?
G = C14.212- (1+4) order 448 = 26·7
21st non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.212- (1+4), C22⋊Q8⋊16D7, C4⋊C4.194D14, (Q8×Dic7)⋊14C2, D14⋊3Q8⋊21C2, Dic7.Q8⋊21C2, (C2×C28).61C23, (C2×Q8).130D14, C22⋊C4.19D14, C4.Dic14⋊24C2, D14.20(C4○D4), C28.211(C4○D4), (C2×C14).183C24, D14.D4.2C2, (C22×C4).245D14, C4.100(D4⋊2D7), D14⋊C4.129C22, C23.D14⋊23C2, Dic7⋊C4.31C22, C4⋊Dic7.375C22, (Q8×C14).113C22, (C2×Dic7).93C23, C23.122(C22×D7), C22.204(C23×D7), C23.21D14⋊29C2, (C22×C14).211C23, (C22×C28).259C22, C7⋊6(C22.46C24), (C4×Dic7).111C22, (C22×D7).204C23, C23.D7.123C22, C2.22(Q8.10D14), (D7×C4⋊C4)⋊30C2, C2.54(D7×C4○D4), (C4×C7⋊D4).9C2, C4⋊C4⋊7D7⋊27C2, (C7×C22⋊Q8)⋊19C2, C14.166(C2×C4○D4), C2.47(C2×D4⋊2D7), (C2×C4×D7).101C22, (C2×C4).53(C22×D7), (C7×C4⋊C4).164C22, (C2×C7⋊D4).130C22, (C7×C22⋊C4).38C22, SmallGroup(448,1092)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 828 in 214 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×7], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, D7 [×2], C14 [×3], C14, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4, C22×C4 [×3], C2×D4, C2×Q8, Dic7 [×7], C28 [×2], C28 [×5], D14 [×2], D14 [×2], C2×C14, C2×C14 [×3], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C22.D4 [×2], C42.C2 [×3], C42⋊2C2 [×2], C4×D7 [×6], C2×Dic7 [×3], C2×Dic7 [×4], C7⋊D4 [×2], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7, C22×C14, C22.46C24, C4×Dic7, C4×Dic7 [×4], Dic7⋊C4, Dic7⋊C4 [×6], C4⋊Dic7 [×2], C4⋊Dic7 [×4], D14⋊C4, D14⋊C4 [×2], C23.D7, C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×C4×D7, C2×C4×D7 [×2], C2×C7⋊D4, C22×C28, Q8×C14, C23.D14 [×2], D14.D4 [×2], Dic7.Q8 [×2], C4.Dic14, D7×C4⋊C4, C4⋊C4⋊7D7 [×2], C23.21D14, C4×C7⋊D4, Q8×Dic7, D14⋊3Q8, C7×C22⋊Q8, C14.212- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D7 [×7], C22.46C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, Q8.10D14, D7×C4○D4, C14.212- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=a7b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a7b-1, bd=db, be=eb, cd=dc, 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)
(1 166 19 149)(2 167 20 150)(3 168 21 151)(4 155 22 152)(5 156 23 153)(6 157 24 154)(7 158 25 141)(8 159 26 142)(9 160 27 143)(10 161 28 144)(11 162 15 145)(12 163 16 146)(13 164 17 147)(14 165 18 148)(29 139 54 119)(30 140 55 120)(31 127 56 121)(32 128 43 122)(33 129 44 123)(34 130 45 124)(35 131 46 125)(36 132 47 126)(37 133 48 113)(38 134 49 114)(39 135 50 115)(40 136 51 116)(41 137 52 117)(42 138 53 118)(57 222 71 205)(58 223 72 206)(59 224 73 207)(60 211 74 208)(61 212 75 209)(62 213 76 210)(63 214 77 197)(64 215 78 198)(65 216 79 199)(66 217 80 200)(67 218 81 201)(68 219 82 202)(69 220 83 203)(70 221 84 204)(85 189 102 172)(86 190 103 173)(87 191 104 174)(88 192 105 175)(89 193 106 176)(90 194 107 177)(91 195 108 178)(92 196 109 179)(93 183 110 180)(94 184 111 181)(95 185 112 182)(96 186 99 169)(97 187 100 170)(98 188 101 171)
