?
G = C14.1182+ (1+4) order 448 = 26·7
27th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1182+ (1+4), C7⋊D4⋊2Q8, C28⋊Q8⋊26C2, C7⋊5(D4⋊3Q8), C22⋊Q8⋊12D7, D14.9(C2×Q8), C4⋊C4.192D14, (Q8×Dic7)⋊13C2, C22.1(Q8×D7), D14⋊2Q8⋊28C2, D14⋊3Q8⋊17C2, Dic7.Q8⋊18C2, (C2×C28).58C23, (C2×Q8).129D14, C22⋊C4.60D14, Dic7.11(C2×Q8), C28.210(C4○D4), C4.99(D4⋊2D7), C14.37(C22×Q8), (C2×C14).179C24, D14⋊C4.24C22, Dic7⋊4D4.2C2, (C22×C4).241D14, C2.36(D4⋊8D14), C22⋊Dic14⋊25C2, C4⋊Dic7.374C22, (Q8×C14).110C22, (C2×Dic7).90C23, C23.192(C22×D7), C22.200(C23×D7), Dic7⋊C4.118C22, (C22×C28).258C22, (C22×C14).207C23, (C4×Dic7).108C22, (C22×D7).201C23, C23.D7.119C22, (C2×Dic14).160C22, (C22×Dic7).120C22, (D7×C4⋊C4)⋊27C2, C2.20(C2×Q8×D7), (C4×C7⋊D4).8C2, (C2×C14).8(C2×Q8), (C2×C4⋊Dic7)⋊42C2, C14.90(C2×C4○D4), (C7×C22⋊Q8)⋊15C2, (C2×C4×D7).99C22, C2.46(C2×D4⋊2D7), (C7×C4⋊C4).161C22, (C2×C4).184(C22×D7), (C2×C7⋊D4).126C22, (C7×C22⋊C4).34C22, SmallGroup(448,1088)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 956 in 228 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×13], C22, C22 [×2], C22 [×6], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×4], Q8 [×4], C23, C23, D7 [×2], C14 [×3], C14 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4, C22×C4 [×5], C2×D4, C2×Q8, C2×Q8 [×2], Dic7 [×2], Dic7 [×6], C28 [×2], C28 [×5], D14 [×2], D14 [×2], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8, C22⋊Q8 [×5], C42.C2 [×2], C4⋊Q8, Dic14 [×2], C4×D7 [×4], C2×Dic7 [×3], C2×Dic7 [×4], C2×Dic7 [×2], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7, C22×C14, D4⋊3Q8, C4×Dic7, C4×Dic7 [×2], Dic7⋊C4, Dic7⋊C4 [×6], C4⋊Dic7 [×2], C4⋊Dic7 [×4], D14⋊C4, D14⋊C4 [×2], C23.D7, C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14 [×2], C2×C4×D7, C2×C4×D7 [×2], C22×Dic7 [×2], C2×C7⋊D4, C22×C28, Q8×C14, C22⋊Dic14 [×2], Dic7⋊4D4 [×2], C28⋊Q8, Dic7.Q8 [×2], D7×C4⋊C4, D14⋊2Q8 [×2], C2×C4⋊Dic7, C4×C7⋊D4, Q8×Dic7, D14⋊3Q8, C7×C22⋊Q8, C14.1182+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], Q8 [×4], C23 [×15], D7, C2×Q8 [×6], C4○D4 [×2], C24, D14 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), C22×D7 [×7], D4⋊3Q8, D4⋊2D7 [×2], Q8×D7 [×2], C23×D7, C2×D4⋊2D7, C2×Q8×D7, D4⋊8D14, C14.1182+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7, ab=ba, cac=a-1, ad=da, ae=ea, cbc=a7b-1, bd=db, ebe-1=a7b, 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 28 143 33)(2 15 144 34)(3 16 145 35)(4 17 146 36)(5 18 147 37)(6 19 148 38)(7 20 149 39)(8 21 150 40)(9 22 151 41)(10 23 152 42)(11 24 153 29)(12 25 154 30)(13 26 141 31)(14 27 142 32)(43 73 161 58)(44 74 162 59)(45 75 163 60)(46 76 164 61)(47 77 165 62)(48 78 166 63)(49 79 167 64)(50 80 168 65)(51 81 155 66)(52 82 156 67)(53 83 157 68)(54 84 158 69)(55 71 159 70)(56 72 160 57)(85 195 214 200)(86 196 215 201)(87 183 216 202)(88 184 217 203)(89 185 218 204)(90 186 219 205)(91 187 220 206)(92 188 221 207)(93 189 222 208)(94 190 223 209)(95 191 224 210)(96 192 211 197)(97 193 212 198)(98 194 213 199)(99 182 113 130)(100 169 114 131)(101 170 115 132)(102 171 116 133)(103 172 117 134)(104 173 118 135)(105 174 119 136)(106 175 120 137)(107 176 121 138)(108 177 122 139)(109 178 123 140)(110 179 124 127)(111 180 125 128)(112 181 126 129)
