?
G = C14.262- (1+4) order 448 = 26·7
26th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.262- (1+4), C14.592+ (1+4), C22⋊Q8⋊24D7, C28⋊1D4⋊28C2, C28⋊7D4⋊47C2, C4⋊C4.102D14, (C2×Q8).80D14, D14⋊D4⋊27C2, D14⋊2Q8⋊30C2, D14⋊3Q8⋊26C2, D14⋊Q8⋊25C2, Dic7.Q8⋊23C2, C22⋊C4.24D14, D14.5D4⋊24C2, D14.D4⋊28C2, C28.23D4⋊18C2, (C2×C28).177C23, (C2×C14).191C24, D14⋊C4.29C22, (C22×C4).253D14, C4⋊Dic7.48C22, C2.39(D4⋊8D14), C2.61(D4⋊6D14), Dic7.D4⋊28C2, (C2×D28).154C22, Dic7⋊C4.36C22, (Q8×C14).120C22, (C2×Dic7).97C23, (C22×D7).82C23, C23.127(C22×D7), C22.212(C23×D7), C23.D7.37C22, C23.23D14⋊14C2, (C22×C28).319C22, (C22×C14).219C23, C7⋊2(C22.56C24), (C4×Dic7).118C22, C2.27(Q8.10D14), (C2×Dic14).165C22, (C7×C22⋊Q8)⋊27C2, (C2×C4×D7).107C22, (C2×C4).57(C22×D7), (C7×C4⋊C4).171C22, (C2×C7⋊D4).43C22, (C7×C22⋊C4).46C22, SmallGroup(448,1100)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1148 in 220 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22, C22 [×12], C7, C2×C4 [×6], C2×C4 [×9], D4 [×6], Q8 [×2], C23, C23 [×3], D7 [×3], C14 [×3], C14, C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4, C22×C4 [×3], C2×D4 [×6], C2×Q8, C2×Q8, Dic7 [×5], C28 [×6], D14 [×9], C2×C14, C2×C14 [×3], C4⋊D4 [×4], C22⋊Q8, C22⋊Q8 [×3], C22.D4 [×4], C4.4D4 [×2], C42.C2, Dic14, C4×D7 [×3], D28 [×3], C2×Dic7 [×5], C7⋊D4 [×3], C2×C28 [×6], C2×C28, C7×Q8, C22×D7 [×3], C22×C14, C22.56C24, C4×Dic7, Dic7⋊C4 [×5], C4⋊Dic7 [×2], D14⋊C4 [×9], C23.D7, C7×C22⋊C4 [×2], C7×C4⋊C4 [×3], C2×Dic14, C2×C4×D7 [×3], C2×D28 [×3], C2×C7⋊D4 [×3], C22×C28, Q8×C14, D14.D4, D14⋊D4 [×2], Dic7.D4, Dic7.Q8, D14.5D4 [×2], C28⋊1D4, D14⋊Q8, D14⋊2Q8, C23.23D14, C28⋊7D4, D14⋊3Q8, C28.23D4, C7×C22⋊Q8, C14.262- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4) [×2], 2- (1+4), C22×D7 [×7], C22.56C24, C23×D7, D4⋊6D14, Q8.10D14, D4⋊8D14, C14.262- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7b2, ab=ba, cac=dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a7b, cd=dc, ece-1=a7c, ede-1=a7b2d >
►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 76 23 63)(2 77 24 64)(3 78 25 65)(4 79 26 66)(5 80 27 67)(6 81 28 68)(7 82 15 69)(8 83 16 70)(9 84 17 57)(10 71 18 58)(11 72 19 59)(12 73 20 60)(13 74 21 61)(14 75 22 62)(29 100 56 94)(30 101 43 95)(31 102 44 96)(32 103 45 97)(33 104 46 98)(34 105 47 85)(35 106 48 86)(36 107 49 87)(37 108 50 88)(38 109 51 89)(39 110 52 90)(40 111 53 91)(41 112 54 92)(42 99 55 93)(113 182 134 191)(114 169 135 192)(115 170 136 193)(116 171 137 194)(117 172 138 195)(118 173 139 196)(119 174 140 183)(120 175 127 184)(121 176 128 185)(122 177 129 186)(123 178 130 187)(124 179 131 188)(125 180 132 189)(126 181 133 190)(141 199 162 215)(142 200 163 216)(143 201 164 217)(144 202 165 218)(145 203 166 219)(146 204 167 220)(147 205 168 221)(148 206 155 222)(149 207 156 223)(150 208 157 224)(151 209 158 211)(152 210 159 212)(153 197 160 213)(154 198 161 214)