(1 149 8 142)(2 148 9 141)(3 147 10 154)(4 146 11 153)(5 145 12 152)(6 144 13 151)(7 143 14 150)(15 156 22 163)(16 155 23 162)(17 168 24 161)(18 167 25 160)(19 166 26 159)(20 165 27 158)(21 164 28 157)(29 134 36 127)(30 133 37 140)(31 132 38 139)(32 131 39 138)(33 130 40 137)(34 129 41 136)(35 128 42 135)(43 125 50 118)(44 124 51 117)(45 123 52 116)(46 122 53 115)(47 121 54 114)(48 120 55 113)(49 119 56 126)(57 207 64 200)(58 206 65 199)(59 205 66 198)(60 204 67 197)(61 203 68 210)(62 202 69 209)(63 201 70 208)(71 224 78 217)(72 223 79 216)(73 222 80 215)(74 221 81 214)(75 220 82 213)(76 219 83 212)(77 218 84 211)(85 188 92 195)(86 187 93 194)(87 186 94 193)(88 185 95 192)(89 184 96 191)(90 183 97 190)(91 196 98 189)(99 174 106 181)(100 173 107 180)(101 172 108 179)(102 171 109 178)(103 170 110 177)(104 169 111 176)(105 182 112 175)
(1 65 26 72)(2 64 27 71)(3 63 28 84)(4 62 15 83)(5 61 16 82)(6 60 17 81)(7 59 18 80)(8 58 19 79)(9 57 20 78)(10 70 21 77)(11 69 22 76)(12 68 23 75)(13 67 24 74)(14 66 25 73)(29 106 47 96)(30 105 48 95)(31 104 49 94)(32 103 50 93)(33 102 51 92)(34 101 52 91)(35 100 53 90)(36 99 54 89)(37 112 55 88)(38 111 56 87)(39 110 43 86)(40 109 44 85)(41 108 45 98)(42 107 46 97)(113 185 140 175)(114 184 127 174)(115 183 128 173)(116 196 129 172)(117 195 130 171)(118 194 131 170)(119 193 132 169)(120 192 133 182)(121 191 134 181)(122 190 135 180)(123 189 136 179)(124 188 137 178)(125 187 138 177)(126 186 139 176)(141 207 165 217)(142 206 166 216)(143 205 167 215)(144 204 168 214)(145 203 155 213)(146 202 156 212)(147 201 157 211)(148 200 158 224)(149 199 159 223)(150 198 160 222)(151 197 161 221)(152 210 162 220)(153 209 163 219)(154 208 164 218)
(1 133 19 113)(2 134 20 114)(3 135 21 115)(4 136 22 116)(5 137 23 117)(6 138 24 118)(7 139 25 119)(8 140 26 120)(9 127 27 121)(10 128 28 122)(11 129 15 123)(12 130 16 124)(13 131 17 125)(14 132 18 126)(29 158 54 141)(30 159 55 142)(31 160 56 143)(32 161 43 144)(33 162 44 145)(34 163 45 146)(35 164 46 147)(36 165 47 148)(37 166 48 149)(38 167 49 150)(39 168 50 151)(40 155 51 152)(41 156 52 153)(42 157 53 154)(57 191 71 174)(58 192 72 175)(59 193 73 176)(60 194 74 177)(61 195 75 178)(62 196 76 179)(63 183 77 180)(64 184 78 181)(65 185 79 182)(66 186 80 169)(67 187 81 170)(68 188 82 171)(69 189 83 172)(70 190 84 173)(85 220 102 203)(86 221 103 204)(87 222 104 205)(88 223 105 206)(89 224 106 207)(90 211 107 208)(91 212 108 209)(92 213 109 210)(93 214 110 197)(94 215 111 198)(95 216 112 199)(96 217 99 200)(97 218 100 201)(98 219 101 202)
