Copied to
clipboard

?

G = C14.472+ (1+4)  order 448 = 26·7

47th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C14.472+ (1+4), C4⋊D4⋊22D7, C28⋊7D4⋊34C2, C28⋊2D4⋊27C2, C28⋊D4⋊19C2, C4⋊C4.185D14, (D4×Dic7)⋊26C2, (C2×D4).95D14, D28⋊C4⋊22C2, (C2×C28).45C23, C22⋊C4.11D14, C4.Dic14⋊21C2, Dic7⋊D4⋊17C2, D14.D4⋊22C2, C28.206(C4○D4), C4.69(D4⋊2D7), (C2×C14).163C24, D14⋊C4.17C22, (C22×C4).230D14, C2.31(D4⋊8D14), C2.49(D4⋊6D14), C23.23(C22×D7), (D4×C14).128C22, (C2×D28).145C22, C22.D28⋊13C2, Dic7⋊C4.21C22, C4⋊Dic7.210C22, (C4×Dic7).99C22, (C2×Dic7).80C23, (C22×D7).70C23, C22.184(C23×D7), C23.21D14⋊28C2, (C22×C14).191C23, (C22×C28).246C22, C7⋊4(C22.34C24), C23.D7.113C22, (C22×Dic7).115C22, (C7×C4⋊D4)⋊25C2, C14.87(C2×C4○D4), (C2×C4×D7).89C22, C2.42(C2×D4⋊2D7), (C2×C4).41(C22×D7), (C7×C4⋊C4).150C22, (C2×C7⋊D4).35C22, (C7×C22⋊C4).19C22, SmallGroup(448,1072)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C14.472+ (1+4)
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C2×C4×D7 — D14.D4 — C14.472+ (1+4)
Lower central
C7 — C2×C14 — C14.472+ (1+4)

Subgroups: 1164 in 240 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×12], D4 [×12], C23, C23 [×2], C23 [×2], D7 [×2], C14 [×3], C14 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×7], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×7], Dic7 [×6], C28 [×2], C28 [×3], D14 [×6], C2×C14, C2×C14 [×9], C42⋊C2, C4×D4 [×2], C4⋊D4, C4⋊D4 [×5], C22.D4 [×4], C42.C2, C4⋊1D4, C4×D7 [×2], D28 [×2], C2×Dic7 [×4], C2×Dic7 [×2], C2×Dic7 [×2], C7⋊D4 [×6], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×4], C22×D7 [×2], C22×C14, C22×C14 [×2], C22.34C24, C4×Dic7 [×2], Dic7⋊C4 [×2], C4⋊Dic7 [×3], C4⋊Dic7 [×2], D14⋊C4 [×4], C23.D7 [×2], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×C4×D7 [×2], C2×D28, C22×Dic7 [×2], C2×C7⋊D4 [×6], C22×C28, D4×C14, D4×C14 [×2], D14.D4 [×2], C22.D28 [×2], C4.Dic14, D28⋊C4, C23.21D14, C28⋊7D4, D4×Dic7, C28⋊2D4 [×2], Dic7⋊D4 [×2], C28⋊D4, C7×C4⋊D4, C14.472+ (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, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D4⋊6D14, D4⋊8D14, C14.472+ (1+4)

Generators and relations
 G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=a7b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=a7b-1, dbd-1=a7b, be=eb, dcd-1=ece=a7c, ede=a7b2d >

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)

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

(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)(15 134 22 127)(16 135 23 128)(17 136 24 129)(18 137 25 130)(19 138 26 131)(20 139 27 132)(21 140 28 133)(29 143 36 150)(30 144 37 151)(31 145 38 152)(32 146 39 153)(33 147 40 154)(34 148 41 141)(35 149 42 142)(43 157 50 164)(44 158 51 165)(45 159 52 166)(46 160 53 167)(47 161 54 168)(48 162 55 155)(49 163 56 156)(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)(71 187 78 194)(72 188 79 195)(73 189 80 196)(74 190 81 183)(75 191 82 184)(76 192 83 185)(77 193 84 186)(85 209 92 202)(86 210 93 203)(87 197 94 204)(88 198 95 205)(89 199 96 206)(90 200 97 207)(91 201 98 208)(99 216 106 223)(100 217 107 224)(101 218 108 211)(102 219 109 212)(103 220 110 213)(104 221 111 214)(105 222 112 215)

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

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

Matrix representation ►G ⊆ GL6(𝔽29)

2800000
0280000
008800
0021300
000088
0000213
,
010000
100000
0021800
00112700
0000218
00001127
,
1200000
0170000
00140170
00014017
00140150
00014015
,
0280000
100000
0010270
00328232
0000280
0000261
,
0280000
2800000
0010270
0001027
0000280
0000028

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,8,21,0,0,0,0,8,3,0,0,0,0,0,0,8,21,0,0,0,0,8,3],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,11,0,0,0,0,18,27,0,0,0,0,0,0,2,11,0,0,0,0,18,27],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,14,0,14,0,0,0,0,14,0,14,0,0,17,0,15,0,0,0,0,17,0,15],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,1,3,0,0,0,0,0,28,0,0,0,0,27,23,28,26,0,0,0,2,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,27,0,28,0,0,0,0,27,0,28] >;
 

64 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M7A7B7C14A···14I14J···14O14P···14U28A···28L28M···28R
order122222222444444444444477714···1414···1414···1428···2828···28
size111144428282244414141414282828282222···24···48···84···48···8

64 irreducible representations

dim1111111111112222224444
type++++++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2D7C4○D4D14D14D14D142+ (1+4)D4⋊2D7D4⋊6D14D4⋊8D14
kernelC14.472+ (1+4)D14.D4C22.D28C4.Dic14D28⋊C4C23.21D14C28⋊7D4D4×Dic7C28⋊2D4Dic7⋊D4C28⋊D4C7×C4⋊D4C4⋊D4C28C22⋊C4C4⋊C4C22×C4C2×D4C14C4C2C2
# reps1221111122113463392666

In GAP, Magma, Sage, TeX 

C_{14}._{47}2_+^{(1+4)}
 
% in TeX
 
G:=Group("C14.47ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(448,1072);
 
// by ID
 
G=gap.SmallGroup(448,1072);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,675,570,297,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=a^7*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