?
G = C14.1482+ (1+4) order 448 = 26·7
57th non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1482+ (1+4), (C7×Q8)⋊19D4, C7⋊5(Q8⋊6D4), C28⋊2D4⋊44C2, C28⋊D4⋊31C2, C28⋊7D4⋊40C2, (Q8×Dic7)⋊30C2, Q8⋊10(C7⋊D4), C28.268(C2×D4), Dic7⋊9(C4○D4), (C2×D4).239D14, (C2×Q8).211D14, (C2×C14).320C24, (C2×C28).562C23, D14⋊C4.91C22, (C22×C4).290D14, C14.170(C22×D4), C2.72(D4⋊8D14), (C2×D28).185C22, (D4×C14).278C22, C4⋊Dic7.261C22, (Q8×C14).246C22, C23.141(C22×D7), C22.329(C23×D7), Dic7⋊C4.175C22, (C22×C28).298C22, (C22×C14).246C23, (C2×Dic7).166C23, (C4×Dic7).177C22, (C22×D7).141C23, C23.D7.138C22, (C2×C4○D4)⋊12D7, (C4×C7⋊D4)⋊31C2, C4.74(C2×C7⋊D4), (C14×C4○D4)⋊12C2, C2.107(D7×C4○D4), (C2×Q8⋊2D7)⋊19C2, C14.219(C2×C4○D4), (C2×C4×D7).171C22, C2.43(C22×C7⋊D4), (C2×C4).642(C22×D7), (C2×C7⋊D4).143C22, SmallGroup(448,1287)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1460 in 312 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×7], C22, C22 [×18], C7, C2×C4, C2×C4 [×3], C2×C4 [×17], D4 [×24], Q8 [×4], C23 [×3], C23 [×3], D7 [×3], C14 [×3], C14 [×3], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4 [×3], C2×D4 [×12], C2×Q8, C4○D4 [×8], Dic7 [×2], Dic7 [×4], C28 [×6], C28, D14 [×9], C2×C14, C2×C14 [×9], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C4⋊1D4 [×3], C2×C4○D4, C2×C4○D4, C4×D7 [×6], D28 [×6], C2×Dic7 [×2], C2×Dic7 [×3], C7⋊D4 [×12], C2×C28, C2×C28 [×3], C2×C28 [×6], C7×D4 [×6], C7×Q8 [×4], C22×D7 [×3], C22×C14 [×3], Q8⋊6D4, C4×Dic7 [×3], Dic7⋊C4, C4⋊Dic7 [×3], D14⋊C4 [×3], C23.D7 [×3], C2×C4×D7 [×3], C2×D28 [×3], Q8⋊2D7 [×4], C2×C7⋊D4 [×9], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×4], C4×C7⋊D4 [×3], C28⋊7D4 [×3], C28⋊2D4 [×3], C28⋊D4 [×3], Q8×Dic7, C2×Q8⋊2D7, C14×C4○D4, C14.1482+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2+ (1+4), C7⋊D4 [×4], C22×D7 [×7], Q8⋊6D4, C2×C7⋊D4 [×6], C23×D7, D7×C4○D4, D4⋊8D14, C22×C7⋊D4, C14.1482+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ece=a7c, ede=a7b2d >
►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 97 16 99)(2 98 17 100)(3 85 18 101)(4 86 19 102)(5 87 20 103)(6 88 21 104)(7 89 22 105)(8 90 23 106)(9 91 24 107)(10 92 25 108)(11 93 26 109)(12 94 27 110)(13 95 28 111)(14 96 15 112)(29 71 51 60)(30 72 52 61)(31 73 53 62)(32 74 54 63)(33 75 55 64)(34 76 56 65)(35 77 43 66)(36 78 44 67)(37 79 45 68)(38 80 46 69)(39 81 47 70)(40 82 48 57)(41 83 49 58)(42 84 50 59)(113 220 130 208)(114 221 131 209)(115 222 132 210)(116 223 133 197)(117 224 134 198)(118 211 135 199)(119 212 136 200)(120 213 137 201)(121 214 138 202)(122 215 139 203)(123 216 140 204)(124 217 127 205)(125 218 128 206)(126 219 129 207)(141 180 155 195)(142 181 156 196)(143 182 157 183)(144 169 158 184)(145 170 159 185)(146 171 160 186)(147 172 161 187)(148 173 162 188)(149 174 163 189)(150 175 164 190)(151 176 165 191)(152 177 166 192)(153 178 167 193)(154 179 168 194)