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), (1,166,19,149)(2,167,20,150)(3,168,21,151)(4,155,22,152)(5,156,23,153)(6,157,24,154)(7,158,25,141)(8,159,26,142)(9,160,27,143)(10,161,28,144)(11,162,15,145)(12,163,16,146)(13,164,17,147)(14,165,18,148)(29,139,54,119)(30,140,55,120)(31,127,56,121)(32,128,43,122)(33,129,44,123)(34,130,45,124)(35,131,46,125)(36,132,47,126)(37,133,48,113)(38,134,49,114)(39,135,50,115)(40,136,51,116)(41,137,52,117)(42,138,53,118)(57,222,71,205)(58,223,72,206)(59,224,73,207)(60,211,74,208)(61,212,75,209)(62,213,76,210)(63,214,77,197)(64,215,78,198)(65,216,79,199)(66,217,80,200)(67,218,81,201)(68,219,82,202)(69,220,83,203)(70,221,84,204)(85,189,102,172)(86,190,103,173)(87,191,104,174)(88,192,105,175)(89,193,106,176)(90,194,107,177)(91,195,108,178)(92,196,109,179)(93,183,110,180)(94,184,111,181)(95,185,112,182)(96,186,99,169)(97,187,100,170)(98,188,101,171), (1,149,8,142)(2,148,9,141)(3,147,10,154)(4,146,11,153)(5,145,12,152)(6,144,13,151)(7,143,14,150)(15,156,22,163)(16,155,23,162)(17,168,24,161)(18,167,25,160)(19,166,26,159)(20,165,27,158)(21,164,28,157)(29,134,36,127)(30,133,37,140)(31,132,38,139)(32,131,39,138)(33,130,40,137)(34,129,41,136)(35,128,42,135)(43,125,50,118)(44,124,51,117)(45,123,52,116)(46,122,53,115)(47,121,54,114)(48,120,55,113)(49,119,56,126)(57,207,64,200)(58,206,65,199)(59,205,66,198)(60,204,67,197)(61,203,68,210)(62,202,69,209)(63,201,70,208)(71,224,78,217)(72,223,79,216)(73,222,80,215)(74,221,81,214)(75,220,82,213)(76,219,83,212)(77,218,84,211)(85,188,92,195)(86,187,93,194)(87,186,94,193)(88,185,95,192)(89,184,96,191)(90,183,97,190)(91,196,98,189)(99,174,106,181)(100,173,107,180)(101,172,108,179)(102,171,109,178)(103,170,110,177)(104,169,111,176)(105,182,112,175), (1,65,26,72)(2,64,27,71)(3,63,28,84)(4,62,15,83)(5,61,16,82)(6,60,17,81)(7,59,18,80)(8,58,19,79)(9,57,20,78)(10,70,21,77)(11,69,22,76)(12,68,23,75)(13,67,24,74)(14,66,25,73)(29,106,47,96)(30,105,48,95)(31,104,49,94)(32,103,50,93)(33,102,51,92)(34,101,52,91)(35,100,53,90)(36,99,54,89)(37,112,55,88)(38,111,56,87)(39,110,43,86)(40,109,44,85)(41,108,45,98)(42,107,46,97)(113,185,140,175)(114,184,127,174)(115,183,128,173)(116,196,129,172)(117,195,130,171)(118,194,131,170)(119,193,132,169)(120,192,133,182)(121,191,134,181)(122,190,135,180)(123,189,136,179)(124,188,137,178)(125,187,138,177)(126,186,139,176)(141,207,165,217)(142,206,166,216)(143,205,167,215)(144,204,168,214)(145,203,155,213)(146,202,156,212)(147,201,157,211)(148,200,158,224)(149,199,159,223)(150,198,160,222)(151,197,161,221)(152,210,162,220)(153,209,163,219)(154,208,164,218), (1,133,19,113)(2,134,20,114)(3,135,21,115)(4,136,22,116)(5,137,23,117)(6,138,24,118)(7,139,25,119)(8,140,26,120)(9,127,27,121)(10,128,28,122)(11,129,15,123)(12,130,16,124)(13,131,17,125)(14,132,18,126)(29,158,54,141)(30,159,55,142)(31,160,56,143)(32,161,43,144)(33,162,44,145)(34,163,45,146)(35,164,46,147)(36,165,47,148)(37,166,48,149)(38,167,49,150)(39,168,50,151)(40,155,51,152)(41,156,52,153)(42,157,53,154)(57,191,71,174)(58,192,72,175)(59,193,73,176)(60,194,74,177)(61,195,75,178)(62,196,76,179)(63,183,77,180)(64,184,78,181)(65,185,79,182)(66,186,80,169)(67,187,81,170)(68,188,82,171)(69,189,83,172)(70,190,84,173)(85,220,102,203)(86,221,103,204)(87,222,104,205)(88,223,105,206)(89,224,106,207)(90,211,107,208)(91,212,108,209)(92,213,109,210)(93,214,110,197)(94,215,111,198)(95,216,112,199)(96,217,99,200)(97,218,100,201)(98,219,101,202)>;
