183)(86 196)(87 195)(88 194)(89 193)(90 192)(91 191)(92 190)(93 189)(94 188)(95 187)(96 186)(97 185)(98 184)(99 180)(100 179)(101 178)(102 177)(103 176)(104 175)(105 174)(106 173)(107 172)(108 171)(109 170)(110 169)(111 182)(112 181)
(1 60 19 83)(2 59 20 82)(3 58 21 81)(4 57 22 80)(5 70 23 79)(6 69 24 78)(7 68 25 77)(8 67 26 76)(9 66 27 75)(10 65 28 74)(11 64 15 73)(12 63 16 72)(13 62 17 71)(14 61 18 84)(29 99 49 94)(30 112 50 93)(31 111 51 92)(32 110 52 91)(33 109 53 90)(34 108 54 89)(35 107 55 88)(36 106 56 87)(37 105 43 86)(38 104 44 85)(39 103 45 98)(40 102 46 97)(41 101 47 96)(42 100 48 95)(113 185 129 177)(114 184 130 176)(115 183 131 175)(116 196 132 174)(117 195 133 173)(118 194 134 172)(119 193 135 171)(120 192 136 170)(121 191 137 169)(122 190 138 182)(123 189 139 181)(124 188 140 180)(125 187 127 179)(126 186 128 178)(141 201 161 212)(142 200 162 211)(143 199 163 224)(144 198 164 223)(145 197 165 222)(146 210 166 221)(147 209 167 220)(148 208 168 219)(149 207 155 218)(150 206 156 217)(151 205 157 216)(152 204 158 215)(153 203 159 214)(154 202 160 213)
(1 133 19 117)(2 134 20 118)(3 135 21 119)(4 136 22 120)(5 137 23 121)(6 138 24 122)(7 139 25 123)(8 140 26 124)(9 127 27 125)(10 128 28 126)(11 129 15 113)(12 130 16 114)(13 131 17 115)(14 132 18 116)(29 159 49 153)(30 160 50 154)(31 161 51 141)(32 162 52 142)(33 163 53 143)(34 164 54 144)(35 165 55 145)(36 166 56 146)(37 167 43 147)(38 168 44 148)(39 155 45 149)(40 156 46 150)(41 157 47 151)(42 158 48 152)(57 192 80 170)(58 193 81 171)(59 194 82 172)(60 195 83 173)(61 196 84 174)(62 183 71 175)(63 184 72 176)(64 185 73 177)(65 186 74 178)(66 187 75 179)(67 188 76 180)(68 189 77 181)(69 190 78 182)(70 191 79 169)(85 219 104 208)(86 220 105 209)(87 221 106 210)(88 222 107 197)(89 223 108 198)(90 224 109 199)(91 211 110 200)(92 212 111 201)(93 213 112 202)(94 214 99 203)(95 215 100 204)(96 216 101 205)(97 217 102 206)(98 218 103 207)
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), (1,159,19,153)(2,160,20,154)(3,161,21,141)(4,162,22,142)(5,163,23,143)(6,164,24,144)(7,165,25,145)(8,166,26,146)(9,167,27,147)(10,168,28,148)(11,155,15,149)(12,156,16,150)(13,157,17,151)(14,158,18,152)(29,133,49,117)(30,134,50,118)(31,135,51,119)(32,136,52,120)(33,137,53,121)(34,138,54,122)(35,139,55,123)(36,140,56,124)(37,127,43,125)(38,128,44,126)(39,129,45,113)(40,130,46,114)(41,131,47,115)(42,132,48,116)(57,218,80,207)(58,219,81,208)(59,220,82,209)(60,221,83,210)(61,222,84,197)(62,223,71,198)(63,224,72,199)(64,211,73,200)(65,212,74,201)(66,213,75,202)(67,214,76,203)(68,215,77,204)(69,216,78,205)(70,217,79,206)(85,193,104,171)(86,194,105,172)(87,195,106,173)(88,196,107,174)(89,183,108,175)(90,184,109,176)(91,185,110,177)(92,186,111,178)(93,187,112,179)(94,188,99,180)(95,189,100,181)(96,190,101,182)(97,191,102,169)(98,192,103,170), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,154