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G = C14.662+ (1+4)  order 448 = 26·7

66th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C14.662+ (1+4), C28⋊1D4⋊31C2, C28⋊7D4⋊22C2, C28⋊D4⋊21C2, C4⋊C4.200D14, D14⋊D4⋊34C2, (C2×D4).103D14, Dic7.Q8⋊29C2, C22⋊C4.30D14, Dic7⋊4D4⋊24C2, Dic7⋊D4⋊23C2, D14.D4⋊36C2, D14.5D4⋊30C2, (C2×C14).207C24, (C2×C28).182C23, (C22×C4).261D14, C22.D4⋊12D7, C2.68(D4⋊6D14), C2.44(D4⋊8D14), C23.31(C22×D7), D14⋊C4.109C22, Dic7.10(C4○D4), (C2×D28).158C22, (D4×C14).145C22, C22.D28⋊22C2, Dic7⋊C4.45C22, C4⋊Dic7.230C22, (C22×C14).39C23, (C22×D7).88C23, C22.228(C23×D7), (C22×C28).117C22, C7⋊5(C22.34C24), (C2×Dic7).247C23, (C4×Dic7).126C22, C23.D7.128C22, (C22×Dic7).133C22, (C4×C7⋊D4)⋊9C2, C2.69(D7×C4○D4), C4⋊C4⋊7D7⋊34C2, C14.181(C2×C4○D4), (C2×C4×D7).115C22, (C2×C4).69(C22×D7), (C7×C4⋊C4).180C22, (C2×C7⋊D4).51C22, (C7×C22.D4)⋊15C2, (C7×C22⋊C4).55C22, SmallGroup(448,1116)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C14.662+ (1+4)
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D14.5D4 — C14.662+ (1+4)
Lower central
C7 — C2×C14 — C14.662+ (1+4)

Subgroups: 1260 in 240 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×5], C4 [×11], C22, C22 [×15], C7, C2×C4 [×5], C2×C4 [×11], D4 [×12], C23 [×2], C23 [×3], D7 [×3], C14 [×3], C14 [×2], C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×9], Dic7 [×2], Dic7 [×4], C28 [×5], D14 [×9], C2×C14, C2×C14 [×6], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4, C22.D4 [×3], C42.C2, C4⋊1D4, C4×D7 [×4], D28 [×3], C2×Dic7 [×5], C2×Dic7, C7⋊D4 [×8], C2×C28 [×5], C2×C28, C7×D4, C22×D7 [×3], C22×C14 [×2], C22.34C24, C4×Dic7 [×2], Dic7⋊C4 [×4], C4⋊Dic7 [×2], D14⋊C4 [×6], C23.D7, C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×C4×D7 [×3], C2×D28 [×3], C22×Dic7, C2×C7⋊D4 [×6], C22×C28, D4×C14, Dic7⋊4D4, D14.D4, D14⋊D4 [×3], C22.D28, Dic7.Q8, C4⋊C4⋊7D7, D14.5D4, C28⋊1D4, C4×C7⋊D4, C28⋊7D4, Dic7⋊D4, C28⋊D4, C7×C22.D4, C14.662+ (1+4)

Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D7 [×7], C22.34C24, C23×D7, D4⋊6D14, D7×C4○D4, D4⋊8D14, C14.662+ (1+4)

Generators and relations
 G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, ebe=a7b, cd=dc, ce=ec, ede=b2d >

Smallest permutation representation
►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)

(1 183 24 181)(2 184 25 182)(3 185 26 169)(4 186 27 170)(5 187 28 171)(6 188 15 172)(7 189 16 173)(8 190 17 174)(9 191 18 175)(10 192 19 176)(11 193 20 177)(12 194 21 178)(13 195 22 179)(14 196 23 180)(29 216 43 210)(30 217 44 197)(31 218 45 198)(32 219 46 199)(33 220 47 200)(34 221 48 201)(35 222 49 202)(36 223 50 203)(37 224 51 204)(38 211 52 205)(39 212 53 206)(40 213 54 207)(41 214 55 208)(42 215 56 209)(57 138 83 123)(58 139 84 124)(59 140 71 125)(60 127 72 126)(61 128 73 113)(62 129 74 114)(63 130 75 115)(64 131 76 116)(65 132 77 117)(66 133 78 118)(67 134 79 119)(68 135 80 120)(69 136 81 121)(70 137 82 122)(85 160 100 149)(86 161 101 150)(87 162 102 151)(88 163 103 152)(89 164 104 153)(90 165 105 154)(91 166 106 141)(92 167 107 142)(93 168 108 143)(94 155 109 144)(95 156 110 145)(96 157 111 146)(97 158 112 147)(98 159 99 148)

