?
G = C14.1052- (1+4) order 448 = 26·7
60th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1052- (1+4), (C2×C28).220D4, C28.429(C2×D4), (C2×D4).235D14, (C2×Q8).193D14, Dic7⋊Q8⋊32C2, C28.48D4⋊48C2, C28.17D4⋊30C2, (C2×C14).311C24, (C2×C28).650C23, (C22×C4).285D14, C14.163(C22×D4), (C22×Dic14)⋊22C2, (D4×C14).314C22, Dic7⋊C4.92C22, C4⋊Dic7.320C22, (Q8×C14).240C22, C22.322(C23×D7), C23.208(C22×D7), C23.21D14⋊35C2, C23.18D14⋊32C2, (C22×C14).237C23, (C22×C28).320C22, C7⋊7(C23.38C23), (C2×Dic7).161C23, (C4×Dic7).174C22, C2.69(D4.10D14), C23.D7.133C22, (C2×Dic14).310C22, (C22×Dic7).166C22, C4.32(C2×C7⋊D4), (C2×C4○D4).11D7, (C2×C14).79(C2×D4), (C14×C4○D4).12C2, (C2×C4).97(C7⋊D4), C2.36(C22×C7⋊D4), C22.22(C2×C7⋊D4), (C2×C4).249(C22×D7), SmallGroup(448,1278)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 980 in 270 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C7, C2×C4 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×10], C23, C23 [×2], C14, C14 [×2], C14 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic7 [×8], C28 [×4], C28 [×2], C2×C14, C2×C14 [×2], C2×C14 [×8], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic14 [×8], C2×Dic7 [×8], C2×Dic7 [×4], C2×C28 [×2], C2×C28 [×6], C2×C28 [×4], C7×D4 [×6], C7×Q8 [×2], C22×C14, C22×C14 [×2], C23.38C23, C4×Dic7 [×2], Dic7⋊C4 [×8], C4⋊Dic7 [×2], C23.D7 [×10], C2×Dic14 [×4], C2×Dic14 [×4], C22×Dic7 [×2], C22×C28, C22×C28 [×2], D4×C14, D4×C14 [×2], Q8×C14, C7×C4○D4 [×4], C28.48D4 [×4], C23.21D14, C23.18D14 [×4], C28.17D4 [×2], Dic7⋊Q8 [×2], C22×Dic14, C14×C4○D4, C14.1052- (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, D4.10D14 [×2], C22×C7⋊D4, C14.1052- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=b2, e2=a7b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=a7b-1, dbd-1=ebe-1=a7b, dcd-1=a7c, ce=ec, 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 20 224 46)(2 19 211 45)(3 18 212 44)(4 17 213 43)(5 16 214 56)(6 15 215 55)(7 28 216 54)(8 27 217 53)(9 26 218 52)(10 25 219 51)(11 24 220 50)(12 23 221 49)(13 22 222 48)(14 21 223 47)(29 167 130 209)(30 166 131 208)(31 165 132 207)(32 164 133 206)(33 163 134 205)(34 162 135 204)(35 161 136 203)(36 160 137 202)(37 159 138 201)(38 158 139 200)(39 157 140 199)(40 156 127 198)(41 155 128 197)(42 168 129 210)(57 98 194 169)(58 97 195 182)(59 96 196 181)(60 95 183 180)(61 94 184 179)(62 93 185 178)(63 92 186 177)(64 91 187 176)(65 90 188 175)(66 89 189 174)(67 88 190 173)(68 87 191 172)(69 86 192 171)(70 85 193 170)(71 122 102 152)(72 121 103 151)(73 120 104 150)(74 119 105 149)(75 118 106 148)(76 117 107 147)(77 116 108 146)(78 115 109 145)(79 114 110 144)(80 113 111 143)(81 126 112 142)(82 125 99 141)(83 124 100 154)(84 123 101 153)