(1 194)(2 193)(3 192)(4 191)(5 190)(6 189)(7 188)(8 187)(9 186)(10 185)(11 184)(12 183)(13 196)(14 195)(15 179)(16 178)(17 177)(18 176)(19 175)(20 174)(21 173)(22 172)(23 171)(24 170)(25 169)(26 182)(27 181)(28 180)(29 214)(30 213)(31 212)(32 211)(33 224)(34 223)(35 222)(36 221)(37 220)(38 219)(39 218)(40 217)(41 216)(42 215)(43 197)(44 210)(45 209)(46 208)(47 207)(48 206)(49 205)(50 204)(51 203)(52 202)(53 201)(54 200)(55 199)(56 198)(57 122)(58 121)(59 120)(60 119)(61 118)(62 117)(63 116)(64 115)(65 114)(66 113)(67 126)(68 125)(69 124)(70 123)(71 128)(72 127)(73 140)(74 139)(75 138)(76 137)(77 136)(78 135)(79 134)(80 133)(81 132)(82 131)(83 130)(84 129)(85 149)(86 148)(87 147)(88 146)(89 145)(90 144)(91 143)(92 142)(93 141)(94 154)(95 153)(96 152)(97 151)(98 150)(99 162)(100 161)(101 160)(102 159)(103 158)(104 157)(105 156)(106 155)(107 168)(108 167)(109 166)(110 165)(111 164)(112 163)
(1 152 23 159)(2 151 24 158)(3 150 25 157)(4 149 26 156)(5 148 27 155)(6 147 28 168)(7 146 15 167)(8 145 16 166)(9 144 17 165)(10 143 18 164)(11 142 19 163)(12 141 20 162)(13 154 21 161)(14 153 22 160)(29 132 56 125)(30 131 43 124)(31 130 44 123)(32 129 45 122)(33 128 46 121)(34 127 47 120)(35 140 48 119)(36 139 49 118)(37 138 50 117)(38 137 51 116)(39 136 52 115)(40 135 53 114)(41 134 54 113)(42 133 55 126)(57 211 84 209)(58 224 71 208)(59 223 72 207)(60 222 73 206)(61 221 74 205)(62 220 75 204)(63 219 76 203)(64 218 77 202)(65 217 78 201)(66 216 79 200)(67 215 80 199)(68 214 81 198)(69 213 82 197)(70 212 83 210)(85 182 105 191)(86 181 106 190)(87 180 107 189)(88 179 108 188)(89 178 109 187)(90 177 110 186)(91 176 111 185)(92 175 112 184)(93 174 99 183)(94 173 100 196)(95 172 101 195)(96 171 102 194)(97 170 103 193)(98 169 104 192)
(1 83 16 63)(2 84 17 64)(3 71 18 65)(4 72 19 66)(5 73 20 67)(6 74 21 68)(7 75 22 69)(8 76 23 70)(9 77 24 57)(10 78 25 58)(11 79 26 59)(12 80 27 60)(13 81 28 61)(14 82 15 62)(29 107 49 94)(30 108 50 95)(31 109 51 96)(32 110 52 97)(33 111 53 98)(34 112 54 85)(35 99 55 86)(36 100 56 87)(37 101 43 88)(38 102 44 89)(39 103 45 90)(40 104 46 91)(41 105 47 92)(42 106 48 93)(113 184 127 182)(114 185 128 169)(115 186 129 170)(116 187 130 171)(117 188 131 172)(118 189 132 173)(119 190 133 174)(120 191 134 175)(121 192 135 176)(122 193 136 177)(123 194 137 178)(124 195 138 179)(125 196 139 180)(126 183 140 181)(141 222 155 199)(142 223 156 200)(143 224 157 201)(144 211 158 202)(145 212 159 203)(146 213 160 204)(147 214 161 205)(148 215 162 206)(149 216 163 207)(150 217 164 208)(151 218 165 209)(152 219 166 210)(153 220 167 197)(154 221 168 198)