(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 39)(16 38)(17 37)(18 36)(19 35)(20 34)(21 33)(22 32)(23 31)(24 30)(25 29)(26 42)(27 41)(28 40)(43 47)(44 46)(48 56)(49 55)(50 54)(51 53)(57 71)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(85 219)(86 218)(87 217)(88 216)(89 215)(90 214)(91 213)(92 212)(93 211)(94 224)(95 223)(96 222)(97 221)(98 220)(99 107)(100 106)(101 105)(102 104)(108 112)(109 111)(113 121)(114 120)(115 119)(116 118)(122 126)(123 125)(127 172)(128 171)(129 170)(130 169)(131 182)(132 181)(133 180)(134 179)(135 178)(136 177)(137 176)(138 175)(139 174)(140 173)(141 145)(142 144)(146 154)(147 153)(148 152)(149 151)(155 157)(158 168)(159 167)(160 166)(161 165)(162 164)(183 191)(184 190)(185 189)(186 188)(192 196)(193 195)(197 201)(198 200)(202 210)(203 209)(204 208)(205 207)
(1 156 143 52)(2 157 144 53)(3 158 145 54)(4 159 146 55)(5 160 147 56)(6 161 148 43)(7 162 149 44)(8 163 150 45)(9 164 151 46)(10 165 152 47)(11 166 153 48)(12 167 154 49)(13 168 141 50)(14 155 142 51)(15 68 34 83)(16 69 35 84)(17 70 36 71)(18 57 37 72)(19 58 38 73)(20 59 39 74)(21 60 40 75)(22 61 41 76)(23 62 42 77)(24 63 29 78)(25 64 30 79)(26 65 31 80)(27 66 32 81)(28 67 33 82)(85 128 214 180)(86 129 215 181)(87 130 216 182)(88 131 217 169)(89 132 218 170)(90 133 219 171)(91 134 220 172)(92 135 221 173)(93 136 222 174)(94 137 223 175)(95 138 224 176)(96 139 211 177)(97 140 212 178)(98 127 213 179)(99 202 113 183)(100 203 114 184)(101 204 115 185)(102 205 116 186)(103 206 117 187)(104 207 118 188)(105 208 119 189)(106 209 120 190)(107 210 121 191)(108 197 122 192)(109 198 123 193)(110 199 124 194)(111 200 125 195)(112 201 126 196)
(1 187 8 194)(2 188 9 195)(3 189 10 196)(4 190 11 183)(5 191 12 184)(6 192 13 185)(7 193 14 186)(15 214 22 221)(16 215 23 222)(17 216 24 223)(18 217 25 224)(19 218 26 211)(20 219 27 212)(21 220 28 213)(29 94 36 87)(30 95 37 88)(31 96 38 89)(32 97 39 90)(33 98 40 91)(34 85 41 92)(35 86 42 93)(43 108 50 101)(44 109 51 102)(45 110 52 103)(46 111 53 104)(47 112 54 105)(48 99 55 106)(49 100 56 107)(57 131 64 138)(58 132 65 139)(59 133 66 140)(60 134 67 127)(61 135 68 128)(62 136 69 129)(63 137 70 130)(71 182 78 175)(72 169 79 176)(73 170 80 177)(74 171 81 178)(75 172 82 179)(76 173 83 180)(77 174 84 181)(113 159 120 166)(114 160 121 167)(115 161 122 168)(116 162 123 155)(117 163 124 156)(118 164 125 157)(119 165 126 158)(141 204 148 197)(142 205 149 198)(143 206 150 199)(144 207 151 200)(145 208 152 201)(146 209 153 202)(147 210 154 203)
