?
G = C14.172- (1+4) order 448 = 26·7
17th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.172- (1+4), C14.1162+ (1+4), (C4×D7)⋊3D4, C22⋊Q8⋊6D7, C4⋊C4.96D14, D14.4(C2×D4), C4.188(D4×D7), C28⋊1D4⋊24C2, C28⋊7D4⋊36C2, C28.233(C2×D4), D14⋊D4⋊24C2, D14⋊3Q8⋊15C2, D14⋊Q8⋊18C2, (C2×C28).53C23, (C2×Q8).125D14, C22⋊C4.15D14, Dic7.48(C2×D4), C14.75(C22×D4), (C2×C14).173C24, D14⋊C4.22C22, (C22×C4).235D14, C2.33(D4⋊8D14), (C2×D28).148C22, Dic7⋊C4.26C22, C4⋊Dic7.214C22, (Q8×C14).106C22, (C2×Dic7).88C23, C22.194(C23×D7), C23.118(C22×D7), (C22×C14).201C23, (C22×C28).253C22, C7⋊4(C22.31C24), (C22×D7).195C23, C2.18(Q8.10D14), (C2×Dic14).247C22, C2.48(C2×D4×D7), (D7×C4⋊C4)⋊25C2, (C7×C22⋊Q8)⋊9C2, (C2×C4○D28)⋊23C2, (C2×Q8⋊2D7)⋊6C2, (C2×C4×D7).94C22, (C7×C4⋊C4).157C22, (C2×C4).591(C22×D7), (C2×C7⋊D4).121C22, (C7×C22⋊C4).28C22, SmallGroup(448,1082)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1564 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×10], C22, C22 [×16], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×18], D4 [×16], Q8 [×4], C23, C23 [×4], D7 [×5], C14 [×3], C14, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×5], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8, C2×Q8, C4○D4 [×8], Dic7 [×2], Dic7 [×3], C28 [×2], C28 [×5], D14 [×2], D14 [×11], C2×C14, C2×C14 [×3], C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8, C22⋊Q8 [×3], C2×C4○D4 [×2], Dic14 [×2], C4×D7 [×4], C4×D7 [×8], D28 [×10], C2×Dic7 [×2], C2×Dic7 [×2], C7⋊D4 [×6], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7 [×2], C22×D7 [×2], C22×C14, C22.31C24, Dic7⋊C4 [×4], C4⋊Dic7, D14⋊C4 [×6], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7 [×2], C2×C4×D7 [×4], C2×D28 [×2], C2×D28 [×4], C4○D28 [×4], Q8⋊2D7 [×4], C2×C7⋊D4 [×2], C2×C7⋊D4 [×2], C22×C28, Q8×C14, D14⋊D4 [×4], D7×C4⋊C4, C28⋊1D4, C28⋊1D4 [×2], D14⋊Q8 [×2], C28⋊7D4, D14⋊3Q8, C7×C22⋊Q8, C2×C4○D28, C2×Q8⋊2D7, C14.172- (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- (1+4), C22×D7 [×7], C22.31C24, D4×D7 [×2], C23×D7, C2×D4×D7, Q8.10D14, D4⋊8D14, C14.172- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=a7b2, 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 >
►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)
(1 167 21 153)(2 168 22 154)(3 155 23 141)(4 156 24 142)(5 157 25 143)(6 158 26 144)(7 159 27 145)(8 160 28 146)(9 161 15 147)(10 162 16 148)(11 163 17 149)(12 164 18 150)(13 165 19 151)(14 166 20 152)(29 140 54 121)(30 127 55 122)(31 128 56 123)(32 129 43 124)(33 130 44 125)(34 131 45 126)(35 132 46 113)(36 133 47 114)(37 134 48 115)(38 135 49 116)(39 136 50 117)(40 137 51 118)(41 138 52 119)(42 139 53 120)(57 213 82 201)(58 214 83 202)(59 215 84 203)(60 216 71 204)(61 217 72 205)(62 218 73 206)(63 219 74 207)(64 220 75 208)(65 221 76 209)(66 222 77 210)(67 223 78 197)(68 224 79 198)(69 211 80 199)(70 212 81 200)(85 184 104 181)(86 185 105 182)(87 186 106 169)(88 187 107 170)(89 188 108 171)(90 189 109 172)(91 190 110 173)(92 191 111 174)(93 192 112 175)(94 193 99 176)(95 194 100 177)(96 195 101 178)(97 196 102 179)(98 183 103 180)