(1 123 8 116)(2 124 9 117)(3 125 10 118)(4 126 11 119)(5 113 12 120)(6 114 13 121)(7 115 14 122)(15 129 22 136)(16 130 23 137)(17 131 24 138)(18 132 25 139)(19 133 26 140)(20 134 27 127)(21 135 28 128)(29 145 36 152)(30 146 37 153)(31 147 38 154)(32 148 39 141)(33 149 40 142)(34 150 41 143)(35 151 42 144)(43 156 50 163)(44 157 51 164)(45 158 52 165)(46 159 53 166)(47 160 54 167)(48 161 55 168)(49 162 56 155)(57 174 64 181)(58 175 65 182)(59 176 66 169)(60 177 67 170)(61 178 68 171)(62 179 69 172)(63 180 70 173)(71 192 78 185)(72 193 79 186)(73 194 80 187)(74 195 81 188)(75 196 82 189)(76 183 83 190)(77 184 84 191)(85 207 92 200)(86 208 93 201)(87 209 94 202)(88 210 95 203)(89 197 96 204)(90 198 97 205)(91 199 98 206)(99 212 106 219)(100 213 107 220)(101 214 108 221)(102 215 109 222)(103 216 110 223)(104 217 111 224)(105 218 112 211)

(1 107 24 92)(2 106 25 91)(3 105 26 90)(4 104 27 89)(5 103 28 88)(6 102 15 87)(7 101 16 86)(8 100 17 85)(9 99 18 98)(10 112 19 97)(11 111 20 96)(12 110 21 95)(13 109 22 94)(14 108 23 93)(29 61 43 73)(30 60 44 72)(31 59 45 71)(32 58 46 84)(33 57 47 83)(34 70 48 82)(35 69 49 81)(36 68 50 80)(37 67 51 79)(38 66 52 78)(39 65 53 77)(40 64 54 76)(41 63 55 75)(42 62 56 74)(113 216 128 210)(114 215 129 209)(115 214 130 208)(116 213 131 207)(117 212 132 206)(118 211 133 205)(119 224 134 204)(120 223 135 203)(121 222 136 202)(122 221 137 201)(123 220 138 200)(124 219 139 199)(125 218 140 198)(126 217 127 197)(141 182 166 184)(142 181 167 183)(143 180 168 196)(144 179 155 195)(145 178 156 194)(146 177 157 193)(147 176 158 192)(148 175 159 191)(149 174 160 190)(150 173 161 189)(151 172 162 188)(152 171 163 187)(153 170 164 186)(154 169 165 185)

(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 29)(12 30)(13 31)(14 32)(15 52)(16 53)(17 54)(18 55)(19 56)(20 43)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 109)(72 110)(73 111)(74 112)(75 99)(76 100)(77 101)(78 102)(79 103)(80 104)(81 105)(82 106)(83 107)(84 108)(113 153)(114 154)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 147)(122 148)(123 149)(124 150)(125 151)(126 152)(127 163)(128 164)(129 165)(130 166)(131 167)(132 168)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 161)(140 162)(169 209)(170 210)(171 197)(172 198)(173 199)(174 200)(175 201)(176 202)(177 203)(178 204)(179 205)(180 206)(181 207)(182 208)(183 213)(184 214)(185 215)(186 216)(187 217)(188 218)(189 219)(190 220)(191 221)(192 222)(193 223)(194 224)(195 211)(196 212)
 