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), (1,28,143,33)(2,15,144,34)(3,16,145,35)(4,17,146,36)(5,18,147,37)(6,19,148,38)(7,20,149,39)(8,21,150,40)(9,22,151,41)(10,23,152,42)(11,24,153,29)(12,25,154,30)(13,26,141,31)(14,27,142,32)(43,73,161,58)(44,74,162,59)(45,75,163,60)(46,76,164,61)(47,77,165,62)(48,78,166,63)(49,79,167,64)(50,80,168,65)(51,81,155,66)(52,82,156,67)(53,83,157,68)(54,84,158,69)(55,71,159,70)(56,72,160,57)(85,195,214,200)(86,196,215,201)(87,183,216,202)(88,184,217,203)(89,185,218,204)(90,186,219,205)(91,187,220,206)(92,188,221,207)(93,189,222,208)(94,190,223,209)(95,191,224,210)(96,192,211,197)(97,193,212,198)(98,194,213,199)(99,182,113,130)(100,169,114,131)(101,170,115,132)(102,171,116,133)(103,172,117,134)(104,173,118,135)(105,174,119,136)(106,175,120,137)(107,176,121,138)(108,177,122,139)(109,178,123,140)(110,179,124,127)(111,180,125,128)(112,181,126,129), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,42)(27,41)(28,40)(43,47)(44,46)(48,56)(49,55)(50,54)(51,53)(57,71)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,219)(86,218)(87,217)(88,216)(89,215)(90,214)(91,213)(92,212)(93,211)(94,224)(95,223)(96,222)(97,221)(98,220)(99,107)(100,106)(101,105)(102,104)(108,112)(109,111)(113,121)(114,120)(115,119)(116,118)(122,126)(123,125)(127,172)(128,171)(129,170)(130,169)(131,182)(132,181)(133,180)(134,179)(135,178)(136,177)(137,176)(138,175)(139,174)(140,173)(141,145)(142,144)(146,154)(147,153)(148,152)(149,151)(155,157)(158,168)(159,167)(160,166)(161,165)(162,164)(183,191)(184,190)(185,189)(186,188)(192,196)(193,195)(197,201)(198,200)(202,210)(203,209)(204,208)(205,207), (1,156,143,52)(2,157,144,53)(3,158,145,54)(4,159,146,55)(5,160,147,56)(6,161,148,43)(7,162,149,44)(8,163,150,45)(9,164,151,46)(10,165,152,47)(11,166,153,48)(12,167,154,49)(13,168,141,50)(14,155,142,51)(15,68,34,83)(16,69,35,84)(17,70,36,71)(18,57,37,72)(19,58,38,73)(20,59,39,74)(21,60,40,75)(22,61,41,76)(23,62,42,77)(24,63,29,78)(25,64,30,79)(26,65,31,80)(27,66,32,81)(28,67,33,82)(85,128,214,180)(86,129,215,181)(87,130,216,182)(88,131,217,169)(89,132,218,170)(90,133,219,171)(91,134,220,172)(92,135,221,173)(93,136,222,174)(94,137,223,175)(95,138,224,176)(96,139,211,177)(97,140,212,178)(98,127,213,179)(99,202,113,183)(100,203,114,184)(101,204,115,185)(102,205,116,186)(103,206,117,187)(104,207,118,188)(105,208,119,189)(106,209,120,190)(107,210,121,191)(108,197,122,192)(109,198,123,193)(110,199,124,194)(111,200,125,195)(112,201,126,196), (1,187,8,194)(2,188,9,195)(3,189,10,196)(4,190,11,183)(5,191,12,184)(6,192,13,185)(7,193,14,186)(15,214,22,221)(16,215,23,222)(17,216,24,223)(18,217,25,224)(19,218,26,211)(20,219,27,212)(21,220,28,213)(29,94,36,87)(30,95,37,88)(31,96,38,89)(32,97,39,90)(33,98,40,91)(34,85,41,92)(35,86,42,93)(43,108,50,101)(44,109,51,102)(45,110,52,103)(46,111,53,104)(47,112,54,105)(48,99,55,106)(49,100,56,107)(57,131,64,138)(58,132,65,139)(59,133,66,140)(60,134,67,127)(61,135,68,128)(62,136,69,129)(63,137,70,130)(71,182,78,175)(72,169,79,176)(73,170,80,177)(74,171,81,178)(75,172,82,179)(76,173,83,180)(77,174,84,181)(113,159,120,166)(114,160,121,167)(115,161,122,168)(116,162,123,155)(117,163,124,156)(118,164,125,157)(119,165,126,158)(141,204,148,197)(142,205,149,198)(143,206,150,199)(144,207,151,200)(145,208,152,201)(146,209,153,202)(147,210,154,203)>;