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), (1,76,23,63)(2,77,24,64)(3,78,25,65)(4,79,26,66)(5,80,27,67)(6,81,28,68)(7,82,15,69)(8,83,16,70)(9,84,17,57)(10,71,18,58)(11,72,19,59)(12,73,20,60)(13,74,21,61)(14,75,22,62)(29,100,56,94)(30,101,43,95)(31,102,44,96)(32,103,45,97)(33,104,46,98)(34,105,47,85)(35,106,48,86)(36,107,49,87)(37,108,50,88)(38,109,51,89)(39,110,52,90)(40,111,53,91)(41,112,54,92)(42,99,55,93)(113,182,134,191)(114,169,135,192)(115,170,136,193)(116,171,137,194)(117,172,138,195)(118,173,139,196)(119,174,140,183)(120,175,127,184)(121,176,128,185)(122,177,129,186)(123,178,130,187)(124,179,131,188)(125,180,132,189)(126,181,133,190)(141,199,162,215)(142,200,163,216)(143,201,164,217)(144,202,165,218)(145,203,166,219)(146,204,167,220)(147,205,168,221)(148,206,155,222)(149,207,156,223)(150,208,157,224)(151,209,158,211)(152,210,159,212)(153,197,160,213)(154,198,161,214), (1,194)(2,193)(3,192)(4,191)(5,190)(6,189)(7,188)(8,187)(9,186)(10,185)(11,184)(12,183)(13,196)(14,195)(15,179)(16,178)(17,177)(18,176)(19,175)(20,174)(21,173)(22,172)(23,171)(24,170)(25,169)(26,182)(27,181)(28,180)(29,214)(30,213)(31,212)(32,211)(33,224)(34,223)(35,222)(36,221)(37,220)(38,219)(39,218)(40,217)(41,216)(42,215)(43,197)(44,210)(45,209)(46,208)(47,207)(48,206)(49,205)(50,204)(51,203)(52,202)(53,201)(54,200)(55,199)(56,198)(57,122)(58,121)(59,120)(60,119)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,126)(68,125)(69,124)(70,123)(71,128)(72,127)(73,140)(74,139)(75,138)(76,137)(77,136)(78,135)(79,134)(80,133)(81,132)(82,131)(83,130)(84,129)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(91,143)(92,142)(93,141)(94,154)(95,153)(96,152)(97,151)(98,150)(99,162)(100,161)(101,160)(102,159)(103,158)(104,157)(105,156)(106,155)(107,168)(108,167)(109,166)(110,165)(111,164)(112,163), (1,152,23,159)(2,151,24,158)(3,150,25,157)(4,149,26,156)(5,148,27,155)(6,147,28,168)(7,146,15,167)(8,145,16,166)(9,144,17,165)(10,143,18,164)(11,142,19,163)(12,141,20,162)(13,154,21,161)(14,153,22,160)(29,132,56,125)(30,131,43,124)(31,130,44,123)(32,129,45,122)(33,128,46,121)(34,127,47,120)(35,140,48,119)(36,139,49,118)(37,138,50,117)(38,137,51,116)(39,136,52,115)(40,135,53,114)(41,134,54,113)(42,133,55,126)(57,211,84,209)(58,224,71,208)(59,223,72,207)(60,222,73,206)(61,221,74,205)(62,220,75,204)(63,219,76,203)(64,218,77,202)(65,217,78,201)(66,216,79,200)(67,215,80,199)(68,214,81,198)(69,213,82,197)(70,212,83,210)(85,182,105,191)(86,181,106,190)(87,180,107,189)(88,179,108,188)(89,178,109,187)(90,177,110,186)(91,176,111,185)(92,175,112,184)(93,174,99,183)(94,173,100,196)(95,172,101,195)(96,171,102,194)(97,170,103,193)(98,169,104,192), (1,83,16,63)(2,84,17,64)(3,71,18,65)(4,72,19,66)(5,73,20,67)(6,