(1 63 8 70)(2 62 9 69)(3 61 10 68)(4 60 11 67)(5 59 12 66)(6 58 13 65)(7 57 14 64)(15 175 22 182)(16 174 23 181)(17 173 24 180)(18 172 25 179)(19 171 26 178)(20 170 27 177)(21 169 28 176)(29 100 36 107)(30 99 37 106)(31 112 38 105)(32 111 39 104)(33 110 40 103)(34 109 41 102)(35 108 42 101)(43 88 50 95)(44 87 51 94)(45 86 52 93)(46 85 53 92)(47 98 54 91)(48 97 55 90)(49 96 56 89)(71 135 78 128)(72 134 79 127)(73 133 80 140)(74 132 81 139)(75 131 82 138)(76 130 83 137)(77 129 84 136)(113 164 120 157)(114 163 121 156)(115 162 122 155)(116 161 123 168)(117 160 124 167)(118 159 125 166)(119 158 126 165)(141 208 148 201)(142 207 149 200)(143 206 150 199)(144 205 151 198)(145 204 152 197)(146 203 153 210)(147 202 154 209)(183 220 190 213)(184 219 191 212)(185 218 192 211)(186 217 193 224)(187 216 194 223)(188 215 195 222)(189 214 196 221)
(1 158 224 200)(2 159 211 201)(3 160 212 202)(4 161 213 203)(5 162 214 204)(6 163 215 205)(7 164 216 206)(8 165 217 207)(9 166 218 208)(10 167 219 209)(11 168 220 210)(12 155 221 197)(13 156 222 198)(14 157 223 199)(15 127 55 40)(16 128 56 41)(17 129 43 42)(18 130 44 29)(19 131 45 30)(20 132 46 31)(21 133 47 32)(22 134 48 33)(23 135 49 34)(24 136 50 35)(25 137 51 36)(26 138 52 37)(27 139 53 38)(28 140 54 39)(57 113 194 143)(58 114 195 144)(59 115 196 145)(60 116 183 146)(61 117 184 147)(62 118 185 148)(63 119 186 149)(64 120 187 150)(65 121 188 151)(66 122 189 152)(67 123 190 153)(68 124 191 154)(69 125 192 141)(70 126 193 142)(71 96 102 181)(72 97 103 182)(73 98 104 169)(74 85 105 170)(75 86 106 171)(76 87 107 172)(77 88 108 173)(78 89 109 174)(79 90 110 175)(80 91 111 176)(81 92 112 177)(82 93 99 178)(83 94 100 179)(84 95 101 180)
(1 31 217 139)(2 30 218 138)(3 29 219 137)(4 42 220 136)(5 41 221 135)(6 40 222 134)(7 39 223 133)(8 38 224 132)(9 37 211 131)(10 36 212 130)(11 35 213 129)(12 34 214 128)(13 33 215 127)(14 32 216 140)(15 163 48 198)(16 162 49 197)(17 161 50 210)(18 160 51 209)(19 159 52 208)(20 158 53 207)(21 157 54 206)(22 156 55 205)(23 155 56 204)(24 168 43 203)(25 167 44 202)(26 166 45 201)(27 165 46 200)(28 164 47 199)(57 104 187 80)(58 103 188 79)(59 102 189 78)(60 101 190 77)(61 100 191 76)(62 99 192 75)(63 112 193 74)(64 111 194 73)(65 110 195 72)(66 109 196 71)(67 108 183 84)(68 107 184 83)(69 106 185 82)(70 105 186 81)(85 142 177 119)(86 141 178 118)(87 154 179 117)(88 153 180 116)(89 152 181 115)(90 151 182 114)(91 150 169 113)(92 149 170 126)(93 148 171 125)(94 147 172 124)(95 146 173 123)(96 145 174 122)(97 144 175 121)(98 143 176 120)
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), (1,20,224,46)(2,19,211,45)(3,18,212,44)(4,17,213,43)(5,16,214,56)(6,15,215,55)(7,28,216,54)(8,27,217,53)(9,26,218,52)(10,25,219,51)(11,24,220,50)(12,23,221,49)(13,22,222,48)(14,21,223,47)(29,167,130,209)(30,166,131,208)(31,165,132,207)(32,164,133,206)(33,163,134,205)(34,162,135,204)(35,161,136,203)(36,160,137,202)(37,159,138,201)(38,158,139,200)(39,157,140,199)(40,156,127,198)(41,155,128,197)(42,168,129,210)(57,98,194,169)(58,97,195,182)(59,96,196,181)(60,95,183,180)(61,94,184,179)(62,93,185,178)(63,92,186,177)(64,91,187,176)(65,90,188,175)(66,89,189,174)(67,88,190,173)(68,87,191,172)(69,86,192,171)(70,85,193,170)(71,122,102,152)(72,121,103,151)(73,120,104,150)(74,119,105,149)(75,118