(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)(15 159)(16 158)(17 157)(18 156)(19 155)(20 168)(21 167)(22 166)(23 165)(24 164)(25 163)(26 162)(27 161)(28 160)(29 128)(30 127)(31 140)(32 139)(33 138)(34 137)(35 136)(36 135)(37 134)(38 133)(39 132)(40 131)(41 130)(42 129)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 126)(52 125)(53 124)(54 123)(55 122)(56 121)(57 205)(58 204)(59 203)(60 202)(61 201)(62 200)(63 199)(64 198)(65 197)(66 210)(67 209)(68 208)(69 207)(70 206)(71 214)(72 213)(73 212)(74 211)(75 224)(76 223)(77 222)(78 221)(79 220)(80 219)(81 218)(82 217)(83 216)(84 215)(85 194)(86 193)(87 192)(88 191)(89 190)(90 189)(91 188)(92 187)(93 186)(94 185)(95 184)(96 183)(97 196)(98 195)(99 182)(100 181)(101 180)(102 179)(103 178)(104 177)(105 176)(106 175)(107 174)(108 173)(109 172)(110 171)(111 170)(112 169)
(1 59 28 77)(2 58 15 76)(3 57 16 75)(4 70 17 74)(5 69 18 73)(6 68 19 72)(7 67 20 71)(8 66 21 84)(9 65 22 83)(10 64 23 82)(11 63 24 81)(12 62 25 80)(13 61 26 79)(14 60 27 78)(29 110 47 98)(30 109 48 97)(31 108 49 96)(32 107 50 95)(33 106 51 94)(34 105 52 93)(35 104 53 92)(36 103 54 91)(37 102 55 90)(38 101 56 89)(39 100 43 88)(40 99 44 87)(41 112 45 86)(42 111 46 85)(113 191 139 181)(114 190 140 180)(115 189 127 179)(116 188 128 178)(117 187 129 177)(118 186 130 176)(119 185 131 175)(120 184 132 174)(121 183 133 173)(122 196 134 172)(123 195 135 171)(124 194 136 170)(125 193 137 169)(126 192 138 182)(141 208 162 213)(142 207 163 212)(143 206 164 211)(144 205 165 224)(145 204 166 223)(146 203 167 222)(147 202 168 221)(148 201 155 220)(149 200 156 219)(150 199 157 218)(151 198 158 217)(152 197 159 216)(153 210 160 215)(154 209 161 214)
(1 127 21 122)(2 128 22 123)(3 129 23 124)(4 130 24 125)(5 131 25 126)(6 132 26 113)(7 133 27 114)(8 134 28 115)(9 135 15 116)(10 136 16 117)(11 137 17 118)(12 138 18 119)(13 139 19 120)(14 140 20 121)(29 166 54 152)(30 167 55 153)(31 168 56 154)(32 155 43 141)(33 156 44 142)(34 157 45 143)(35 158 46 144)(36 159 47 145)(37 160 48 146)(38 161 49 147)(39 162 50 148)(40 163 51 149)(41 164 52 150)(42 165 53 151)(57 194 82 177)(58 195 83 178)(59 196 84 179)(60 183 71 180)(61 184 72 181)(62 185 73 182)(63 186 74 169)(64 187 75 170)(65 188 76 171)(66 189 77 172)(67 190 78 173)(68 191 79 174)(69 192 80 175)(70 193 81 176)(85 217 104 205)(86 218 105 206)(87 219 106 207)(88 220 107 208)(89 221 108 209)(90 222 109 210)(91 223 110 197)(92 224 111 198)(93 211 112 199)(94 212 99 200)(95 213 100 201)(96 214 101 202)(97 215 102 203)(98 216 103 204)
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), (1,167,21,153)(2,168,22,154)(3,155,23,141)(4,156,24,142)(5,157,25,143)(6,158,26,144)(7,159,27,145)(8,160,28,146)(9,161,15,147)(10,162,16,148)(11,163,17,149)(12,164,18,150)(13,165,19,151)(14,166,20,152)(29,140,54,121)(30,127,55,122)(31,128,56,123)(32,129,43,124)(33,130,44,125)(34,131,45,126)(35,132,46,113)(36,133,47,114)(37,134,48,115)(38,135,49,116)(39,136,50,117)(40,137,51,118)(41,138,52,119)(42,139,53,120)(57,213,82,201)(58,214,83,202)(59,215