74,21,68)(7,75,22,69)(8,76,23,70)(9,77,24,57)(10,78,25,58)(11,79,26,59)(12,80,27,60)(13,81,28,61)(14,82,15,62)(29,107,49,94)(30,108,50,95)(31,109,51,96)(32,110,52,97)(33,111,53,98)(34,112,54,85)(35,99,55,86)(36,100,56,87)(37,101,43,88)(38,102,44,89)(39,103,45,90)(40,104,46,91)(41,105,47,92)(42,106,48,93)(113,184,127,182)(114,185,128,169)(115,186,129,170)(116,187,130,171)(117,188,131,172)(118,189,132,173)(119,190,133,174)(120,191,134,175)(121,192,135,176)(122,193,136,177)(123,194,137,178)(124,195,138,179)(125,196,139,180)(126,183,140,181)(141,222,155,199)(142,223,156,200)(143,224,157,201)(144,211,158,202)(145,212,159,203)(146,213,160,204)(147,214,161,205)(148,215,162,206)(149,216,163,207)(150,217,164,208)(151,218,165,209)(152,219,166,210)(153,220,167,197)(154,221,168,198)>;
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), (1,76,23,63)(2,77,24,64)(3,78,25,65)(4,79,26,66)(5,80,27,67)(6,81,28,68)(7,82,15,69)(8,83,16,70)(9,84,17,57)(10,71,18,58)(11,72,19,59)(12,73,20,60)(13,74,21,61)(14,75,22,62)(29,100,56,94)(30,101,43,95)(31,102,44,96)(32,103,45,97)(33,104,46,98)(34,105,47,85)(35,106,48,86)(36,107,49,87)(37,108,50,88)(38,109,51,89)(39,110,52,90)(40,111,53,91)(41,112,54,92)(42,99,55,93)(113,182,134,191)(114,169,135,192)(115,170,136,193)(116,171,137,194)(117,172,138,195)(118,173,139,196)(119,174,140,183)(120,175,127,184)(121,176,128,185)(122,177,129,186)(123,178,130,187)(124,179,131,188)(125,180,132,189)(126,181,133,190)(141,199,162,215)(142,200,163,216)(143,201,164,217)(144,202,165,218)(145,203,166,219)(146,204,167,220)(147,205,168,221)(148,206,155,222)(149,207,156,223)(150,208,157,224)(151,209,158,211)(152,210,159,212)(153,197,160,213)(154,198,161,214), (1,194)(2,193)(3,192)(4,191)(5,190)(6,189)(7,188)(8,187)(9,186)(10,185)(11,184)(12,183)(13,196)(14,195)(15,179)(16,178)(17,177)(18,176)(19,175)(20,174)(21,173)(22,172)(23,171)(24,170)(25,169)(26,182)(27,181)(28,180)(29,214)(30,213)(31,212)(32,211)(33,224)(34,223)(35,222)(36,221)(37,220)(38,219)(39,218)(40,217)(41,216)(42,215)(43,197)(44,210)(45,209)(46,208)(47,207)(48,206)(49,205)(50,204)(51,203)(52,202)(53,201)(54,200)(55,199)(56,198)(57,122)(58,121)(59,120)(60,119)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,126)(68,125)(69,124)(70,123)(71,128)(72,127)(73,140)(74,139)(75,138)(76,137)(77,136)(78,135)(79,134)(80,133)(81,132)(82,131)(83,130)(84,129)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(91,143)(92,142)(93,141)(94,154)(95,153)(96,152)(97,151)(98,150)(99,162)(100,161)(101,160)(102,159)(103,158)(104,157)(105,156)(106,155)(107,168)(108,167)(109,166)(110,165)(111,164)(112,163), (1,152,23,159)(2,151,24,158)(3,150,25,157)(4,149,26,156)(5,148,27,155)(6,147,28,168)(7,146,15,167)(8,145,16,166)(9,144,17,165)(10,143