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), (1,28,143,33)(2,15,144,34)(3,16,145,35)(4,17,146,36)(5,18,147,37)(6,19,148,38)(7,20,149,39)(8,21,150,40)(9,22,151,41)(10,23,152,42)(11,24,153,29)(12,25,154,30)(13,26,141,31)(14,27,142,32)(43,73,161,58)(44,74,162,59)(45,75,163,60)(46,76,164,61)(47,77,165,62)(48,78,166,63)(49,79,167,64)(50,80,168,65)(51,81,155,66)(52,82,156,67)(53,83,157,68)(54,84,158,69)(55,71,159,70)(56,72,160,57)(85,195,214,200)(86,196,215,201)(87,183,216,202)(88,184,217,203)(89,185,218,204)(90,186,219,205)(91,187,220,206)(92,188,221,207)(93,189,222,208)(94,190,223,209)(95,191,224,210)(96,192,211,197)(97,193,212,198)(98,194,213,199)(99,182,113,130)(100,169,114,131)(101,170,115,132)(102,171,116,133)(103,172,117,134)(104,173,118,135)(105,174,119,136)(106,175,120,137)(107,176,121,138)(108,177,122,139)(109,178,123,140)(110,179,124,127)(111,180,125,128)(112,181,126,129), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,42)(27,41)(28,40)(43,47)(44,46)(48,56)(49,55)(50,54)(51,53)(57,71)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79)(64,78)(65,77)(66,76)(67,75)(68,74)(69,73)(70,72)(85,219)(86,218)(87,217)(88,216)(89,215)(90,214)(91,213)(92,212)(93,211)(94,224)(95,223)(96,222)(97,221)(98,220)(99,107)(100,106)(101,105)(102,104)(108,112)(109,111)(113,121)(114,120)(115,119)(116,118)(122,126)(123,125)(127,172)(128,171)(129,170)(130,169)(131,182)(132,181)(133,180)(134,179)(135,178)(136,177)(137,176)(138,175)(139,174)(140,173)(141,145)(142,144)(146,154)(147,153)(148,152)(149,151)(155,157)(158,168)(159,167)(160,166)(161,165)(162,164)(183,191)(184,190)(185,189)(186,188)(192,196)(193,195)(197,201)(198,200)(202,210)(203,209)(204,208)(205,207), (1,156,143,52)(2,157,144,53)(3,158,145,54)(4,159,146,55)(5,160,147,56)(6,161,148,43)(7,162,149,44)(8,163,150,45)(9,164,151,46)(10,165,152,47)(11,166,153,48)(12,167,154,49)(13,168,141,50)(14,155,142,51)(15,68,34,83)(16,69,35,84)(17,70,36,71)(18,57,37,72)(19,58,38,73)(20,59,39,74)(21,60,40,75)(22,61,41,76)(23,62,42,77)(24,63,29,78)(25,64,30,79)(26,65,31,80)(27,66,32,81)(28,67,33,82)(85,128,214,180)(86,129,215,181)(87,130,216,182)(88,131,217,169)(89,132,218,170)(90,133,219,171)(91,134,220,172)(92,135,221,173)(93,136,222,174)(94,137,223,175)(95,138,224,176)(96,139,211,177)(97,140,212,178)(98,127,213,179)(99,202,113,183)(100,203,114,184)(101,204,115,185)(102,205,116,186)(103,206,117,187)(104,207,118,188)(105,208,119,189)(106,209,120,190)(107,210,121,191)(108,197,122,192)(109,198,123,193)(110,199,124,194)(111,200,125,195)(112,201,126,196), (1,187,8,194)(2,188,9,195)(3,189,10,196)(4,190,11,183)(5,191,12,184)(6,192,13,185)(7,193,14,186)(15,214,22,221)(16,215,23,222)(17,216,24,223)(18,217,25,224