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), (1,183,24,181)(2,184,25,182)(3,185,26,169)(4,186,27,170)(5,187,28,171)(6,188,15,172)(7,189,16,173)(8,190,17,174)(9,191,18,175)(10,192,19,176)(11,193,20,177)(12,194,21,178)(13,195,22,179)(14,196,23,180)(29,216,43,210)(30,217,44,197)(31,218,45,198)(32,219,46,199)(33,220,47,200)(34,221,48,201)(35,222,49,202)(36,223,50,203)(37,224,51,204)(38,211,52,205)(39,212,53,206)(40,213,54,207)(41,214,55,208)(42,215,56,209)(57,138,83,123)(58,139,84,124)(59,140,71,125)(60,127,72,126)(61,128,73,113)(62,129,74,114)(63,130,75,115)(64,131,76,116)(65,132,77,117)(66,133,78,118)(67,134,79,119)(68,135,80,120)(69,136,81,121)(70,137,82,122)(85,160,100,149)(86,161,101,150)(87,162,102,151)(88,163,103,152)(89,164,104,153)(90,165,105,154)(91,166,106,141)(92,167,107,142)(93,168,108,143)(94,155,109,144)(95,156,110,145)(96,157,111,146)(97,158,112,147)(98,159,99,148), (1,123,8,116)(2,124,9,117)(3,125,10,118)(4,126,11,119)(5,113,12,120)(6,114,13,121)(7,115,14,122)(15,129,22,136)(16,130,23,137)(17,131,24,138)(18,132,25,139)(19,133,26,140)(20,134,27,127)(21,135,28,128)(29,145,36,152)(30,146,37,153)(31,147,38,154)(32,148,39,141)(33,149,40,142)(34,150,41,143)(35,151,42,144)(43,156,50,163)(44,157,51,164)(45,158,52,165)(46,159,53,166)(47,160,54,167)(48,161,55,168)(49,162,56,155)(57,174,64,181)(58,175,65,182)(59,176,66,169)(60,177,67,170)(61,178,68,171)(62,179,69,172)(63,180,70,173)(71,192,78,185)(72,193,79,186)(73,194,80,187)(74,195,81,188)(75,196,82,189)(76,183,83,190)(77,184,84,191)(85,207,92,200)(86,208,93,201)(87,209,94,202)(88,210,95,203)(89,197,96,204)(90,198,97,205)(91,199,98,206)(99,212,106,219)(100,213,107,220)(101,214,108,221)(102,215,109,222)(103,216,110,223)(104,217,111,224)(105,218,112,211), (1,107,24,92)(2,106,25,91)(3,105,26,90)(4,104,27,89)(5,103,28,88)(6,102,15,87)(7,101,16,86)(8,100,17,85)(9,99,18,98)(10,112,19,97)(11,111,20,96)(12,110,21,95)(13,109,22,94)(14,108,23,93)(29,61,43,73)(30,60,44,72)(31,59,45,71)(32,58,46,84)(33,57,47,83)(34,70,48,82)(35,69,49,81)(36,68,50,80)(37,67,51,79)(38,66,52,78)(39,65,53,77)(40,64,54,76)(41,63,55,75)(42,62,56,74)(113,216,128,210)(114,215,129,209)(115,214,130,208)(116,213,131,207)(117,212,132,206)(118,211,133,205)(119,224,134,204)(120,223,135,203)(121,222,136,202)(122,221,137,201)(123,220,138,200)(124,219,139,199)(125,218,140,198)(126,217,127,197)(141,182,166,184)(142,181,167,183)(143,180,168,196)(144,179,155,195)(145,178,156,194)(146,177,157,193)(147,176,158,192)(148,175,159,191)(149,174,160,190)(150,173,161,189)(151,172,162,188)(152,171,163,187)(153,170,164,186)(154,169,165,185), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,29)(12,30)(13,31)(14,32)(15,52)(16,53)(17,54)(18,55)(19,56)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,109)(72,110)(73,111)(74,112)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(113,153)(114,154)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,163)(128,164)(129,165)(130,166)(131,167)(132,168)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(169,209)(170,210)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,211)(196,212)>;
 