,18,164)(11,142,19,163)(12,141,20,162)(13,154,21,161)(14,153,22,160)(29,132,56,125)(30,131,43,124)(31,130,44,123)(32,129,45,122)(33,128,46,121)(34,127,47,120)(35,140,48,119)(36,139,49,118)(37,138,50,117)(38,137,51,116)(39,136,52,115)(40,135,53,114)(41,134,54,113)(42,133,55,126)(57,211,84,209)(58,224,71,208)(59,223,72,207)(60,222,73,206)(61,221,74,205)(62,220,75,204)(63,219,76,203)(64,218,77,202)(65,217,78,201)(66,216,79,200)(67,215,80,199)(68,214,81,198)(69,213,82,197)(70,212,83,210)(85,182,105,191)(86,181,106,190)(87,180,107,189)(88,179,108,188)(89,178,109,187)(90,177,110,186)(91,176,111,185)(92,175,112,184)(93,174,99,183)(94,173,100,196)(95,172,101,195)(96,171,102,194)(97,170,103,193)(98,169,104,192), (1,83,16,63)(2,84,17,64)(3,71,18,65)(4,72,19,66)(5,73,20,67)(6,74,21,68)(7,75,22,69)(8,76,23,70)(9,77,24,57)(10,78,25,58)(11,79,26,59)(12,80,27,60)(13,81,28,61)(14,82,15,62)(29,107,49,94)(30,108,50,95)(31,109,51,96)(32,110,52,97)(33,111,53,98)(34,112,54,85)(35,99,55,86)(36,100,56,87)(37,101,43,88)(38,102,44,89)(39,103,45,90)(40,104,46,91)(41,105,47,92)(42,106,48,93)(113,184,127,182)(114,185,128,169)(115,186,129,170)(116,187,130,171)(117,188,131,172)(118,189,132,173)(119,190,133,174)(120,191,134,175)(121,192,135,176)(122,193,136,177)(123,194,137,178)(124,195,138,179)(125,196,139,180)(126,183,140,181)(141,222,155,199)(142,223,156,200)(143,224,157,201)(144,211,158,202)(145,212,159,203)(146,213,160,204)(147,214,161,205)(148,215,162,206)(149,216,163,207)(150,217,164,208)(151,218,165,209)(152,219,166,210)(153,220,167,197)(154,221,168,198) );
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)], [(1,76,23,63),(2,77,24,64),(3,78,25,65),(4,79,26,66),(5,80,27,67),(6,81,28,68),(7,82,15,69),(8,83,16,70),(9,84,17,57),(10,71,18,58),(11,72,19,59),(12,73,20,60),(13,74,21,61),(14,75,22,62),(29,100,56,94),(30,101,43,95),(31,102,44,96),(32,103,45,97),(33,104,46,98),(34,105,47,85),(35,106,48,86),(36,107,49,87),(37,108,50,88),(38,109,51,89),(39,110,52,90),(40,111,53,91),(41,112,54,92),(42,99,55,93),(113,182,134,191),(114,169,135,192),(115,170,136,193),(116,171,137,194),(117,172,138,195),(118,173,139,196),(119,174,140,183),(120,175,127,184),(121,176,128,185),(122,177,129,186),(123,178,130,187),(124,179,131,188),(125,180,132,189),(126,181,133,190),(141,199,162,215),(142,200,163,216),(143,201,164,217),(144,202,165,218),(145,203,166,219),(146,204,167,220),(147,205,168,221),(148,206,155,222),(149,207,156,223),(150,208,157,224),(151,209,158,211),(152,210,159,212),(153,197,160,213),(154,198,161,214)], [(1,194),(2,193),(3,192),(4,191),(5,190),(6,189),(7,188),(8,187),(9,186),(10,185),(11,184),(12,183),(13,196),(14,195),(15,179),(16,178),(17,177),(18,176),(19,175),(20,174),(21,173),(22,172),(23,171),(24,170