)(19,218,26,211)(20,219,27,212)(21,220,28,213)(29,94,36,87)(30,95,37,88)(31,96,38,89)(32,97,39,90)(33,98,40,91)(34,85,41,92)(35,86,42,93)(43,108,50,101)(44,109,51,102)(45,110,52,103)(46,111,53,104)(47,112,54,105)(48,99,55,106)(49,100,56,107)(57,131,64,138)(58,132,65,139)(59,133,66,140)(60,134,67,127)(61,135,68,128)(62,136,69,129)(63,137,70,130)(71,182,78,175)(72,169,79,176)(73,170,80,177)(74,171,81,178)(75,172,82,179)(76,173,83,180)(77,174,84,181)(113,159,120,166)(114,160,121,167)(115,161,122,168)(116,162,123,155)(117,163,124,156)(118,164,125,157)(119,165,126,158)(141,204,148,197)(142,205,149,198)(143,206,150,199)(144,207,151,200)(145,208,152,201)(146,209,153,202)(147,210,154,203) );
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)], [(1,28,143,33),(2,15,144,34),(3,16,145,35),(4,17,146,36),(5,18,147,37),(6,19,148,38),(7,20,149,39),(8,21,150,40),(9,22,151,41),(10,23,152,42),(11,24,153,29),(12,25,154,30),(13,26,141,31),(14,27,142,32),(43,73,161,58),(44,74,162,59),(45,75,163,60),(46,76,164,61),(47,77,165,62),(48,78,166,63),(49,79,167,64),(50,80,168,65),(51,81,155,66),(52,82,156,67),(53,83,157,68),(54,84,158,69),(55,71,159,70),(56,72,160,57),(85,195,214,200),(86,196,215,201),(87,183,216,202),(88,184,217,203),(89,185,218,204),(90,186,219,205),(91,187,220,206),(92,188,221,207),(93,189,222,208),(94,190,223,209),(95,191,224,210),(96,192,211,197),(97,193,212,198),(98,194,213,199),(99,182,113,130),(100,169,114,131),(101,170,115,132),(102,171,116,133),(103,172,117,134),(104,173,118,135),(105,174,119,136),(106,175,120,137),(107,176,121,138),(108,177,122,139),(109,178,123,140),(110,179,124,127),(111,180,125,128),(112,181,126,129)], [(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(15,39),(16,38),(17,37),(18,36),(19,35),(20,34),(21,33),(22,32),(23,31),(24,30),(25,29),(26,42),(27,41),(28,40),(43,47),(44,46),(48,56),(49,55),(50,54),(51,53),(57,71),(58,84),(59,83),(60,82),(61,81),(62,80),(63,79),(64,78),(65,77),(66,76),(67,75),(68,74),(69,73),(70,72),(85,219),(86,218),(87,217),(88,216),(89,215),(90,214),(91,213),(92,212),(93,211),(94,224),(95,223),(96,222),(97,221),(98,220),(99,107),(100,106),(101,105),(102,104),(108,112),(109,111),(113,121),(114,120),(115,119),(116,118),(122,126),(123,125),(127,172),(128,171),(129,170),(130,169),(131,182),(132,181),(133,180),(134,179),(135,178),(136,177),(137,176),(138,175),(139,174),(140,173),(141,145),(142,144),(146,154),(147,153),(148,152),(149,151),(155,157),(158,168),(159,167),(160,166),(161,165),(162,164),(183,191),(184,190),(185,189),(186,188),(192,196),(193,195),(197,201),(198,200),(202,210),(203,209),(204,208),(205,207)], [(1,156,143,52),(2,157,144,53),(3,158,145,54),(4,159,146,55),(5,160,147,56),(6,161,148,43),(7,162,149,44),(8,163,150,45),(9,164,151,46),(10,165,152,47),(11,166,153,48),(12,167,154,49),(13,168,141,50),(14,155,142,51),(15,68,34,83),(16,69,35,84),(17,70,36,71),(18,57,37,72),(19,58,38,73),(20,59,39,74),(21,60,40,75),(22,61,41,76