,84,203)(60,216,71,204)(61,217,72,205)(62,218,73,206)(63,219,74,207)(64,220,75,208)(65,221,76,209)(66,222,77,210)(67,223,78,197)(68,224,79,198)(69,211,80,199)(70,212,81,200)(85,184,104,181)(86,185,105,182)(87,186,106,169)(88,187,107,170)(89,188,108,171)(90,189,109,172)(91,190,110,173)(92,191,111,174)(93,192,112,175)(94,193,99,176)(95,194,100,177)(96,195,101,178)(97,196,102,179)(98,183,103,180), (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)(15,159)(16,158)(17,157)(18,156)(19,155)(20,168)(21,167)(22,166)(23,165)(24,164)(25,163)(26,162)(27,161)(28,160)(29,128)(30,127)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,205)(58,204)(59,203)(60,202)(61,201)(62,200)(63,199)(64,198)(65,197)(66,210)(67,209)(68,208)(69,207)(70,206)(71,214)(72,213)(73,212)(74,211)(75,224)(76,223)(77,222)(78,221)(79,220)(80,219)(81,218)(82,217)(83,216)(84,215)(85,194)(86,193)(87,192)(88,191)(89,190)(90,189)(91,188)(92,187)(93,186)(94,185)(95,184)(96,183)(97,196)(98,195)(99,182)(100,181)(101,180)(102,179)(103,178)(104,177)(105,176)(106,175)(107,174)(108,173)(109,172)(110,171)(111,170)(112,169), (1,59,28,77)(2,58,15,76)(3,57,16,75)(4,70,17,74)(5,69,18,73)(6,68,19,72)(7,67,20,71)(8,66,21,84)(9,65,22,83)(10,64,23,82)(11,63,24,81)(12,62,25,80)(13,61,26,79)(14,60,27,78)(29,110,47,98)(30,109,48,97)(31,108,49,96)(32,107,50,95)(33,106,51,94)(34,105,52,93)(35,104,53,92)(36,103,54,91)(37,102,55,90)(38,101,56,89)(39,100,43,88)(40,99,44,87)(41,112,45,86)(42,111,46,85)(113,191,139,181)(114,190,140,180)(115,189,127,179)(116,188,128,178)(117,187,129,177)(118,186,130,176)(119,185,131,175)(120,184,132,174)(121,183,133,173)(122,196,134,172)(123,195,135,171)(124,194,136,170)(125,193,137,169)(126,192,138,182)(141,208,162,213)(142,207,163,212)(143,206,164,211)(144,205,165,224)(145,204,166,223)(146,203,167,222)(147,202,168,221)(148,201,155,220)(149,200,156,219)(150,199,157,218)(151,198,158,217)(152,197,159,216)(153,210,160,215)(154,209,161,214), (1,127,21,122)(2,128,22,123)(3,129,23,124)(4,130,24,125)(5,131,25,126)(6,132,26,113)(7,133,27,114)(8,134,28,115)(9,135,15,116)(10,136,16,117)(11,137,17,118)(12,138,18,119)(13,139,19,120)(14,140,20,121)(29,166,54,152)(30,167,55,153)(31,168,56,154)(32,155,43,141)(33,156,44,142)(34,157,45,143)(35,158,46,144)(36,159,47,145)(37,160,48,146)(38,161,49,147)(39,162,50,148)(40,163,51,149)(41,164,52,150)(42,165,53,151)(57,194,82,177)(58,195,83,178)(59,196,84,179)(60,183,71,180)(61,184,72,181)(62,185,73,182)(63,186,74,169)(64,187,75,170)(65,188,76,171)(66,189,77,172)(67,190,78,173)(68,191,79,174)(69,192,80,175)(70,193,81,176)(85,217,104,205)(86,218,105,206)(87,219,106,207)(88,220,107,208)(89,221,108,209)(90,222,109,210)(91,223,110,197)(92,224,111,198)(93,211,112,199)(94,212,99,200)(95,213,100,201)(96,214,101,202)(97,215,102,203)(98,216,103,204)>;