),(25,169),(26,182),(27,181),(28,180),(29,214),(30,213),(31,212),(32,211),(33,224),(34,223),(35,222),(36,221),(37,220),(38,219),(39,218),(40,217),(41,216),(42,215),(43,197),(44,210),(45,209),(46,208),(47,207),(48,206),(49,205),(50,204),(51,203),(52,202),(53,201),(54,200),(55,199),(56,198),(57,122),(58,121),(59,120),(60,119),(61,118),(62,117),(63,116),(64,115),(65,114),(66,113),(67,126),(68,125),(69,124),(70,123),(71,128),(72,127),(73,140),(74,139),(75,138),(76,137),(77,136),(78,135),(79,134),(80,133),(81,132),(82,131),(83,130),(84,129),(85,149),(86,148),(87,147),(88,146),(89,145),(90,144),(91,143),(92,142),(93,141),(94,154),(95,153),(96,152),(97,151),(98,150),(99,162),(100,161),(101,160),(102,159),(103,158),(104,157),(105,156),(106,155),(107,168),(108,167),(109,166),(110,165),(111,164),(112,163)], [(1,152,23,159),(2,151,24,158),(3,150,25,157),(4,149,26,156),(5,148,27,155),(6,147,28,168),(7,146,15,167),(8,145,16,166),(9,144,17,165),(10,143,18,164),(11,142,19,163),(12,141,20,162),(13,154,21,161),(14,153,22,160),(29,132,56,125),(30,131,43,124),(31,130,44,123),(32,129,45,122),(33,128,46,121),(34,127,47,120),(35,140,48,119),(36,139,49,118),(37,138,50,117),(38,137,51,116),(39,136,52,115),(40,135,53,114),(41,134,54,113),(42,133,55,126),(57,211,84,209),(58,224,71,208),(59,223,72,207),(60,222,73,206),(61,221,74,205),(62,220,75,204),(63,219,76,203),(64,218,77,202),(65,217,78,201),(66,216,79,200),(67,215,80,199),(68,214,81,198),(69,213,82,197),(70,212,83,210),(85,182,105,191),(86,181,106,190),(87,180,107,189),(88,179,108,188),(89,178,109,187),(90,177,110,186),(91,176,111,185),(92,175,112,184),(93,174,99,183),(94,173,100,196),(95,172,101,195),(96,171,102,194),(97,170,103,193),(98,169,104,192)], [(1,83,16,63),(2,84,17,64),(3,71,18,65),(4,72,19,66),(5,73,20,67),(6,74,21,68),(7,75,22,69),(8,76,23,70),(9,77,24,57),(10,78,25,58),(11,79,26,59),(12,80,27,60),(13,81,28,61),(14,82,15,62),(29,107,49,94),(30,108,50,95),(31,109,51,96),(32,110,52,97),(33,111,53,98),(34,112,54,85),(35,99,55,86),(36,100,56,87),(37,101,43,88),(38,102,44,89),(39,103,45,90),(40,104,46,91),(41,105,47,92),(42,106,48,93),(113,184,127,182),(114,185,128,169),(115,186,129,170),(116,187,130,171),(117,188,131,172),(118,189,132,173),(119,190,133,174),(120,191,134,175),(121,192,135,176),(122,193,136,177),(123,194,137,178),(124,195,138,179),(125,196,139,180),(126,183,140,181),(141,222,155,199),(142,223,156,200),(143,224,157,201),(144,211,158,202),(145,212,159,203),(146,213,160,204),(147,214,161,205),(148,215,162,206),(149,216,163,207),(150,217,164,208),(151,218,165,209),(152,219,166,210),(153,220,167,197),(154,221,168,198)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 8 | 16 | 0 | 0 | 0 | 0 | 0 |