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), (1,166,19,149)(2,167,20,150)(3,168,21,151)(4,155,22,152)(5,156,23,153)(6,157,24,154)(7,158,25,141)(8,159,26,142)(9,160,27,143)(10,161,28,144)(11,162,15,145)(12,163,16,146)(13,164,17,147)(14,165,18,148)(29,139,54,119)(30,140,55,120)(31,127,56,121)(32,128,43,122)(33,129,44,123)(34,130,45,124)(35,131,46,125)(36,132,47,126)(37,133,48,113)(38,134,49,114)(39,135,50,115)(40,136,51,116)(41,137,52,117)(42,138,53,118)(57,222,71,205)(58,223,72,206)(59,224,73,207)(60,211,74,208)(61,212,75,209)(62,213,76,210)(63,214,77,197)(64,215,78,198)(65,216,79,199)(66,217,80,200)(67,218,81,201)(68,219,82,202)(69,220,83,203)(70,221,84,204)(85,189,102,172)(86,190,103,173)(87,191,104,174)(88,192,105,175)(89,193,106,176)(90,194,107,177)(91,195,108,178)(92,196,109,179)(93,183,110,180)(94,184,111,181)(95,185,112,182)(96,186,99,169)(97,187,100,170)(98,188,101,171), (1,149,8,142)(2,148,9,141)(3,147,10,154)(4,146,11,153)(5,145,12,152)(6,144,13,151)(7,143,14,150)(15,156,22,163)(16,155,23,162)(17,168,24,161)(18,167,25,160)(19,166,26,159)(20,165,27,158)(21,164,28,157)(29,134,36,127)(30,133,37,140)(31,132,38,139)(32,131,39,138)(33,130,40,137)(34,129,41,136)(35,128,42,135)(43,125,50,118)(44,124,51,117)(45,123,52,116)(46,122,53,115)(47,121,54,114)(48,120,55,113)(49,119,56,126)(57,207,64,200)(58,206,65,199)(59,205,66,198)(60,204,67,197)(61,203,68,210)(62,202,69,209)(63,201,70,208)(71,224,78,217)(72,223,79,216)(73,222,80,215)(74,221,81,214)(75,220,82,213)(76,219,83,212)(77,218,84,211)(85,188,92,195)(86,187,93,194)(87,186,94,193)(88,185,95,192)(89,184,96,191)(90,183,97,190)(91,196,98,189)(99,174,106,181)(100,173,107,180)(101,172,108,179)(102,171,109,178)(103,170,110,177)(104,169,111,176)(105,182,112,175), (1,65,26,72)(2,64,27,71)(3,63,28,84)(4,62,15,83)(5,61,16,82)(6,60,17,81)(7,59,18,80)(8,58,19,79)(9,57,20,78)(10,70,21,77)(11,69,22,76)(12,68,23,75)(13,67,24,74)(14,66,25,73)(29,106,47,96)(30,105,48,95)(31,104,49,94)(32,103,50,93)(33,102,51,92)(34,101,52,91)(35,100,53,90)(36,99,54,89)(37,112,55,88)(38,111,56,87)(39,110,43,86)(40,109,44,85)(41,108,45,98)(42,107,46,97)(113,185,140,175)(114,184,127,174)(115,183,128,173)(116,196,129,172)(117,195,130,171)(118,194,131,170)(119,193,132,169)(120,192,133,182)(121,191,134,181)(122,190,135,180)(123,189,136,179)(124,188,137,178)(125,187,138,177)(126,186,139,176)(141,207,165,217)(142,206,166,216)(143,205,167,215)(144,204,168,214)(145,203,155,213)(146,202,156,212)(147,201,157,211)(148,200,158,224)(149,199,159,223)(150,198,160