)(15,163)(16,162)(17,161)(18,160)(19,159)(20,158)(21,157)(22,156)(23,155)(24,168)(25,167)(26,166)(27,165)(28,164)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,123)(44,122)(45,121)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,126)(55,125)(56,124)(57,199)(58,198)(59,197)(60,210)(61,209)(62,208)(63,207)(64,206)(65,205)(66,204)(67,203)(68,202)(69,201)(70,200)(71,219)(72,218)(73,217)(74,216)(75,215)(76,214)(77,213)(78,212)(79,211)(80,224)(81,223)(82,222)(83,221)(84,220)(85,183)(86,196)(87,195)(88,194)(89,193)(90,192)(91,191)(92,190)(93,189)(94,188)(95,187)(96,186)(97,185)(98,184)(99,180)(100,179)(101,178)(102,177)(103,176)(104,175)(105,174)(106,173)(107,172)(108,171)(109,170)(110,169)(111,182)(112,181), (1,60,19,83)(2,59,20,82)(3,58,21,81)(4,57,22,80)(5,70,23,79)(6,69,24,78)(7,68,25,77)(8,67,26,76)(9,66,27,75)(10,65,28,74)(11,64,15,73)(12,63,16,72)(13,62,17,71)(14,61,18,84)(29,99,49,94)(30,112,50,93)(31,111,51,92)(32,110,52,91)(33,109,53,90)(34,108,54,89)(35,107,55,88)(36,106,56,87)(37,105,43,86)(38,104,44,85)(39,103,45,98)(40,102,46,97)(41,101,47,96)(42,100,48,95)(113,185,129,177)(114,184,130,176)(115,183,131,175)(116,196,132,174)(117,195,133,173)(118,194,134,172)(119,193,135,171)(120,192,136,170)(121,191,137,169)(122,190,138,182)(123,189,139,181)(124,188,140,180)(125,187,127,179)(126,186,128,178)(141,201,161,212)(142,200,162,211)(143,199,163,224)(144,198,164,223)(145,197,165,222)(146,210,166,221)(147,209,167,220)(148,208,168,219)(149,207,155,218)(150,206,156,217)(151,205,157,216)(152,204,158,215)(153,203,159,214)(154,202,160,213), (1,133,19,117)(2,134,20,118)(3,135,21,119)(4,136,22,120)(5,137,23,121)(6,138,24,122)(7,139,25,123)(8,140,26,124)(9,127,27,125)(10,128,28,126)(11,129,15,113)(12,130,16,114)(13,131,17,115)(14,132,18,116)(29,159,49,153)(30,160,50,154)(31,161,51,141)(32,162,52,142)(33,163,53,143)(34,164,54,144)(35,165,55,145)(36,166,56,146)(37,167,43,147)(38,168,44,148)(39,155,45,149)(40,156,46,150)(41,157,47,151)(42,158,48,152)(57,192,80,170)(58,193,81,171)(59,194,82,172)(60,195,83,173)(61,196,84,174)(62,183,71,175)(63,184,72,176)(64,185,73,177)(65,186,74,178)(66,187,75,179)(67,188,76,180)(68,189,77,181)(69,190,78,182)(70,191,79,169)(85,219,104,208)(86,220,105,209)(87,221,106,210)(88,222,107,197)(89,223,108,198)(90,224,109,199)(91,211,110,200)(92,212,111,201)(93,213,112,202)(94,214,99,203)(95,215,100,204)(96,216,101,205)(97,217,102,206)(98,218,103,207)>;
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), (1,159,19,153)(2,160,20,154)(3,161,21,141)(4,162,22,142)(5,163,23,143)(6,164,24,144)(7,165,25,145)(8,166,26,146)(9,167,27,147)(10,168,28,148)(11,155,15,149)(12,156,16,150)(13,157,17,151)(14,158,18,152)(29,133,49,117)(30,134,50,118)(31,135,51,119)(32,136,52,120)(33,137,53,121)(34,138,54,122)(35,139,55,123)(36,140,56,124)(37,127,43,125)(38,128,44,126)(39,129,45,113)(40,130,46,114)(41,131,47,115)(42,132,48