(1 119)(2 120)(3 121)(4 122)(5 123)(6 124)(7 125)(8 126)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 127)(22 128)(23 129)(24 130)(25 131)(26 132)(27 133)(28 134)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 141)(39 142)(40 143)(41 144)(42 145)(43 166)(44 167)(45 168)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 161)(53 162)(54 163)(55 164)(56 165)(57 182)(58 169)(59 170)(60 171)(61 172)(62 173)(63 174)(64 175)(65 176)(66 177)(67 178)(68 179)(69 180)(70 181)(71 186)(72 187)(73 188)(74 189)(75 190)(76 191)(77 192)(78 193)(79 194)(80 195)(81 196)(82 183)(83 184)(84 185)(85 202)(86 203)(87 204)(88 205)(89 206)(90 207)(91 208)(92 209)(93 210)(94 197)(95 198)(96 199)(97 200)(98 201)(99 212)(100 213)(101 214)(102 215)(103 216)(104 217)(105 218)(106 219)(107 220)(108 221)(109 222)(110 223)(111 224)(112 211)
(1 52 16 30)(2 53 17 31)(3 54 18 32)(4 55 19 33)(5 56 20 34)(6 43 21 35)(7 44 22 36)(8 45 23 37)(9 46 24 38)(10 47 25 39)(11 48 26 40)(12 49 27 41)(13 50 28 42)(14 51 15 29)(57 109 82 93)(58 110 83 94)(59 111 84 95)(60 112 71 96)(61 99 72 97)(62 100 73 98)(63 101 74 85)(64 102 75 86)(65 103 76 87)(66 104 77 88)(67 105 78 89)(68 106 79 90)(69 107 80 91)(70 108 81 92)(113 155 130 141)(114 156 131 142)(115 157 132 143)(116 158 133 144)(117 159 134 145)(118 160 135 146)(119 161 136 147)(120 162 137 148)(121 163 138 149)(122 164 139 150)(123 165 140 151)(124 166 127 152)(125 167 128 153)(126 168 129 154)(169 223 184 197)(170 224 185 198)(171 211 186 199)(172 212 187 200)(173 213 188 201)(174 214 189 202)(175 215 190 203)(176 216 191 204)(177 217 192 205)(178 218 193 206)(179 219 194 207)(180 220 195 208)(181 221 196 209)(182 222 183 210)
(1 154)(2 153)(3 152)(4 151)(5 150)(6 149)(7 148)(8 147)(9 146)(10 145)(11 144)(12 143)(13 142)(14 141)(15 155)(16 168)(17 167)(18 166)(19 165)(20 164)(21 163)(22 162)(23 161)(24 160)(25 159)(26 158)(27 157)(28 156)(29 120)(30 119)(31 118)(32 117)(33 116)(34 115)(35 114)(36 113)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 131)(44 130)(45 129)(46 128)(47 127)(48 140)(49 139)(50 138)(51 137)(52 136)(53 135)(54 134)(55 133)(56 132)(57 204)(58 203)(59 202)(60 201)(61 200)(62 199)(63 198)(64 197)(65 210)(66 209)(67 208)(68 207)(69 206)(70 205)(71 213)(72 212)(73 211)(74 224)(75 223)(76 222)(77 221)(78 220)(79 219)(80 218)(81 217)(82 216)(83 215)(84 214)(85 177)(86 176)(87 175)(88 174)(89 173)(90 172)(91 171)(92 170)(93 169)(94 182)(95 181)(96 180)(97 179)(98 178)(99 194)(100 193)(101 192)(102 191)(103 190)(104 189)(105 188)(106 187)(107 186)(108 185)(109 184)(110 183)(111 196)(112 195)