,106,148)(76,117,107,147)(77,116,108,146)(78,115,109,145)(79,114,110,144)(80,113,111,143)(81,126,112,142)(82,125,99,141)(83,124,100,154)(84,123,101,153), (1,63,8,70)(2,62,9,69)(3,61,10,68)(4,60,11,67)(5,59,12,66)(6,58,13,65)(7,57,14,64)(15,175,22,182)(16,174,23,181)(17,173,24,180)(18,172,25,179)(19,171,26,178)(20,170,27,177)(21,169,28,176)(29,100,36,107)(30,99,37,106)(31,112,38,105)(32,111,39,104)(33,110,40,103)(34,109,41,102)(35,108,42,101)(43,88,50,95)(44,87,51,94)(45,86,52,93)(46,85,53,92)(47,98,54,91)(48,97,55,90)(49,96,56,89)(71,135,78,128)(72,134,79,127)(73,133,80,140)(74,132,81,139)(75,131,82,138)(76,130,83,137)(77,129,84,136)(113,164,120,157)(114,163,121,156)(115,162,122,155)(116,161,123,168)(117,160,124,167)(118,159,125,166)(119,158,126,165)(141,208,148,201)(142,207,149,200)(143,206,150,199)(144,205,151,198)(145,204,152,197)(146,203,153,210)(147,202,154,209)(183,220,190,213)(184,219,191,212)(185,218,192,211)(186,217,193,224)(187,216,194,223)(188,215,195,222)(189,214,196,221), (1,158,224,200)(2,159,211,201)(3,160,212,202)(4,161,213,203)(5,162,214,204)(6,163,215,205)(7,164,216,206)(8,165,217,207)(9,166,218,208)(10,167,219,209)(11,168,220,210)(12,155,221,197)(13,156,222,198)(14,157,223,199)(15,127,55,40)(16,128,56,41)(17,129,43,42)(18,130,44,29)(19,131,45,30)(20,132,46,31)(21,133,47,32)(22,134,48,33)(23,135,49,34)(24,136,50,35)(25,137,51,36)(26,138,52,37)(27,139,53,38)(28,140,54,39)(57,113,194,143)(58,114,195,144)(59,115,196,145)(60,116,183,146)(61,117,184,147)(62,118,185,148)(63,119,186,149)(64,120,187,150)(65,121,188,151)(66,122,189,152)(67,123,190,153)(68,124,191,154)(69,125,192,141)(70,126,193,142)(71,96,102,181)(72,97,103,182)(73,98,104,169)(74,85,105,170)(75,86,106,171)(76,87,107,172)(77,88,108,173)(78,89,109,174)(79,90,110,175)(80,91,111,176)(81,92,112,177)(82,93,99,178)(83,94,100,179)(84,95,101,180), (1,31,217,139)(2,30,218,138)(3,29,219,137)(4,42,220,136)(5,41,221,135)(6,40,222,134)(7,39,223,133)(8,38,224,132)(9,37,211,131)(10,36,212,130)(11,35,213,129)(12,34,214,128)(13,33,215,127)(14,32,216,140)(15,163,48,198)(16,162,49,197)(17,161,50,210)(18,160,51,209)(19,159,52,208)(20,158,53,207)(21,157,54,206)(22,156,55,205)(23,155,56,204)(24,168,43,203)(25,167,44,202)(26,166,45,201)(27,165,46,200)(28,164,47,199)(57,104,187,80)(58,103,188,79)(59,102,189,78)(60,101,190,77)(61,100,191,76)(62,99,192,75)(63,112,193,74)(64,111,194,73)(65,110,195,72)(66,109,196,71)(67,108,183,84)(68,107,184,83)(69,106,185,82)(70,105,186,81)(85,142,177,119)(86,141,178,118)(87,154,179,117)(88,153,180,116)(89,152,181,115)(90,151,182,114)(91,150,169,113)(92,149,170,126)(93,148,171,125)(94,147,172,124)(95,146,173,123)(96,145,174,122)(97,144,175,121)(98,143,176,120)>;