| 15 | 25 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 19 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 7 |
| 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 7 | 9 | 11 | 0 | 0 | 0 | 0 |
| 15 | 11 | 11 | 20 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 27 |
| 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 13 | 16 | 25 | 8 | 0 | 0 | 0 | 0 |
| 11 | 14 | 14 | 13 | 0 | 0 | 0 | 0 |
| 5 | 2 | 14 | 25 | 0 | 0 | 0 | 0 |
| 3 | 5 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 22 | 1 | 22 | 1 |
| 8 | 23 | 0 | 23 | 0 | 0 | 0 | 0 |
| 27 | 24 | 21 | 8 | 0 | 0 | 0 | 0 |
| 0 | 9 | 13 | 24 | 0 | 0 | 0 | 0 |
| 8 | 26 | 8 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 15 | 11 | 1 |
| 0 | 0 | 0 | 0 | 10 | 9 | 20 | 18 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 20 |
| 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 24 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 21 | 20 | 18 | 0 | 0 | 0 | 0 |
| 10 | 15 | 18 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 27 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(29))| [20,0,14,15,0,0,0,0,0,20,8,25,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,19,10,0,0,0,0,0,0,19,7],[2,24,4,15,0,0,0,0,1,27,7,11,0,0,0,0,0,0,9,11,0,0,0,0,0,0,11,20,0,0,0,0,0,0,0,0,28,0,1,0,0,0,0,0,0,28,0,1,0,0,0,0,27,0,1,0,0,0,0,0,0,27,0,1],[13,11,5,3,0,0,0,0,16,14,2,5,0,0,0,0,25,14,14,0,0,0,0,0,8,13,25,17,0,0,0,0,0,0,0,0,1,7,28,22,0,0,0,0,0,28,0,1,0,0,0,0,0,0,28,22,0,0,0,0,0,0,0,1],[8,27,0,8,0,0,0,0,23,24,9,26,0,0,0,0,0,21,13,8,0,0,0,0,23,8,24,13,0,0,0,0,0,0,0,0,20,10,0,0,0,0,0,0,15,9,0,0,0,0,0,0,11,20,9,19,0,0,0,0,1,18,14,20],[2,24,19,10,0,0,0,0,1,27,21,15,0,0,0,0,0,0,20,18,0,0,0,0,0,0,18,9,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,27,0,1,0,0,0,0,0,0,27,0,1] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4K | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 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 | 28 | 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⋊8D14 |
| kernel | C14.262- (1+4) | D14.D4 | D14⋊D4 | Dic7.D4 | Dic7.Q8 | D14.5D4 | C28⋊1D4 | D14⋊Q8 | D14⋊2Q8 | C23.23D14 | C28⋊7D4 | D14⋊3Q8 | C28.23D4 | C7×C22⋊Q8 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 6 | 9 | 3 | 3 | 2 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{26}2_-^{(1+4)} % in TeX
G:=Group("C14.26ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1100);
// by ID
G=gap.SmallGroup(448,1100);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,219,100,1571,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7*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=e*b*e^-1=a^7*b,c*d=d*c,e*c*e^-1=a^7*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations
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