,116)(57,218,80,207)(58,219,81,208)(59,220,82,209)(60,221,83,210)(61,222,84,197)(62,223,71,198)(63,224,72,199)(64,211,73,200)(65,212,74,201)(66,213,75,202)(67,214,76,203)(68,215,77,204)(69,216,78,205)(70,217,79,206)(85,193,104,171)(86,194,105,172)(87,195,106,173)(88,196,107,174)(89,183,108,175)(90,184,109,176)(91,185,110,177)(92,186,111,178)(93,187,112,179)(94,188,99,180)(95,189,100,181)(96,190,101,182)(97,191,102,169)(98,192,103,170), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,154)(15,163)(16,162)(17,161)(18,160)(19,159)(20,158)(21,157)(22,156)(23,155)(24,168)(25,167)(26,166)(27,165)(28,164)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,123)(44,122)(45,121)(46,120)(47,119)(48,118)(49,117)(50,116)(51,115)(52,114)(53,113)(54,126)(55,125)(56,124)(57,199)(58,198)(59,197)(60,210)(61,209)(62,208)(63,207)(64,206)(65,205)(66,204)(67,203)(68,202)(69,201)(70,200)(71,219)(72,218)(73,217)(74,216)(75,215)(76,214)(77,213)(78,212)(79,211)(80,224)(81,223)(82,222)(83,221)(84,220)(85,183)(86,196)(87,195)(88,194)(89,193)(90,192)(91,191)(92,190)(93,189)(94,188)(95,187)(96,186)(97,185)(98,184)(99,180)(100,179)(101,178)(102,177)(103,176)(104,175)(105,174)(106,173)(107,172)(108,171)(109,170)(110,169)(111,182)(112,181), (1,60,19,83)(2,59,20,82)(3,58,21,81)(4,57,22,80)(5,70,23,79)(6,69,24,78)(7,68,25,77)(8,67,26,76)(9,66,27,75)(10,65,28,74)(11,64,15,73)(12,63,16,72)(13,62,17,71)(14,61,18,84)(29,99,49,94)(30,112,50,93)(31,111,51,92)(32,110,52,91)(33,109,53,90)(34,108,54,89)(35,107,55,88)(36,106,56,87)(37,105,43,86)(38,104,44,85)(39,103,45,98)(40,102,46,97)(41,101,47,96)(42,100,48,95)(113,185,129,177)(114,184,130,176)(115,183,131,175)(116,196,132,174)(117,195,133,173)(118,194,134,172)(119,193,135,171)(120,192,136,170)(121,191,137,169)(122,190,138,182)(123,189,139,181)(124,188,140,180)(125,187,127,179)(126,186,128,178)(141,201,161,212)(142,200,162,211)(143,199,163,224)(144,198,164,223)(145,197,165,222)(146,210,166,221)(147,209,167,220)(148,208,168,219)(149,207,155,218)(150,206,156,217)(151,205,157,216)(152,204,158,215)(153,203,159,214)(154,202,160,213), (1,133,19,117)(2,134,20,118)(3,135,21,119)(4,136,22,120)(5,137,23,121)(6,138,24,122)(7,139,25,123)(8,140,26,124)(9,127,27,125)(10,128,28,126)(11,129,15,113)(12,130,16,114)(13,131,17,115)(14,132,18,116)(29,159,49,153)(30,160,50,154)(31,161,51,141)(32,162,52,142)(33,163,53,143)(34,164,54,144)(35,165,55,145)(36,166,56,146)(37,167,43,147)(38,168,44,148)(39,155,45,149)(40,156,46,150)(41,157,47,151)(42,158,48,152)(57,192,80,170)(58,193,81,171)(59,194,82,172)(60,195,83,173)(61,196,84,174)(62,183,71,175)(63,184,72,176)(64,185,73,177)(65,186,74,178)(66,187,75,179)(67,188,76,180)(68,189,77,181)(69,190,78,182)(70,191,79,169)(85,219,104,208)(86,220,105,209)(87,221,106,210)(88,222,107,197)(89,223,108,198)(90,224,109,199)(91,211,110,200)(92,212,111,201)(93,213,112,202)(94,214,99,203)(95,215,100,204)(96,216,101,205)(97,217,102,206)(98,218,103,207) );