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), (1,20,224,46)(2,19,211,45)(3,18,212,44)(4,17,213,43)(5,16,214,56)(6,15,215,55)(7,28,216,54)(8,27,217,53)(9,26,218,52)(10,25,219,51)(11,24,220,50)(12,23,221,49)(13,22,222,48)(14,21,223,47)(29,167,130,209)(30,166,131,208)(31,165,132,207)(32,164,133,206)(33,163,134,205)(34,162,135,204)(35,161,136,203)(36,160,137,202)(37,159,138,201)(38,158,139,200)(39,157,140,199)(40,156,127,198)(41,155,128,197)(42,168,129,210)(57,98,194,169)(58,97,195,182)(59,96,196,181)(60,95,183,180)(61,94,184,179)(62,93,185,178)(63,92,186,177)(64,91,187,176)(65,90,188,175)(66,89,189,174)(67,88,190,173)(68,87,191,172)(69,86,192,171)(70,85,193,170)(71,122,102,152)(72,121,103,151)(73,120,104,150)(74,119,105,149)(75,118,106,148)(76,117,107,147)(77,116,108,146)(78,115,109,145)(79,114,110,144)(80,113,111,143)(81,126,112,142)(82,125,99,141)(83,124,100,154)(84,123,101,153), (1,63,8,70)(2,62,9,69)(3,61,10,68)(4,60,11,67)(5,59,12,66)(6,58,13,65)(7,57,14,64)(15,175,22,182)(16,174,23,181)(17,173,24,180)(18,172,25,179)(19,171,26,178)(20,170,27,177)(21,169,28,176)(29,100,36,107)(30,99,37,106)(31,112,38,105)(32,111,39,104)(33,110,40,103)(34,109,41,102)(35,108,42,101)(43,88,50,95)(44,87,51,94)(45,86,52,93)(46,85,53,92)(47,98,54,91)(48,97,55,90)(49,96,56,89)(71,135,78,128)(72,134,79,127)(73,133,80,140)(74,132,81,139)(75,131,82,138)(76,130,83,137)(77,129,84,136)(113,164,120,157)(114,163,121,156)(115,162,122,155)(116,161,123,168)(117,160,124,167)(118,159,125,166)(119,158,126,165)(141,208,148,201)(142,207,149,200)(143,206,150,199)(144,205,151,198)(145,204,152,197)(146,203,153,210)(147,202,154,209)(183,220,190,213)(184,219,191,212)(185,218,192,211)(186,217,193,224)(187,216,194,223)(188,215,195,222)(189,214,196,221), (1,158,224,200)(2,159,211,201)(3,160,212,202)(4,161,213,203)(5,162,214,204)(6,163,215,205)(7,164,216,206)(8,165,217,207)(9,166,218,208)(10,167,219,209)(11,168,220,210)(12,155,221,197)(13,156,222,198)(14,157,223,199)(15,127,55,40)(16,128,56,41)(17,129,43,42)(18,130,44,29)(19,131,45,30)(20,132,46,31)(21,133,47,32)(22,134,48,33)(23,135,49,34)(24,136,50,35)(25,137,51,36)(26,138,52,37)(27,139,53,38)(28,140,54,39)(57,113,194,143)(58,114,195,144)(59,115,196,145)(60,116,183,146)(61,117,184,147)(62,118,185,148)(63,119,186,149)(64,120,187,150)(65,121,188,151)(66,122,189,152)(67,123,190,153)(68,124,191,154)(69,125,192,141)(70,126,193,142)(71,96,102,181)(72,97,103,182)(73,98,104,169)(74,85,105,170)(75,86,106,171)(76,87,107,172)(77,88,108,173)(78,89,109,174)(79,90,110,175)(80,91,111,176)(81,92,112,177)(82,93,99,178)(83,94,100,179)(84,95,101,180), (1,31,217,139)(2,30,218,138)(3,29,219,137)(4,42,220,136)(5,41,221,135)(6,40,222,134)(7,39,223,133)(8,38,224,132)(9,37,211,131)(10,36,212,130)(11,35,213,129)(12,34,214,128)(13,33,215,127)(14,32,216,140)(15,163,48,198)(16,162,49,197)(17,161,50,210)(18,160,51,209)(19,159,52,208)(20,158,53,207)(21,157,54,206)(22,156,55,205)(23,155,56,204)(24,168,43,203)(25,167,44,202)(26,166,45,201)(27,165,46,200)(28,164,47,199)(57,104,187,80)(58,103,188,79)(59,102,189,78)(60,101,190,77)(61,100,191,76)(62,99,192,75)(63,112,193,74)(64,111,194,73)(65,110,195,72)(66,109,196,71)(67,108,183,84)(68,107,184,83)(69,106,185,82)(70,105,186,81)(85,142,177,119)(86,141,178,118)(87,154,179,117)(88,153,180,116)(89,152,181,115)(90,151,182,114)(91,150,169,113)(92,149,170,126)(93,148,171,125)(94,147,172,124)(95,146,173,123)(96,145,174,122)(97,144,175,121)(98,143,176,120) );