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), (1,167,21,153)(2,168,22,154)(3,155,23,141)(4,156,24,142)(5,157,25,143)(6,158,26,144)(7,159,27,145)(8,160,28,146)(9,161,15,147)(10,162,16,148)(11,163,17,149)(12,164,18,150)(13,165,19,151)(14,166,20,152)(29,140,54,121)(30,127,55,122)(31,128,56,123)(32,129,43,124)(33,130,44,125)(34,131,45,126)(35,132,46,113)(36,133,47,114)(37,134,48,115)(38,135,49,116)(39,136,50,117)(40,137,51,118)(41,138,52,119)(42,139,53,120)(57,213,82,201)(58,214,83,202)(59,215,84,203)(60,216,71,204)(61,217,72,205)(62,218,73,206)(63,219,74,207)(64,220,75,208)(65,221,76,209)(66,222,77,210)(67,223,78,197)(68,224,79,198)(69,211,80,199)(70,212,81,200)(85,184,104,181)(86,185,105,182)(87,186,106,169)(88,187,107,170)(89,188,108,171)(90,189,109,172)(91,190,110,173)(92,191,111,174)(93,192,112,175)(94,193,99,176)(95,194,100,177)(96,195,101,178)(97,196,102,179)(98,183,103,180), (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)(15,159)(16,158)(17,157)(18,156)(19,155)(20,168)(21,167)(22,166)(23,165)(24,164)(25,163)(26,162)(27,161)(28,160)(29,128)(30,127)(31,140)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,126)(52,125)(53,124)(54,123)(55,122)(56,121)(57,205)(58,204)(59,203)(60,202)(61,201)(62,200)(63,199)(64,198)(65,197)(66,210)(67,209)(68,208)(69,207)(70,206)(71,214)(72,213)(73,212)(74,211)(75,224)(76,223)(77,222)(78,221)(79,220)(80,219)(81,218)(82,217)(83,216)(84,215)(85,194)(86,193)(87,192)(88,191)(89,190)(90,189)(91,188)(92,187)(93,186)(94,185)(95,184)(96,183)(97,196)(98,195)(99,182)(100,181)(101,180)(102,179)(103,178)(104,177)(105,176)(106,175)(107,174)(108,173)(109,172)(110,171)(111,170)(112,169), (1,59,28,77)(2,58,15,76)(3,57,16,75)(4,70,17,74)(5,69,18,73)(6,68,19,72)(7,67,20,71)(8,66,21,84)(9,65,22,83)(10,64,23,82)(11,63,24,81)(12,62,25,80)(13,61,26,79)(14,60,27,78)(29,110,47,98)(30,109,48,97)(31,108,49,96)(32,107,50,95)(33,106,51,94)(34,105,52,93)(35,104,53,92)(36,103,54,91)(37,102,55,90)(38,101,56,89)(39,100,43,88)(40,99,44,87)(41,112,45,86)(42,111,46,85)(113,191,139,181)(114,190,140,180)(115,189,127,179)(116,188,128,178)(117,187,129,177)(118,186,130,176)(119,185,131,175)(120,184,132,174)(121,183,133,173)(122,196,134,172)(123,195,135,171)(124,194,136,170)(125,193,137,169)(126,192,138,182)(141,208,162,213)(142,207,163,212)(143,206,164,211)(144,205,165,224)(145,204,166,223)(146,203,167,222)(147,202,168,221)(148,201,155,220)(149,200,156,219)(150,199,157,218)(151,198,158,217)(152,197,159,216)(153,210,160,215)(154,209,161,214), (1,127,21,122)(2,128,22,123)(3,129,23,124)(4,130,24,125)(5,131,25,126)(6,132,26,113)(7,133,27,114)(8,134,28,115)(9,135,15,116)(10,136,16,117)(11,137,17,118)(12,138,18,119)(13,139,19,120)(14,140,20,121)(29,166,54,152)(30,167,55,153)(31,168,56,154)(32,155,43,141)(33,156,44,142)(34,157,45,143)(35,158,46,144)(36,159,47,145)(37,160,48,146)(38,161,49,147)(39,162,50,148)(40,163,51,149)(41,164,52,150)(42,165,53,151)(57,194,82,177)(58,195,83,178)(59,196,84,179)(60,183,71,180)(61,184,72,181)(62,185,73,182)(63,186,74,169)(64,187,75,170)(65,188,76,171)(66,189,77,172)(67,190,78,173)(68,191,79,174)(69,192,80,175)(70,193,81,176)(85,217,104,205)(86,218,105,206)(87,219,106,207)(88,220,107,208)(89,221,108,209)(90,222,109,210)(91,223,110,197)(92,224,111,198)(93,211,112,199)(94,212,99,200)(95,213,100,201)(96,214,101,202)(97,215,102,203)(98,216,103,204) );
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)], [(1,167,21,153),(2,168,22,154),(3,155,23,141),(4,156,24,142),(5,157,25,143),(6,158,26,144),(7,159,27,145),(8,160,28,146),(9,161,15,147),(10,162,16,148),(11,163,17,149),(12,164,18,150),(13,165,19,151),(14,166,20,152),(29,140,54,121),(30,127,55,122),(31,128,56,123),(32,129,43,124),(33,130,44,125),(34,131,45,126),(35,132,46,113),(36,133,47,114),(37,134,48,115),(38,135,49,116),(39,136,50,117),(40,137,51,118),(41,138,52,119),(42,139,53,120),(57,213,82,201),(58,214,83,202),(59,215,84,203),(60,216,71,204),(61,217,72,205),(62,218,73,206),(63,219,74,207),(64,220,75,208),(65,221,76,209),(66,222,77,210),(67,223,78,197),(68,224,79,198),(69,211,80,199),(70,212,81,200),(85,184,104,181),(86,185,105,182),(87,186,106,169),(88,187,107,170),(89,188,108,171),(90,189,109,172),(91,190,110,173),(92,191,111,174),(93,192,112,175),(94,193,99,176),(95,194,100,177),(96,195,101,178),(97,196,102,179),(98,183,103,180)], [(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),(15,159),(16,158),(17,157),(18,156),(19,155),(20,168),(21,167),(22,166),(23,165),(24,164),(25,163),(26,162),(27,161),(28,160),(29,128),(30,127),(31,140),(32,139),(33,138),(34,137),(35,136),(36,135),(37,134),(38,133),(39,132),(40,131),(41,130),(42,129),(43,120),(44,119),(45,118),(46,117),(47,116),(48,115),(49,114),(50,113),(51,126),(52,125),(53,124),(54,123),(55,122),(56,121),(57,205),(58,204),(59,203),(60,202),(61,201),(62,200),(63,199),(64,198),(65,197),(66,210),(67,209),(68,208),(69,207),(70,206),(71,214),(72,213),(73,212),(74,211),(75,224),(76,223),(77,222),(78,221),(79,220),(80,219),(81,218),(82,217),(83,216),(84,215),(85,194),(86,193),(87,192),(88,191),(89,190),(90,189),(91,188),(92,187),(93,186),(94,185),(95,184),(96,183),(97,196),(98,195),(99,182),(100,181),(101,180),(102,179),(103,178),(104,177),(105,176),(106,175),(107,174),(108,173),(109,172),(110,171),(111,170),(112,169)], [(1,59,28,77),(2,58,15,76),(3,57,16,75),(4,70,17,74),(5,69,18,73),(6,68,19,72),(7,67,20,71),(8,66,21,84),(9,65,22,83),(10,64,23,82),(11,63,24,81),(12,62,25,80),(13,61,26,79),(14,60,27,78),(29,110,47,98),(30,109,48,97),(31,108,49,96),(32,107,50,95),(33,106,51,94),(34,105,52,93),(35,104,53,92),(36,103,54,91),(37,102,55,90),(38,101,56,89),(39,100,43,88),(40,99,44,87),(41,112,45,86),(42,111,46,85),(113,191,139,181),(114,190,140,180),(115,189,127,179),(116,188,128,178),(117,187,129,177),(118,186,130,176),(119,185,131,175),(120,184,132,174),(121,183,133,173),(122,196,134,172),(123,195,135,171),(124,194,136,170),(125,193,137,169),(126,192,138,182),(141,208,162,213),(142,207,163,212),(143,206,164,211),(144,205,165,224),(145,204,166,223),(146,203,167,222),(147,202,168,221),(148,201,155,220),(149,200,156,219),(150,199,157,218),(151,198,158,217),(152,197,159,216),(153,210,160,215),(154,209,161,214)], [(1,127,21,122),(2,128,22,123),(3,129,23,124),(4,130,24,125),(5,131,25,126),(6,132,26,113),(7,133,27,114),(8,134,28,115),(9,135,15,116),(10,136,16,117),(11,137,17,118),(12,138,18,119),(