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), (1,183,24,181)(2,184,25,182)(3,185,26,169)(4,186,27,170)(5,187,28,171)(6,188,15,172)(7,189,16,173)(8,190,17,174)(9,191,18,175)(10,192,19,176)(11,193,20,177)(12,194,21,178)(13,195,22,179)(14,196,23,180)(29,216,43,210)(30,217,44,197)(31,218,45,198)(32,219,46,199)(33,220,47,200)(34,221,48,201)(35,222,49,202)(36,223,50,203)(37,224,51,204)(38,211,52,205)(39,212,53,206)(40,213,54,207)(41,214,55,208)(42,215,56,209)(57,138,83,123)(58,139,84,124)(59,140,71,125)(60,127,72,126)(61,128,73,113)(62,129,74,114)(63,130,75,115)(64,131,76,116)(65,132,77,117)(66,133,78,118)(67,134,79,119)(68,135,80,120)(69,136,81,121)(70,137,82,122)(85,160,100,149)(86,161,101,150)(87,162,102,151)(88,163,103,152)(89,164,104,153)(90,165,105,154)(91,166,106,141)(92,167,107,142)(93,168,108,143)(94,155,109,144)(95,156,110,145)(96,157,111,146)(97,158,112,147)(98,159,99,148), (1,123,8,116)(2,124,9,117)(3,125,10,118)(4,126,11,119)(5,113,12,120)(6,114,13,121)(7,115,14,122)(15,129,22,136)(16,130,23,137)(17,131,24,138)(18,132,25,139)(19,133,26,140)(20,134,27,127)(21,135,28,128)(29,145,36,152)(30,146,37,153)(31,147,38,154)(32,148,39,141)(33,149,40,142)(34,150,41,143)(35,151,42,144)(43,156,50,163)(44,157,51,164)(45,158,52,165)(46,159,53,166)(47,160,54,167)(48,161,55,168)(49,162,56,155)(57,174,64,181)(58,175,65,182)(59,176,66,169)(60,177,67,170)(61,178,68,171)(62,179,69,172)(63,180,70,173)(71,192,78,185)(72,193,79,186)(73,194,80,187)(74,195,81,188)(75,196,82,189)(76,183,83,190)(77,184,84,191)(85,207,92,200)(86,208,93,201)(87,209,94,202)(88,210,95,203)(89,197,96,204)(90,198,97,205)(91,199,98,206)(99,212,106,219)(100,213,107,220)(101,214,108,221)(102,215,109,222)(103,216,110,223)(104,217,111,224)(105,218,112,211), (1,107,24,92)(2,106,25,91)(3,105,26,90)(4,104,27,89)(5,103,28,88)(6,102,15,87)(7,101,16,86)(8,100,17,85)(9,99,18,98)(10,112,19,97)(11,111,20,96)(12,110,21,95)(13,109,22,94)(14,108,23,93)(29,61,43,73)(30,60,44,72)(31,59,45,71)(32,58,46,84)(33,57,47,83)(34,70,48,82)(35,69,49,81)(36,68,50,80)(37,67,51,79)(38,66,52,78)(39,65,53,77)(40,64,54,76)(41,63,55,75)(42,62,56,74)(113,216,128,210)(114,215,129,209)(115,214,130,208)(116,213,131,207)(117,212,132,206)(118,211,133,205)(119,224,134,204)(120,223,135,203)(121,222,136,202)(122,221,137,201)(123,220,138,200)(124,219,139,199)(125,218,140,198)(126,217,127,197)(141,182,166,184)(142,181,167,183)(143,180,168,196)(144,179,155,195)(145,178,156,194)(146,177,157,193)(147,176,158,192)(148,175,159,191)(149,174,160,190)(150,173,161,189)(151,172,162,188)(152,171,163,187)(153,170,164,186)(154,169,165,185), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,29)(12,30)(13,31)(14,32)(15,52)(16,53)(17,54)(18,55)(19,56)(20,43)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,109)(72,110)(73,111)(74,112)(75,99)(76,100)(77,101)(78,102)(79,103)(80,104)(81,105)(82,106)(83,107)(84,108)(113,153)(114,154)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)(121,147)(122,148)(123,149)(124,150)(125,151)(126,152)(127,163)(128,164)(129,165)(130,166)(131,167)(132,168)(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(169,209)(170,210)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208)(183,213)(184,214)(185,215)(186,216)(187,217)(188,218)(189,219)(190,220)(191,221)(192,222)(193,223)(194,224)(195,211)(196,212) );
 