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)], [(1,20,224,46),(2,19,211,45),(3,18,212,44),(4,17,213,43),(5,16,214,56),(6,15,215,55),(7,28,216,54),(8,27,217,53),(9,26,218,52),(10,25,219,51),(11,24,220,50),(12,23,221,49),(13,22,222,48),(14,21,223,47),(29,167,130,209),(30,166,131,208),(31,165,132,207),(32,164,133,206),(33,163,134,205),(34,162,135,204),(35,161,136,203),(36,160,137,202),(37,159,138,201),(38,158,139,200),(39,157,140,199),(40,156,127,198),(41,155,128,197),(42,168,129,210),(57,98,194,169),(58,97,195,182),(59,96,196,181),(60,95,183,180),(61,94,184,179),(62,93,185,178),(63,92,186,177),(64,91,187,176),(65,90,188,175),(66,89,189,174),(67,88,190,173),(68,87,191,172),(69,86,192,171),(70,85,193,170),(71,122,102,152),(72,121,103,151),(73,120,104,150),(74,119,105,149),(75,118,106,148),(76,117,107,147),(77,116,108,146),(78,115,109,145),(79,114,110,144),(80,113,111,143),(81,126,112,142),(82,125,99,141),(83,124,100,154),(84,123,101,153)], [(1,63,8,70),(2,62,9,69),(3,61,10,68),(4,60,11,67),(5,59,12,66),(6,58,13,65),(7,57,14,64),(15,175,22,182),(16,174,23,181),(17,173,24,180),(18,172,25,179),(19,171,26,178),(20,170,27,177),(21,169,28,176),(29,100,36,107),(30,99,37,106),(31,112,38,105),(32,111,39,104),(33,110,40,103),(34,109,41,102),(35,108,42,101),(43,88,50,95),(44,87,51,94),(45,86,52,93),(46,85,53,92),(47,98,54,91),(48,97,55,90),(49,96,56,89),(71,135,78,128),(72,134,79,127),(73,133,80,140),(74,132,81,139),(75,131,82,138),(76,130,83,137),(77,129,84,136),(113,164,120,157),(114,163,121,156),(115,162,122,155),(116,161,123,168),(117,160,124,167),(118,159,125,166),(119,158,126,165),(141,208,148,201),(142,207,149,200),(143,206,150,199),(144,205,151,198),(145,204,152,197),(146,203,153,210),(147,202,154,209),(183,220,190,213),(184,219,191,212),(185,218,192,211),(186,217,193,224),(187,216,194,223),(188,215,195,222),(189,214,196,221)], [(1,158,224,200),(2,159,211,201),(3,160,212,202),(4,161,213,203),(5,162,214,204),(6,163,215,205),(7,164,216,206),(8,165,217,207),(9,166,218,208),(10,167,219,209),(11,168,220,210),(12,155,221,197),(13,156,222,198),(14,157,223,199),(15,127,55,40),(16,128,56,41),(17,129,43,42),(18,130,44,29),(19,131,45,30),(20,132,46,31),(21,133,47,32),(22,134,48,33),(23,135,49,34),(24,136,50,35),(25,137,51,36),(26,138,52,37),(27,139,53,38),(28,140,54,39),(57,113,194,143),(58,114,195,144),(59,115,196,145),(60,116,183,146),(61,117,184,147),(62,118,185,148),(63,119,186,149),(64,120,187,150),(65,121,188,151),(66,122,189,152),(67,123,190,153),(68,124,191,154),(69,125,192,141),(70,126,193,142),(71,96,102,181),(72,97,103,182),(73,98,104,169),(74,85,105,170),(75,86,106,171),(76,87,107,172),(77,88,108,173),(78,89,109,174),(79,90,110,175),(80,91,111,176),(81,92,112,177),(82,93,99,178),(83,94,100,179),(84,95,101,180)], [(1,31,217,139),(2,30,218,138),(3,29,219,137),(4,42,220,136),(5,41,221,135),(6,