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)], [(1,159,19,153),(2,160,20,154),(3,161,21,141),(4,162,22,142),(5,163,23,143),(6,164,24,144),(7,165,25,145),(8,166,26,146),(9,167,27,147),(10,168,28,148),(11,155,15,149),(12,156,16,150),(13,157,17,151),(14,158,18,152),(29,133,49,117),(30,134,50,118),(31,135,51,119),(32,136,52,120),(33,137,53,121),(34,138,54,122),(35,139,55,123),(36,140,56,124),(37,127,43,125),(38,128,44,126),(39,129,45,113),(40,130,46,114),(41,131,47,115),(42,132,48,116),(57,218,80,207),(58,219,81,208),(59,220,82,209),(60,221,83,210),(61,222,84,197),(62,223,71,198),(63,224,72,199),(64,211,73,200),(65,212,74,201),(66,213,75,202),(67,214,76,203),(68,215,77,204),(69,216,78,205),(70,217,79,206),(85,193,104,171),(86,194,105,172),(87,195,106,173),(88,196,107,174),(89,183,108,175),(90,184,109,176),(91,185,110,177),(92,186,111,178),(93,187,112,179),(94,188,99,180),(95,189,100,181),(96,190,101,182),(97,191,102,169),(98,192,103,170)], [(1,153),(2,152),(3,151),(4,150),(5,149),(6,148),(7,147),(8,146),(9,145),(10,144),(11,143),(12,142),(13,141),(14,154),(15,163),(16,162),(17,161),(18,160),(19,159),(20,158),(21,157),(22,156),(23,155),(24,168),(25,167),(26,166),(27,165),(28,164),(29,133),(30,132),(31,131),(32,130),(33,129),(34,128),(35,127),(36,140),(37,139),(38,138),(39,137),(40,136),(41,135),(42,134),(43,123),(44,122),(45,121),(46,120),(47,119),(48,118),(49,117),(50,116),(51,115),(52,114),(53,113),(54,126),(55,125),(56,124),(57,199),(58,198),(59,197),(60,210),(61,209),(62,208),(63,207),(64,206),(65,205),(66,204),(67,203),(68,202),(69,201),(70,200),(71,219),(72,218),(73,217),(74,216),(75,215),(76,214),(77,213),(78,212),(79,211),(80,224),(81,223),(82,222),(83,221),(84,220),(85,183),(86,196),(87,195),(88,194),(89,193),(90,192),(91,191),(92,190),(93,189),(94,188),(95,187),(96,186),(97,185),(98,184),(99,180),(100,179),(101,178),(102,177),(103,176),(104,175),(105,174),(106,173),(107,172),(108,171),(109,170),(110,169),(111,182),(112,181)], [(1,60,19,83),(2,59,20,82),(3,58,21,81),(4,57,22,80),(5,70,23,79),(6,69,24,78),(7,68,25,77),(8,67,26,76),(9,66,27,75),(10,65,28,74),(11,64,15,73),(12,63,16,72),(13,62,17,71),(14,61,18,84),(29,99,49,94),(30,112,50,93),(31,111,51,92),(32,110,52,91),(33,109,53,90),(34,108,54,89),(35,107,55,88),(36,106,56,87),(37,105,43,86),(38,104,44,85),(39,103,45,98),(40,102,46,97),(41,101,47,96),(42,100,48,95),(113,185,129,177),(114,184,130,176),(115,183,131,175),(116,196,132,174),(117,195,133,173),(118,194,134,172),(119,193,135,171),(120,192,136,170),(121,191,137,169),(122,190,138,182),(123,189,139,181),(124,188,140,180),(125,187,127,179),(126,186,128,178),(141,201,161,212),(142,200,162,211),(143,199,163,224),(144,198,164,223),(145,197,165,222),(146,210,166,221),(147,209,167,220),(148,208,168,219),(149,207,155,218),(150,206,156,217),(151,205,157,216),(152,204,