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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⋊7D4⋊22C2, C28⋊D4⋊21C2, C4⋊D28⋊31C2, C4⋊C4.200D14, D14⋊D4⋊34C2, (C2×D4).103D14, Dic7.Q8⋊29C2, C22⋊C4.30D14, Dic7⋊D4⋊23C2, Dic7⋊4D4⋊24C2, D14.5D4⋊30C2, D14.D4⋊36C2, (C2×C28).182C23, (C2×C14).207C24, C22.D4⋊12D7, (C22×C4).261D14, C2.44(D4⋊8D14), C2.68(D4⋊6D14), C23.31(C22×D7), D14⋊C4.109C22, Dic7.10(C4○D4), (D4×C14).145C22, (C2×D28).158C22, 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
Upper central
C1 — C22 — C22.D4

Generators and relations for C14.662+ 1+4
 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 >

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

Smallest permutation representation of C14.662+ 1+4
►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)

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

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

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

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

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+4D4⋊6D14D7×C4○D4D4⋊8D14
kernelC14.662+ 1+4Dic7⋊4D4D14.D4D14⋊D4C22.D28Dic7.Q8C4⋊C4⋊7D7D14.5D4C4⋊D28C4×C7⋊D4C28⋊7D4Dic7⋊D4C28⋊D4C7×C22.D4C22.D4Dic7C22⋊C4C4⋊C4C22×C4C2×D4C14C2C2C2
# reps111311111111113496332666

Matrix representation of C14.662+ 1+4 ►in 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] >;
 

C14.662+ 1+4 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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