40,222,134),(7,39,223,133),(8,38,224,132),(9,37,211,131),(10,36,212,130),(11,35,213,129),(12,34,214,128),(13,33,215,127),(14,32,216,140),(15,163,48,198),(16,162,49,197),(17,161,50,210),(18,160,51,209),(19,159,52,208),(20,158,53,207),(21,157,54,206),(22,156,55,205),(23,155,56,204),(24,168,43,203),(25,167,44,202),(26,166,45,201),(27,165,46,200),(28,164,47,199),(57,104,187,80),(58,103,188,79),(59,102,189,78),(60,101,190,77),(61,100,191,76),(62,99,192,75),(63,112,193,74),(64,111,194,73),(65,110,195,72),(66,109,196,71),(67,108,183,84),(68,107,184,83),(69,106,185,82),(70,105,186,81),(85,142,177,119),(86,141,178,118),(87,154,179,117),(88,153,180,116),(89,152,181,115),(90,151,182,114),(91,150,169,113),(92,149,170,126),(93,148,171,125),(94,147,172,124),(95,146,173,123),(96,145,174,122),(97,144,175,121),(98,143,176,120)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 26 | 3 | 0 | 0 |
| 0 | 0 | 10 | 0 | 10 | 10 |
| 0 | 0 | 22 | 19 | 19 | 22 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 8 | 0 | 0 |
| 0 | 0 | 3 | 27 | 0 | 0 |
| 0 | 0 | 21 | 0 | 8 | 21 |
| 0 | 0 | 19 | 21 | 19 | 21 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 25 | 28 | 21 |
| 0 | 0 | 20 | 22 | 13 | 22 |
| 0 | 0 | 1 | 20 | 4 | 4 |
| 0 | 0 | 8 | 16 | 4 | 4 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 1 |
| 0 | 0 | 7 | 28 | 27 | 7 |
| 0 | 0 | 26 | 17 | 1 | 0 |
| 0 | 0 | 25 | 17 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 21 | 11 | 0 |
| 0 | 0 | 15 | 2 | 1 | 18 |
| 0 | 0 | 21 | 0 | 8 | 21 |
| 0 | 0 | 23 | 8 | 8 | 21 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,26,10,22,0,0,10,3,0,19,0,0,0,0,10,19,0,0,0,0,10,22],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,3,21,19,0,0,8,27,0,21,0,0,0,0,8,19,0,0,0,0,21,21],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,28,20,1,8,0,0,25,22,20,16,0,0,28,13,4,4,0,0,21,22,4,4],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,26,25,0,0,0,28,17,17,0,0,28,27,1,1,0,0,1,7,0,0],[0,28,0,0,0,0,1,0,0,0,0,0,0,0,27,15,21,23,0,0,21,2,0,8,0,0,11,1,8,8,0,0,0,18,21,21] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| 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 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | C7⋊D4 | 2- (1+4) | D4.10D14 |
| kernel | C14.1052- (1+4) | C28.48D4 | C23.21D14 | C23.18D14 | C28.17D4 | Dic7⋊Q8 | C22×Dic14 | C14×C4○D4 | C2×C28 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C14 | C2 |
| # reps | 1 | 4 | 1 | 4 | 2 | 2 | 1 | 1 | 4 | 3 | 9 | 9 | 3 | 24 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{105}2_-^{(1+4)} % in TeX
G:=Group("C14.105ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1278);
// by ID
G=gap.SmallGroup(448,1278);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,184,675,570,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=b^2,e^2=a^7*b^2,b*a*b^-1=c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=e*b*e^-1=a^7*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations
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