,222)(151,197,161,221)(152,210,162,220)(153,209,163,219)(154,208,164,218), (1,133,19,113)(2,134,20,114)(3,135,21,115)(4,136,22,116)(5,137,23,117)(6,138,24,118)(7,139,25,119)(8,140,26,120)(9,127,27,121)(10,128,28,122)(11,129,15,123)(12,130,16,124)(13,131,17,125)(14,132,18,126)(29,158,54,141)(30,159,55,142)(31,160,56,143)(32,161,43,144)(33,162,44,145)(34,163,45,146)(35,164,46,147)(36,165,47,148)(37,166,48,149)(38,167,49,150)(39,168,50,151)(40,155,51,152)(41,156,52,153)(42,157,53,154)(57,191,71,174)(58,192,72,175)(59,193,73,176)(60,194,74,177)(61,195,75,178)(62,196,76,179)(63,183,77,180)(64,184,78,181)(65,185,79,182)(66,186,80,169)(67,187,81,170)(68,188,82,171)(69,189,83,172)(70,190,84,173)(85,220,102,203)(86,221,103,204)(87,222,104,205)(88,223,105,206)(89,224,106,207)(90,211,107,208)(91,212,108,209)(92,213,109,210)(93,214,110,197)(94,215,111,198)(95,216,112,199)(96,217,99,200)(97,218,100,201)(98,219,101,202) );
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)], [(1,166,19,149),(2,167,20,150),(3,168,21,151),(4,155,22,152),(5,156,23,153),(6,157,24,154),(7,158,25,141),(8,159,26,142),(9,160,27,143),(10,161,28,144),(11,162,15,145),(12,163,16,146),(13,164,17,147),(14,165,18,148),(29,139,54,119),(30,140,55,120),(31,127,56,121),(32,128,43,122),(33,129,44,123),(34,130,45,124),(35,131,46,125),(36,132,47,126),(37,133,48,113),(38,134,49,114),(39,135,50,115),(40,136,51,116),(41,137,52,117),(42,138,53,118),(57,222,71,205),(58,223,72,206),(59,224,73,207),(60,211,74,208),(61,212,75,209),(62,213,76,210),(63,214,77,197),(64,215,78,198),(65,216,79,199),(66,217,80,200),(67,218,81,201),(68,219,82,202),(69,220,83,203),(70,221,84,204),(85,189,102,172),(86,190,103,173),(87,191,104,174),(88,192,105,175),(89,193,106,176),(90,194,107,177),(91,195,108,178),(92,196,109,179),(93,183,110,180),(94,184,111,181),(95,185,112,182),(96,186,99,169),(97,187,100,170),(98,188,101,171)], [(1,149,8,142),(2,148,9,141),(3,147,10,154),(4,146,11,153),(5,145,12,152),(6,144,13,151),(7,143,14,150),(15,156,22,163),(16,155,23,162),(17,168,24,161),(18,167,25,160),(19,166,26,159),(20,165,27,158),(21,164,28,157),(29,134,36,127),(30,133,37,140),(31,132,38,139),(32,131,39,138),(33,130,40,137),(34,129,41,136),(35,128,42,135),(43,125,50,118),(44,124,51,117),(45,123,52,116),(46,122,53,115),(47,121,54,114),(48,120,55,113),(49,119,56,126),(57,207,64,200),(58,206,65,199),(59,205,66,198),(60,204,67,197),(61,203,68,210),(62,202,69,209),(63,201,70,208),(71,224,78,217),(72,223,79,216),(73,222,80,215),(74,221,81,214),(75,220,82,213),(76,219,83,212),(77,218,84,211),(85,188,92,195),(86,187,93,194),(87,186,94,193),(88,185,95,192),(89,184,96,191),(90,183,97,190),(91,196,98,189),(99,174,106,181),(100,173,107,180),(101,172,108,179),(102,171,109,178),(103,170,110,177),(104,169,111,176),(105,182,112,175)], [(1,65,26,72),(2,64,27,71),(3,63,28,84),(4,62,15,83),(5,61,16,82),(6,60,17