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)], [(1,183,24,181),(2,184,25,182),(3,185,26,169),(4,186,27,170),(5,187,28,171),(6,188,15,172),(7,189,16,173),(8,190,17,174),(9,191,18,175),(10,192,19,176),(11,193,20,177),(12,194,21,178),(13,195,22,179),(14,196,23,180),(29,216,43,210),(30,217,44,197),(31,218,45,198),(32,219,46,199),(33,220,47,200),(34,221,48,201),(35,222,49,202),(36,223,50,203),(37,224,51,204),(38,211,52,205),(39,212,53,206),(40,213,54,207),(41,214,55,208),(42,215,56,209),(57,138,83,123),(58,139,84,124),(59,140,71,125),(60,127,72,126),(61,128,73,113),(62,129,74,114),(63,130,75,115),(64,131,76,116),(65,132,77,117),(66,133,78,118),(67,134,79,119),(68,135,80,120),(69,136,81,121),(70,137,82,122),(85,160,100,149),(86,161,101,150),(87,162,102,151),(88,163,103,152),(89,164,104,153),(90,165,105,154),(91,166,106,141),(92,167,107,142),(93,168,108,143),(94,155,109,144),(95,156,110,145),(96,157,111,146),(97,158,112,147),(98,159,99,148)], [(1,123,8,116),(2,124,9,117),(3,125,10,118),(4,126,11,119),(5,113,12,120),(6,114,13,121),(7,115,14,122),(15,129,22,136),(16,130,23,137),(17,131,24,138),(18,132,25,139),(19,133,26,140),(20,134,27,127),(21,135,28,128),(29,145,36,152),(30,146,37,153),(31,147,38,154),(32,148,39,141),(33,149,40,142),(34,150,41,143),(35,151,42,144),(43,156,50,163),(44,157,51,164),(45,158,52,165),(46,159,53,166),(47,160,54,167),(48,161,55,168),(49,162,56,155),(57,174,64,181),(58,175,65,182),(59,176,66,169),(60,177,67,170),(61,178,68,171),(62,179,69,172),(63,180,70,173),(71,192,78,185),(72,193,79,186),(73,194,80,187),(74,195,81,188),(75,196,82,189),(76,183,83,190),(77,184,84,191),(85,207,92,200),(86,208,93,201),(87,209,94,202),(88,210,95,203),(89,197,96,204),(90,198,97,205),(91,199,98,206),(99,212,106,219),(100,213,107,220),(101,214,108,221),(102,215,109,222),(103,216,110,223),(104,217,111,224),(105,218,112,211)], [(1,107,24,92),(2,106,25,91),(3,105,26,90),(4,104,27,89),(5,103,28,88),(6,102,15,87),(7,101,16,86),(8,100,17,85),(9,99,18,98),(10,112,19,97),(11,111,20,96),(12,110,21,95),(13,109,22,94),(14,108,23,93),(29,61,43,73),(30,60,44,72),(31,59,45,71),(32,58,46,84),(33,57,47,83),(34,70,48,82),(35,69,49,81),(36,68,50,80),(37,67,51,79),(38,66,52,78),(39,65,53,77),(40,64,54,76),(41,63,55,75),(42,62,56,74),(113,216,128,210),(114,215,129,209),(115,214,130,208),(116,213,131,207),(117,212,132,206),(118,211,133,205),(119,224,134,204),(120,223,135,203),(121,222,136,202),(122,221,137,201),(123,220,138,200),(124,219,139,199),(125,218,140,198),(126,217,127,197),(141,182,166,184),(142,181,167,183),(143,180,168,196),(144,179,155,195),(145,178,156,194),(146,177,157,193),(147,176,158,192),(148,175,159,191),(149,174,160,190),(150,173,161,189),(151,172,162,188),(152,171,163,187),(153,170,164,186),(154,169,165,185)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,29),(12,30),(13,31),(14,32),(15,52),(16,53),(17,54),(18,55),(19,56),(20,43),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,109),(72,110),(73,111),(74,112),(75,99),(76,100),(77,101),(78,102),(79,103),(80,104),(81,105),(82,106),(83,107),(84,108),(113,153),(114,154),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146),(121,147),(122,148),(123,149),(124,150),(125,151),(126,152),(127,163),(128,164),(129,165),(130,166),(131,167),(132,168),(133,155),(134,156),(135,157),(136,158),(137,159),(138,160),(139,161),(140,162),(169,209),(170,210),(171,197),(172,198),(173,199),(174,200),(175,201),(176,202),(177,203),(178,204),(179,205),(180,206),(181,207),(182,208),(183,213),(184,214),(185,215),(186,216),(187,217),(188,218),(189,219),(190,220),(191,221),(192,222),(193,223),(194,224),(195,211),(196,212)])
 