13,139,19,120),(14,140,20,121),(29,166,54,152),(30,167,55,153),(31,168,56,154),(32,155,43,141),(33,156,44,142),(34,157,45,143),(35,158,46,144),(36,159,47,145),(37,160,48,146),(38,161,49,147),(39,162,50,148),(40,163,51,149),(41,164,52,150),(42,165,53,151),(57,194,82,177),(58,195,83,178),(59,196,84,179),(60,183,71,180),(61,184,72,181),(62,185,73,182),(63,186,74,169),(64,187,75,170),(65,188,76,171),(66,189,77,172),(67,190,78,173),(68,191,79,174),(69,192,80,175),(70,193,81,176),(85,217,104,205),(86,218,105,206),(87,219,106,207),(88,220,107,208),(89,221,108,209),(90,222,109,210),(91,223,110,197),(92,224,111,198),(93,211,112,199),(94,212,99,200),(95,213,100,201),(96,214,101,202),(97,215,102,203),(98,216,103,204)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 19 | 21 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 28 | 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 | 28 | 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 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 16 | 0 | 11 |
| 0 | 0 | 0 | 0 | 16 | 8 | 8 | 8 |
| 0 | 0 | 0 | 0 | 11 | 13 | 1 | 15 |
| 0 | 0 | 0 | 0 | 17 | 23 | 20 | 13 |
| 0 | 26 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 16 | 0 | 11 |
| 0 | 0 | 0 | 0 | 10 | 10 | 21 | 8 |
| 0 | 0 | 0 | 0 | 3 | 6 | 28 | 15 |
| 0 | 0 | 0 | 0 | 18 | 28 | 9 | 13 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 2 | 0 |
| 0 | 0 | 0 | 0 | 24 | 4 | 12 | 12 |
| 0 | 0 | 0 | 0 | 18 | 15 | 16 | 16 |
| 0 | 0 | 0 | 0 | 3 | 6 | 5 | 9 |
| 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 5 | 18 | 0 |
| 0 | 0 | 0 | 0 | 13 | 21 | 21 | 21 |
| 0 | 0 | 0 | 0 | 8 | 8 | 14 | 28 |
| 0 | 0 | 0 | 0 | 9 | 1 | 16 | 9 |
G:=sub<GL(8,GF(29))| [19,17,0,0,0,0,0,0,21,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,28,0,0,0,0,0,0,0,0,7,16,11,17,0,0,0,0,16,8,13,23,0,0,0,0,0,8,1,20,0,0,0,0,11,8,15,13],[0,19,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,7,10,3,18,0,0,0,0,16,10,6,28,0,0,0,0,0,21,28,9,0,0,0,0,11,8,15,13],[0,10,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,22,22,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,0,24,18,3,0,0,0,0,7,4,15,6,0,0,0,0,2,12,16,5,0,0,0,0,0,12,16,9],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,14,13,8,9,0,0,0,0,5,21,8,1,0,0,0,0,18,21,14,16,0,0,0,0,0,21,28,9] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 28 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4×D7 | Q8.10D14 | D4⋊8D14 |
| kernel | C14.172- (1+4) | D14⋊D4 | D7×C4⋊C4 | C28⋊1D4 | D14⋊Q8 | C28⋊7D4 | D14⋊3Q8 | C7×C22⋊Q8 | C2×C4○D28 | C2×Q8⋊2D7 | C4×D7 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C4 | C2 | C2 |
| # reps | 1 | 4 | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 6 | 9 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{17}2_-^{(1+4)} % in TeX
G:=Group("C14.17ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1082);
// by ID
G=gap.SmallGroup(448,1082);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=a^7*b^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
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