158,215),(153,203,159,214),(154,202,160,213)], [(1,133,19,117),(2,134,20,118),(3,135,21,119),(4,136,22,120),(5,137,23,121),(6,138,24,122),(7,139,25,123),(8,140,26,124),(9,127,27,125),(10,128,28,126),(11,129,15,113),(12,130,16,114),(13,131,17,115),(14,132,18,116),(29,159,49,153),(30,160,50,154),(31,161,51,141),(32,162,52,142),(33,163,53,143),(34,164,54,144),(35,165,55,145),(36,166,56,146),(37,167,43,147),(38,168,44,148),(39,155,45,149),(40,156,46,150),(41,157,47,151),(42,158,48,152),(57,192,80,170),(58,193,81,171),(59,194,82,172),(60,195,83,173),(61,196,84,174),(62,183,71,175),(63,184,72,176),(64,185,73,177),(65,186,74,178),(66,187,75,179),(67,188,76,180),(68,189,77,181),(69,190,78,182),(70,191,79,169),(85,219,104,208),(86,220,105,209),(87,221,106,210),(88,222,107,197),(89,223,108,198),(90,224,109,199),(91,211,110,200),(92,212,111,201),(93,213,112,202),(94,214,99,203),(95,215,100,204),(96,216,101,205),(97,217,102,206),(98,218,103,207)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 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 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | ··· | 4 | 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 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2- 1+4 | D4×D7 | Q8.10D14 | D4.10D14 |
| kernel | C14.162- 1+4 | D14.D4 | Dic7.D4 | C28⋊Q8 | C4⋊C4⋊7D7 | D14.5D4 | D14⋊2Q8 | C28.48D4 | D14⋊3Q8 | C7×C22⋊Q8 | C2×C4○D28 | C2×Q8×D7 | C4×D7 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 6 | 9 | 3 | 3 | 2 | 6 | 6 | 6 |
Matrix representation of C14.162- 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 19 | 0 | 0 |
| 0 | 0 | 15 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 19 |
| 0 | 0 | 0 | 0 | 15 | 1 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 6 | 0 | 0 |
| 0 | 0 | 20 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 6 |
| 0 | 0 | 0 | 0 | 20 | 13 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 4 | 0 | 0 |
| 0 | 0 | 23 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 4 |
| 0 | 0 | 0 | 0 | 23 | 5 |
| 23 | 21 | 0 | 0 | 0 | 0 |
| 8 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,15,0,0,0,0,19,1,0,0,0,0,0,0,25,15,0,0,0,0,19,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,16,20,0,0,0,0,6,13,0,0,0,0,0,0,16,20,0,0,0,0,6,13],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,24,23,0,0,0,0,4,5,0,0,0,0,0,0,24,23,0,0,0,0,4,5],[23,8,0,0,0,0,21,6,0,0,0,0,0,0,0,0,0,25,0,0,0,0,7,0,0,0,0,4,0,0,0,0,22,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,0,0,0,0,0,17] >;
C14.162- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{16}2_-^{1+4} % in TeX
G:=Group("C14.16ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1081);
// by ID
G=gap.SmallGroup(448,1081);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,100,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,c*a*c=d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations