?
G = C14.442- (1+4) order 448 = 26·7
44th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.442- (1+4), (C22×Q8)⋊8D7, (C2×C28).216D4, C28.261(C2×D4), D14⋊3Q8⋊41C2, (C2×Q8).188D14, Dic7⋊Q8⋊30C2, C28.23D4⋊29C2, (C2×C28).648C23, (C2×C14).308C24, D14⋊C4.77C22, C14.156(C22×D4), (C22×C4).278D14, (C2×D28).279C22, Dic7⋊C4.90C22, C4⋊Dic7.390C22, (Q8×C14).235C22, C22.319(C23×D7), C23.239(C22×D7), C23.23D14⋊29C2, C23.21D14⋊34C2, (C22×C28).440C22, (C22×C14).426C23, C7⋊6(C23.38C23), (C4×Dic7).171C22, (C2×Dic7).159C23, (C22×D7).134C23, C23.D7.132C22, C2.44(Q8.10D14), (C2×Dic14).308C22, (Q8×C2×C14)⋊7C2, C4.99(C2×C7⋊D4), (C2×C4○D28).25C2, (C2×C14).590(C2×D4), (C2×C4).94(C7⋊D4), (C2×C4×D7).165C22, C22.37(C2×C7⋊D4), C2.29(C22×C7⋊D4), (C2×C4).634(C22×D7), (C2×C7⋊D4).138C22, SmallGroup(448,1269)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1076 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C7, C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×10], C23, C23 [×2], D7 [×2], C14, C14 [×2], C14 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×3], C2×Q8 [×4], C2×Q8 [×5], C4○D4 [×4], Dic7 [×6], C28 [×4], C28 [×4], D14 [×6], C2×C14, C2×C14 [×2], C2×C14 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic14 [×2], C4×D7 [×4], D28 [×2], C2×Dic7 [×6], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×8], C2×C28 [×4], C7×Q8 [×8], C22×D7 [×2], C22×C14, C23.38C23, C4×Dic7 [×2], Dic7⋊C4 [×8], C4⋊Dic7 [×2], D14⋊C4 [×8], C23.D7 [×2], C2×Dic14, C2×C4×D7 [×2], C2×D28, C4○D28 [×4], C2×C7⋊D4 [×2], C22×C28, C22×C28 [×2], Q8×C14 [×4], Q8×C14 [×4], C23.21D14, C23.23D14 [×4], Dic7⋊Q8 [×2], D14⋊3Q8 [×4], C28.23D4 [×2], C2×C4○D28, Q8×C2×C14, C14.442- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2- (1+4) [×2], C7⋊D4 [×4], C22×D7 [×7], C23.38C23, C2×C7⋊D4 [×6], C23×D7, Q8.10D14 [×2], C22×C7⋊D4, C14.442- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=a7b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=a7b-1, dbd-1=ebe-1=a7b, dcd-1=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)
(1 157 72 40)(2 156 73 39)(3 155 74 38)(4 168 75 37)(5 167 76 36)(6 166 77 35)(7 165 78 34)(8 164 79 33)(9 163 80 32)(10 162 81 31)(11 161 82 30)(12 160 83 29)(13 159 84 42)(14 158 71 41)(15 196 216 58)(16 195 217 57)(17 194 218 70)(18 193 219 69)(19 192 220 68)(20 191 221 67)(21 190 222 66)(22 189 223 65)(23 188 224 64)(24 187 211 63)(25 186 212 62)(26 185 213 61)(27 184 214 60)(28 183 215 59)(43 96 175 129)(44 95 176 128)(45 94 177 127)(46 93 178 140)(47 92 179 139)(48 91 180 138)(49 90 181 137)(50 89 182 136)(51 88 169 135)(52 87 170 134)(53 86 171 133)(54 85 172 132)(55 98 173 131)(56 97 174 130)(99 123 148 210)(100 122 149 209)(101 121 150 208)(102 120 151 207)(103 119 152 206)(104 118 153 205)(105 117 154 204)(106 116 141 203)(107 115 142 202)(108 114 143 201)(109 113 144 200)(110 126 145 199)(111 125 146 198)(112 124 147 197)
(1 33 8 40)(2 32 9 39)(3 31 10 38)(4 30 11 37)(5 29 12 36)(6 42 13 35)(7 41 14 34)(15 65 22 58)(16 64 23 57)(17 63 24 70)(18 62 25 69)(19 61 26 68)(20 60 27 67)(21 59 28 66)(43 136 50 129)(44 135 51 128)(45 134 52 127)(46 133 53 140)(47 132 54 139)(48 131 55 138)(49 130 56 137)(71 165 78 158)(72 164 79 157)(73 163 80 156)(74 162 81 155)(75 161 82 168)(76 160 83 167)(77 159 84 166)(85 172 92 179)(86 171 93 178)(87 170 94 177)(88 169 95 176)(89 182 96 175)(90 181 97 174)(91 180 98 173)(99 203 106 210)(100 202 107 209)(101 201 108 208)(102 200 109 207)(103 199 110 206)(104 198 111 205)(105 197 112 204)(113 144 120 151)(114 143 121 150)(115 142 122 149)(116 141 123 148)(117 154 124 147)(118 153 125 146)(119 152 126 145)(183 215 190 222)(184 214 191 221)(185 213 192 220)(186 212 193 219)(187 211 194 218)(188 224 195 217)(189 223 196 216)
(1 182 79 43)(2 169 80 44)(3 170 81 45)(4 171 82 46)(5 172 83 47)(6 173 84 48)(7 174 71 49)(8 175 72 50)(9 176 73 51)(10 177 74 52)(11 178 75 53)(12 179 76 54)(13 180 77 55)(14 181 78 56)(15 108 223 150)(16 109 224 151)(17 110 211 152)(18 111 212 153)(19 112 213 154)(20 99 214 141)(21 100 215 142)(22 101 216 143)(23 102 217 144)(24 103 218 145)(25 104 219 146)(26 105 220 147)(27 106 221 148)(28 107 222 149)(29 85 167 139)(30 86 168 140)(31 87 155 127)(32 88 156 128)(33 89 157 129)(34 90 158 130)(35 91 159 131)(36 92 160 132)(37 93 161 133)(38 94 162 134)(39 95 163 135)(40 96 164 136)(41 97 165 137)(42 98 166 138)(57 207 188 113)(58 208 189 114)(59 209 190 115)(60 210 191 116)(61 197 192 117)(62 198 193 118)(63 199 194 119)(64 200 195 120)(65 201 196 121)(66 202 183 122)(67 203 184 123)(68 204 185 124)(69 205 186 125)(70 206 187 126)
(1 153 79 111)(2 154 80 112)(3 141 81 99)(4 142 82 100)(5 143 83 101)(6 144 84 102)(7 145 71 103)(8 146 72 104)(9 147 73 105)(10 148 74 106)(11 149 75 107)(12 150 76 108)(13 151 77 109)(14 152 78 110)(15 179 223 54)(16 180 224 55)(17 181 211 56)(18 182 212 43)(19 169 213 44)(20 170 214 45)(21 171 215 46)(22 172 216 47)(23 173 217 48)(24 174 218 49)(25 175 219 50)(26 176 220 51)(27 177 221 52)(28 178 222 53)(29 114 167 208)(30 115 168 209)(31 116 155 210)(32 117 156 197)(33 118 157 198)(34 119 158 199)(35 120 159 200)(36 121 160 201)(37 122 161 202)(38 123 162 203)(39 124 163 204)(40 125 164 205)(41 126 165 206)(42 113 166 207)(57 98 188 138)(58 85 189 139)(59 86 190 140)(60 87 191 127)(61 88 192 128)(62 89 193 129)(63 90 194 130)(64 91 195 131)(65 92 196 132)(66 93 183 133)(67 94 184 134)(68 95 185 135)(69 96 186 136)(70 97 187 137)
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), (1,157,72,40)(2,156,73,39)(3,155,74,38)(4,168,75,37)(5,167,76,36)(6,166,77,35)(7,165,78,34)(8,164,79,33)(9,163,80,32)(10,162,81,31)(11,161,82,30)(12,160,83,29)(13,159,84,42)(14,158,71,41)(15,196,216,58)(16,195,217,57)(17,194,218,70)(18,193,219,69)(19,192,220,68)(20,191,221,67)(21,190,222,66)(22,189,223,65)(23,188,224,64)(24,187,211,63)(25,186,212,62)(26,185,213,61)(27,184,214,60)(28,183,215,59)(43,96,175,129)(44,95,176,128)(45,94,177,127)(46,93,178,140)(47,92,179,139)(48,91,180,138)(49,90,181,137)(50,89,182,136)(51,88,169,135)(52,87,170,134)(53,86,171,133)(54,85,172,132)(55,98,173,131)(56,97,174,130)(99,123,148,210)(100,122,149,209)(101,121,150,208)(102,120,151,207)(103,119,152,206)(104,118,153,205)(105,117,154,204)(106,116,141,203)(107,115,142,202)(108,114,143,201)(109,113,144,200)(110,126,145,199)(111,125,146,198)(112,124,147,197), (1,33,8,40)(2,32,9,39)(3,31,10,38)(4,30,11,37)(5,29,12,36)(6,42,13,35)(7,41,14,34)(15,65,22,58)(16,64,23,57)(17,63,24,70)(18,62,25,69)(19,61,26,68)(20,60,27,67)(21,59,28,66)(43,136,50,129)(44,135,51,128)(45,134,52,127)(46,133,53,140)(47,132,54,139)(48,131,55,138)(49,130,56,137)(71,165,78,158)(72,164,79,157)(73,163,80,156)(74,162,81,155)(75,161,82,168)(76,160,83,167)(77,159,84,166)(85,172,92,179)(86,171,93,178)(87,170,94,177)(88,169,95,176)(89,182,96,175)(90,181,97,174)(91,180,98,173)(99,203,106,210)(100,202,107,209)(101,201,108,208)(102,200,109,207)(103,199,110,206)(104,198,111,205)(105,197,112,204)(113,144,120,151)(114,143,121,150)(115,142,122,149)(116,141,123,148)(117,154,124,147)(118,153,125,146)(119,152,126,145)(183,215,190,222)(184,214,191,221)(185,213,192,220)(186,212,193,219)(187,211,194,218)(188,224,195,217)(189,223,196,216), (1,182,79,43)(2,169,80,44)(3,170,81,45)(4,171,82,46)(5,172,83,47)(6,173,84,48)(7,174,71,49)(8,175,72,50)(9,176,73,51)(10,177,74,52)(11,178,75,53)(12,179,76,54)(13,180,77,55)(14,181,78,56)(15,108,223,150)(16,109,224,151)(17,110,211,152)(18,111,212,153)(19,112,213,154)(20,99,214,141)(21,100,215,142)(22,101,216,143)(23,102,217,144)(24,103,218,145)(25,104,219,146)(26,105,220,147)(27,106,221,148)(28,107,222,149)(29,85,167,139)(30,86,168,140)(31,87,155,127)(32,88,156,128)(33,89,157,129)(34,90,158,130)(35,91,159,131)(36,92,160,132)(37,93,161,133)(38,94,162,134)(39,95,163,135)(40,96,164,136)(41,97,165,137)(42,98,166,138)(57,207,188,113)(58,208,189,114)(59,209,190,115)(60,210,191,116)(61,197,192,117)(62,198,193,118)(63,199,194,119)(64,200,195,120)(65,201,196,121)(66,202,183,122)(67,203,184,123)(68,204,185,124)(69,205,186,125)(70,206,187,126), (1,153,79,111)(2,154,80,112)(3,141,81,99)(4,142,82,100)(5,143,83,101)(6,144,84,102)(7,145,71,103)(8,146,72,104)(9,147,73,105)(10,148,74,106)(11,149,75,107)(12,150,76,108)(13,151,77,109)(14,152,78,110)(15,179,223,54)(16,180,224,55)(17,181,211,56)(18,182,212,43)(19,169,213,44)(20,170,214,45)(21,171,215,46)(22,172,216,47)(23,173,217,48)(24,174,218,49)(25,175,219,50)(26,176,220,51)(27,177,221,52)(28,178,222,53)(29,114,167,208)(30,115,168,209)(31,116,155,210)(32,117,156,197)(33,118,157,198)(34,119,158,199)(35,120,159,200)(36,121,160,201)(37,122,161,202)(38,123,162,203)(39,124,163,204)(40,125,164,205)(41,126,165,206)(42,113,166,207)(57,98,188,138)(58,85,189,139)(59,86,190,140)(60,87,191,127)(61,88,192,128)(62,89,193,129)(63,90,194,130)(64,91,195,131)(65,92,196,132)(66,93,183,133)(67,94,184,134)(68,95,185,135)(69,96,186,136)(70,97,187,137)>;
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), (1,157,72,40)(2,156,73,39)(3,155,74,38)(4,168,75,37)(5,167,76,36)(6,166,77,35)(7,165,78,34)(8,164,79,33)(9,163,80,32)(10,162,81,31)(11,161,82,30)(12,160,83,29)(13,159,84,42)(14,158,71,41)(15,196,216,58)(16,195,217,57)(17,194,218,70)(18,193,219,69)(19,192,220,68)(20,191,221,67)(21,190,222,66)(22,189,223,65)(23,188,224,64)(24,187,211,63)(25,186,212,62)(26,185,213,61)(27,184,214,60)(28,183,215,59)(43,96,175,129)(44,95,176,128)(45,94,177,127)(46,93,178,140)(47,92,179,139)(48,91,180,138)(49,90,181,137)(50,89,182,136)(51,88,169,135)(52,87,170,134)(53,86,171,133)(54,85,172,132)(55,98,173,131)(56,97,174,130)(99,123,148,210)(100,122,149,209)(101,121,150,208)(102,120,151,207)(103,119,152,206)(104,118,153,205)(105,117,154,204)(106,116,141,203)(107,115,142,202)(108,114,143,201)(109,113,144,200)(110,126,145,199)(111,125,146,198)(112,124,147,197), (1,33,8,40)(2,32,9,39)(3,31,10,38)(4,30,11,37)(5,29,12,36)(6,42,13,35)(7,41,14,34)(15,65,22,58)(16,64,23,57)(17,63,24,70)(18,62,25,69)(19,61,26,68)(20,60,27,67)(21,59,28,66)(43,136,50,129)(44,135,51,128)(45,134,52,127)(46,133,53,140)(47,132,54,139)(48,131,55,138)(49,130,56,137)(71,165,78,158)(72,164,79,157)(73,163,80,156)(74,162,81,155)(75,161,82,168)(76,160,83,167)(77,159,84,166)(85,172,92,179)(86,171,93,178)(87,170,94,177)(88,169,95,176)(89,182,96,175)(90,181,97,174)(91,180,98,173)(99,203,106,210)(100,202,107,209)(101,201,108,208)(102,200,109,207)(103,199,110,206)(104,198,111,205)(105,197,112,204)(113,144,120,151)(114,143,121,150)(115,142,122,149)(116,141,123,148)(117,154,124,147)(118,153,125,146)(119,152,126,145)(183,215,190,222)(184,214,191,221)(185,213,192,220)(186,212,193,219)(187,211,194,218)(188,224,195,217)(189,223,196,216), (1,182,79,43)(2,169,80,44)(3,170,81,45)(4,171,82,46)(5,172,83,47)(6,173,84,48)(7,174,71,49)(8,175,72,50)(9,176,73,51)(10,177,74,52)(11,178,75,53)(12,179,76,54)(13,180,77,55)(14,181,78,56)(15,108,223,150)(16,109,224,151)(17,110,211,152)(18,111,212,153)(19,112,213,154)(20,99,214,141)(21,100,215,142)(22,101,216,143)(23,102,217,144)(24,103,218,145)(25,104,219,146)(26,105,220,147)(27,106,221,148)(28,107,222,149)(29,85,167,139)(30,86,168,140)(31,87,155,127)(32,88,156,128)(33,89,157,129)(34,90,158,130)(35,91,159,131)(36,92,160,132)(37,93,161,133)(38,94,162,134)(39,95,163,135)(40,96,164,136)(41,97,165,137)(42,98,166,138)(57,207,188,113)(58,208,189,114)(59,209,190,115)(60,210,191,116)(61,197,192,117)(62,198,193,118)(63,199,194,119)(64,200,195,120)(65,201,196,121)(66,202,183,122)(67,203,184,123)(68,204,185,124)(69,205,186,125)(70,206,187,126), (1,153,79,111)(2,154,80,112)(3,141,81,99)(4,142,82,100)(5,143,83,101)(6,144,84,102)(7,145,71,103)(8,146,72,104)(9,147,73,105)(10,148,74,106)(11,149,75,107)(12,150,76,108)(13,151,77,109)(14,152,78,110)(15,179,223,54)(16,180,224,55)(17,181,211,56)(18,182,212,43)(19,169,213,44)(20,170,214,45)(21,171,215,46)(22,172,216,47)(23,173,217,48)(24,174,218,49)(25,175,219,50)(26,176,220,51)(27,177,221,52)(28,178,222,53)(29,114,167,208)(30,115,168,209)(31,116,155,210)(32,117,156,197)(33,118,157,198)(34,119,158,199)(35,120,159,200)(36,121,160,201)(37,122,161,202)(38,123,162,203)(39,124,163,204)(40,125,164,205)(41,126,165,206)(42,113,166,207)(57,98,188,138)(58,85,189,139)(59,86,190,140)(60,87,191,127)(61,88,192,128)(62,89,193,129)(63,90,194,130)(64,91,195,131)(65,92,196,132)(66,93,183,133)(67,94,184,134)(68,95,185,135)(69,96,186,136)(70,97,187,137) );
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)], [(1,157,72,40),(2,156,73,39),(3,155,74,38),(4,168,75,37),(5,167,76,36),(6,166,77,35),(7,165,78,34),(8,164,79,33),(9,163,80,32),(10,162,81,31),(11,161,82,30),(12,160,83,29),(13,159,84,42),(14,158,71,41),(15,196,216,58),(16,195,217,57),(17,194,218,70),(18,193,219,69),(19,192,220,68),(20,191,221,67),(21,190,222,66),(22,189,223,65),(23,188,224,64),(24,187,211,63),(25,186,212,62),(26,185,213,61),(27,184,214,60),(28,183,215,59),(43,96,175,129),(44,95,176,128),(45,94,177,127),(46,93,178,140),(47,92,179,139),(48,91,180,138),(49,90,181,137),(50,89,182,136),(51,88,169,135),(52,87,170,134),(53,86,171,133),(54,85,172,132),(55,98,173,131),(56,97,174,130),(99,123,148,210),(100,122,149,209),(101,121,150,208),(102,120,151,207),(103,119,152,206),(104,118,153,205),(105,117,154,204),(106,116,141,203),(107,115,142,202),(108,114,143,201),(109,113,144,200),(110,126,145,199),(111,125,146,198),(112,124,147,197)], [(1,33,8,40),(2,32,9,39),(3,31,10,38),(4,30,11,37),(5,29,12,36),(6,42,13,35),(7,41,14,34),(15,65,22,58),(16,64,23,57),(17,63,24,70),(18,62,25,69),(19,61,26,68),(20,60,27,67),(21,59,28,66),(43,136,50,129),(44,135,51,128),(45,134,52,127),(46,133,53,140),(47,132,54,139),(48,131,55,138),(49,130,56,137),(71,165,78,158),(72,164,79,157),(73,163,80,156),(74,162,81,155),(75,161,82,168),(76,160,83,167),(77,159,84,166),(85,172,92,179),(86,171,93,178),(87,170,94,177),(88,169,95,176),(89,182,96,175),(90,181,97,174),(91,180,98,173),(99,203,106,210),(100,202,107,209),(101,201,108,208),(102,200,109,207),(103,199,110,206),(104,198,111,205),(105,197,112,204),(113,144,120,151),(114,143,121,150),(115,142,122,149),(116,141,123,148),(117,154,124,147),(118,153,125,146),(119,152,126,145),(183,215,190,222),(184,214,191,221),(185,213,192,220),(186,212,193,219),(187,211,194,218),(188,224,195,217),(189,223,196,216)], [(1,182,79,43),(2,169,80,44),(3,170,81,45),(4,171,82,46),(5,172,83,47),(6,173,84,48),(7,174,71,49),(8,175,72,50),(9,176,73,51),(10,177,74,52),(11,178,75,53),(12,179,76,54),(13,180,77,55),(14,181,78,56),(15,108,223,150),(16,109,224,151),(17,110,211,152),(18,111,212,153),(19,112,213,154),(20,99,214,141),(21,100,215,142),(22,101,216,143),(23,102,217,144),(24,103,218,145),(25,104,219,146),(26,105,220,147),(27,106,221,148),(28,107,222,149),(29,85,167,139),(30,86,168,140),(31,87,155,127),(32,88,156,128),(33,89,157,129),(34,90,158,130),(35,91,159,131),(36,92,160,132),(37,93,161,133),(38,94,162,134),(39,95,163,135),(40,96,164,136),(41,97,165,137),(42,98,166,138),(57,207,188,113),(58,208,189,114),(59,209,190,115),(60,210,191,116),(61,197,192,117),(62,198,193,118),(63,199,194,119),(64,200,195,120),(65,201,196,121),(66,202,183,122),(67,203,184,123),(68,204,185,124),(69,205,186,125),(70,206,187,126)], [(1,153,79,111),(2,154,80,112),(3,141,81,99),(4,142,82,100),(5,143,83,101),(6,144,84,102),(7,145,71,103),(8,146,72,104),(9,147,73,105),(10,148,74,106),(11,149,75,107),(12,150,76,108),(13,151,77,109),(14,152,78,110),(15,179,223,54),(16,180,224,55),(17,181,211,56),(18,182,212,43),(19,169,213,44),(20,170,214,45),(21,171,215,46),(22,172,216,47),(23,173,217,48),(24,174,218,49),(25,175,219,50),(26,176,220,51),(27,177,221,52),(28,178,222,53),(29,114,167,208),(30,115,168,209),(31,116,155,210),(32,117,156,197),(33,118,157,198),(34,119,158,199),(35,120,159,200),(36,121,160,201),(37,122,161,202),(38,123,162,203),(39,124,163,204),(40,125,164,205),(41,126,165,206),(42,113,166,207),(57,98,188,138),(58,85,189,139),(59,86,190,140),(60,87,191,127),(61,88,192,128),(62,89,193,129),(63,90,194,130),(64,91,195,131),(65,92,196,132),(66,93,183,133),(67,94,184,134),(68,95,185,135),(69,96,186,136),(70,97,187,137)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 23 | 0 | 9 | 0 |
| 0 | 0 | 24 | 0 | 0 | 9 |
| 12 | 16 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 9 | 27 |
| 0 | 0 | 8 | 0 | 11 | 28 |
| 0 | 0 | 18 | 18 | 1 | 3 |
| 0 | 0 | 25 | 23 | 25 | 17 |
| 17 | 13 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 9 | 27 |
| 0 | 0 | 8 | 0 | 11 | 28 |
| 0 | 0 | 7 | 11 | 1 | 3 |
| 0 | 0 | 20 | 6 | 25 | 17 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 28 | 0 | 0 |
| 0 | 0 | 8 | 6 | 0 | 0 |
| 0 | 0 | 27 | 16 | 5 | 20 |
| 0 | 0 | 8 | 23 | 19 | 24 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 2 | 0 | 0 |
| 0 | 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 28 | 26 | 23 | 11 |
| 0 | 0 | 7 | 12 | 23 | 6 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,13,0,23,24,0,0,0,13,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[12,0,0,0,0,0,16,17,0,0,0,0,0,0,11,8,18,25,0,0,0,0,18,23,0,0,9,11,1,25,0,0,27,28,3,17],[17,0,0,0,0,0,13,12,0,0,0,0,0,0,11,8,7,20,0,0,0,0,11,6,0,0,9,11,1,25,0,0,27,28,3,17],[28,16,0,0,0,0,0,1,0,0,0,0,0,0,23,8,27,8,0,0,28,6,16,23,0,0,0,0,5,19,0,0,0,0,20,24],[1,13,0,0,0,0,0,28,0,0,0,0,0,0,28,28,28,7,0,0,2,1,26,12,0,0,0,0,23,23,0,0,0,0,11,6] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | C7⋊D4 | 2- (1+4) | Q8.10D14 |
| kernel | C14.442- (1+4) | C23.21D14 | C23.23D14 | Dic7⋊Q8 | D14⋊3Q8 | C28.23D4 | C2×C4○D28 | Q8×C2×C14 | C2×C28 | C22×Q8 | C22×C4 | C2×Q8 | C2×C4 | C14 | C2 |
| # reps | 1 | 1 | 4 | 2 | 4 | 2 | 1 | 1 | 4 | 3 | 9 | 12 | 24 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{44}2_-^{(1+4)} % in TeX
G:=Group("C14.44ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1269);
// by ID
G=gap.SmallGroup(448,1269);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,675,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=e^2=a^7*b^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=e*b*e^-1=a^7*b,d*c*d^-1=e*c*e^-1=a^7*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations
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