,81),(7,59,18,80),(8,58,19,79),(9,57,20,78),(10,70,21,77),(11,69,22,76),(12,68,23,75),(13,67,24,74),(14,66,25,73),(29,106,47,96),(30,105,48,95),(31,104,49,94),(32,103,50,93),(33,102,51,92),(34,101,52,91),(35,100,53,90),(36,99,54,89),(37,112,55,88),(38,111,56,87),(39,110,43,86),(40,109,44,85),(41,108,45,98),(42,107,46,97),(113,185,140,175),(114,184,127,174),(115,183,128,173),(116,196,129,172),(117,195,130,171),(118,194,131,170),(119,193,132,169),(120,192,133,182),(121,191,134,181),(122,190,135,180),(123,189,136,179),(124,188,137,178),(125,187,138,177),(126,186,139,176),(141,207,165,217),(142,206,166,216),(143,205,167,215),(144,204,168,214),(145,203,155,213),(146,202,156,212),(147,201,157,211),(148,200,158,224),(149,199,159,223),(150,198,160,222),(151,197,161,221),(152,210,162,220),(153,209,163,219),(154,208,164,218)], [(1,133,19,113),(2,134,20,114),(3,135,21,115),(4,136,22,116),(5,137,23,117),(6,138,24,118),(7,139,25,119),(8,140,26,120),(9,127,27,121),(10,128,28,122),(11,129,15,123),(12,130,16,124),(13,131,17,125),(14,132,18,126),(29,158,54,141),(30,159,55,142),(31,160,56,143),(32,161,43,144),(33,162,44,145),(34,163,45,146),(35,164,46,147),(36,165,47,148),(37,166,48,149),(38,167,49,150),(39,168,50,151),(40,155,51,152),(41,156,52,153),(42,157,53,154),(57,191,71,174),(58,192,72,175),(59,193,73,176),(60,194,74,177),(61,195,75,178),(62,196,76,179),(63,183,77,180),(64,184,78,181),(65,185,79,182),(66,186,80,169),(67,187,81,170),(68,188,82,171),(69,189,83,172),(70,190,84,173),(85,220,102,203),(86,221,103,204),(87,222,104,205),(88,223,105,206),(89,224,106,207),(90,211,107,208),(91,212,108,209),(92,213,109,210),(93,214,110,197),(94,215,111,198),(95,216,112,199),(96,217,99,200),(97,218,100,201),(98,219,101,202)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 19 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 3 |
| 0 | 0 | 0 | 0 | 8 | 21 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,10,19,0,0,0,0,10,22,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,7,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,1,7,0,0,0,0,0,28,0,0,0,0,0,0,8,8,0,0,0,0,3,21],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,5,17] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2- (1+4) | D4⋊2D7 | Q8.10D14 | D7×C4○D4 |
| kernel | C14.212- (1+4) | C23.D14 | D14.D4 | Dic7.Q8 | C4.Dic14 | D7×C4⋊C4 | C4⋊C4⋊7D7 | C23.21D14 | C4×C7⋊D4 | Q8×Dic7 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | C28 | D14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{21}2_-^{(1+4)} % in TeX
G:=Group("C14.21ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1092);
// by ID
G=gap.SmallGroup(448,1092);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,268,1571,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=a^7*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^7*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