),(23,62,42,77),(24,63,29,78),(25,64,30,79),(26,65,31,80),(27,66,32,81),(28,67,33,82),(85,128,214,180),(86,129,215,181),(87,130,216,182),(88,131,217,169),(89,132,218,170),(90,133,219,171),(91,134,220,172),(92,135,221,173),(93,136,222,174),(94,137,223,175),(95,138,224,176),(96,139,211,177),(97,140,212,178),(98,127,213,179),(99,202,113,183),(100,203,114,184),(101,204,115,185),(102,205,116,186),(103,206,117,187),(104,207,118,188),(105,208,119,189),(106,209,120,190),(107,210,121,191),(108,197,122,192),(109,198,123,193),(110,199,124,194),(111,200,125,195),(112,201,126,196)], [(1,187,8,194),(2,188,9,195),(3,189,10,196),(4,190,11,183),(5,191,12,184),(6,192,13,185),(7,193,14,186),(15,214,22,221),(16,215,23,222),(17,216,24,223),(18,217,25,224),(19,218,26,211),(20,219,27,212),(21,220,28,213),(29,94,36,87),(30,95,37,88),(31,96,38,89),(32,97,39,90),(33,98,40,91),(34,85,41,92),(35,86,42,93),(43,108,50,101),(44,109,51,102),(45,110,52,103),(46,111,53,104),(47,112,54,105),(48,99,55,106),(49,100,56,107),(57,131,64,138),(58,132,65,139),(59,133,66,140),(60,134,67,127),(61,135,68,128),(62,136,69,129),(63,137,70,130),(71,182,78,175),(72,169,79,176),(73,170,80,177),(74,171,81,178),(75,172,82,179),(76,173,83,180),(77,174,84,181),(113,159,120,166),(114,160,121,167),(115,161,122,168),(116,162,123,155),(117,163,124,156),(118,164,125,157),(119,165,126,158),(141,204,148,197),(142,205,149,198),(143,206,150,199),(144,207,151,200),(145,208,152,201),(146,209,153,202),(147,210,154,203)])
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 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 27 |
| 0 | 0 | 0 | 0 | 12 | 20 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 2 |
| 0 | 0 | 0 | 0 | 17 | 9 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 22 |
| 0 | 0 | 0 | 0 | 1 | 21 |
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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,9,12,0,0,0,0,27,20],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,7,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,17,0,0,0,0,2,9],[17,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,1,0,0,0,0,22,21] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 14 | 14 | 2 | 2 | 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 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | D4⋊2D7 | Q8×D7 | D4⋊8D14 |
| kernel | C14.1182+ (1+4) | C22⋊Dic14 | Dic7⋊4D4 | C28⋊Q8 | Dic7.Q8 | D7×C4⋊C4 | D14⋊2Q8 | C2×C4⋊Dic7 | C4×C7⋊D4 | Q8×Dic7 | D14⋊3Q8 | C7×C22⋊Q8 | C7⋊D4 | C22⋊Q8 | C28 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{118}2_+^{(1+4)} % in TeX
G:=Group("C14.118ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1088);
// by ID
G=gap.SmallGroup(448,1088);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,100,570,185,192,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=a^7*b^-1,b*d=d*b,e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
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