Matrix representation ►G ⊆ GL8(𝔽29)

2626000000
100000000
0026260000
001000000
0000262100
000082100
00002716021
00001481118
,
2623200000
203020000
270360000
0279260000
00002611022
00002110243
000041013
00006192422
,
170000000
017000000
001700000
000170000
0000216727
0000110115
00001117217
00001231525
,
112516190000
241828130000
161911250000
281324180000
00002217427
00001913234
00005141527
0000821238
,
00100000
00010000
10000000
01000000
000051300
0000162400
00002531113
000028162218

G:=sub<GL(8,GF(29))| [26,10,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,26,10,0,0,0,0,0,0,26,0,0,0,0,0,0,0,0,0,26,8,27,14,0,0,0,0,21,21,16,8,0,0,0,0,0,0,0,11,0,0,0,0,0,0,21,18],[26,20,27,0,0,0,0,0,23,3,0,27,0,0,0,0,2,0,3,9,0,0,0,0,0,2,6,26,0,0,0,0,0,0,0,0,26,21,4,6,0,0,0,0,1,10,1,19,0,0,0,0,10,24,0,24,0,0,0,0,22,3,13,22],[17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,2,11,11,12,0,0,0,0,16,0,17,3,0,0,0,0,7,11,2,15,0,0,0,0,27,5,17,25],[11,24,16,28,0,0,0,0,25,18,19,13,0,0,0,0,16,28,11,24,0,0,0,0,19,13,25,18,0,0,0,0,0,0,0,0,22,19,5,8,0,0,0,0,17,13,14,21,0,0,0,0,4,23,15,23,0,0,0,0,27,4,27,8],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,5,16,25,28,0,0,0,0,13,24,3,16,0,0,0,0,0,0,11,22,0,0,0,0,0,0,13,18] >;
 

64 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M7A7B7C14A···14I14J···14O14P14Q14R28A···28L28M···28U
order122222222444444444444477714···1414···1414141428···2828···28
size111144282828224444141414142828282222···24···48884···48···8

64 irreducible representations

dim111111111111112222224444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D7C4○D4D14D14D14D142+ (1+4)D4⋊6D14D7×C4○D4D4⋊8D14
kernelC14.662+ (1+4)Dic7⋊4D4D14.D4D14⋊D4C22.D28Dic7.Q8C4⋊C4⋊7D7D14.5D4C28⋊1D4C4×C7⋊D4C28⋊7D4Dic7⋊D4C28⋊D4C7×C22.D4C22.D4Dic7C22⋊C4C4⋊C4C22×C4C2×D4C14C2C2C2
# reps111311111111113496332666

In GAP, Magma, Sage, TeX 

C_{14}._{66}2_+^{(1+4)}
 
% in TeX
 
G:=Group("C14.66ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(448,1116);
 
// by ID
 
G=gap.SmallGroup(448,1116);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,184,675,297,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=a^7*b,c*d=d*c,c*e=e*c,e*d*e=b^2*d>;
 
// generators/relations
 

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