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), (1,97,16,99)(2,98,17,100)(3,85,18,101)(4,86,19,102)(5,87,20,103)(6,88,21,104)(7,89,22,105)(8,90,23,106)(9,91,24,107)(10,92,25,108)(11,93,26,109)(12,94,27,110)(13,95,28,111)(14,96,15,112)(29,71,51,60)(30,72,52,61)(31,73,53,62)(32,74,54,63)(33,75,55,64)(34,76,56,65)(35,77,43,66)(36,78,44,67)(37,79,45,68)(38,80,46,69)(39,81,47,70)(40,82,48,57)(41,83,49,58)(42,84,50,59)(113,220,130,208)(114,221,131,209)(115,222,132,210)(116,223,133,197)(117,224,134,198)(118,211,135,199)(119,212,136,200)(120,213,137,201)(121,214,138,202)(122,215,139,203)(123,216,140,204)(124,217,127,205)(125,218,128,206)(126,219,129,207)(141,180,155,195)(142,181,156,196)(143,182,157,183)(144,169,158,184)(145,170,159,185)(146,171,160,186)(147,172,161,187)(148,173,162,188)(149,174,163,189)(150,175,164,190)(151,176,165,191)(152,177,166,192)(153,178,167,193)(154,179,168,194), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,141)(39,142)(40,143)(41,144)(42,145)(43,166)(44,167)(45,168)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,165)(57,182)(58,169)(59,170)(60,171)(61,172)(62,173)(63,174)(64,175)(65,176)(66,177)(67,178)(68,179)(69,180)(70,181)(71,186)(72,187)(73,188)(74,189)(75,190)(76,191)(77,192)(78,193)(79,194)(80,195)(81,196)(82,183)(83,184)(84,185)(85,202)(86,203)(87,204)(88,205)(89,206)(90,207)(91,208)(92,209)(93,210)(94,197)(95,198)(96,199)(97,200)(98,201)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,219)(107,220)(108,221)(109,222)(110,223)(111,224)(112,211), (1,52,16,30)(2,53,17,31)(3,54,18,32)(4,55,19,33)(5,56,20,34)(6,43,21,35)(7,44,22,36)(8,45,23,37)(9,46,24,38)(10,47,25,39)(11,48,26,40)(12,49,27,41)(13,50,28,42)(14,51,15,29)(57,109,82,93)(58,110,83,94)(59,111,84,95)(60,112,71,96)(61,99,72,97)(62,100,73,98)(63,101,74,85)(64,102,75,86)(65,103,76,87)(66,104,77,88)(67,105,78,89)(68,106,79,90)(69,107,80,91)(70,108,81,92)(113,155,130,141)(114,156,131,142)(115,157,132,143)(116,158,133,144)(117,159,134,145)(118,160,135,146)(119,161,136,147)(120,162,137,148)(121,163,138,149)(122,164,139,150)(123,165,140,151)(124,166,127,152)(125,167,128,153)(126,168,129,154)(169,223,184,197)(170,224,185,198)(171,211,186,199)(172,212,187,200)(173,213,188,201)(174,214,189,202)(175,215,190,203)(176,216,191,204)(177,217,192,205)(178,218,193,206)(179,219,194,207)(180,220,195,208)(181,221,196,209)(182,222,183,210), (1,154)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,155)(16,168)(17,167)(18,166)(19,165)(20,164)(21,163)(22,162)(23,161)(24,160)(25,159)(26,158)(27,157)(28,156)(29,120)(30,119)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,131)(44,130)(45,129)(46,128)(47,127)(48,140)(49,139)(50,138)(51,137)(52,136)(53,135)(54,134)(55,133)(56,132)(57,204)(58,203)(59,202)(60,201)(61,200)(62,199)(63,198)(64,197)(65,210)(66,209)(67,208)(68,207)(69,206)(70,205)(71,213)(72,212)(73,211)(74,224)(75,223)(76,222)(77,221)(78,220)(79,219)(80,218)(81,217)(82,216)(83,215)(84,214)(85,177)(86,176)(87,175)(88,174)(89,173)(90,172)(91,171)(92,170)(93,169)(94,182)(95,181)(96,180)(97,179)(98,178)(99,194)(100,193)(101,192)(102,191)(103,190)(104,189)(105,188)(106,187)(107,186)(108,185)(109,184)(110,183)(111,196)(112,195)>;
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), (1,97,16,99)(2,98,17,100)(3,85,18,101)(4,86,19,102)(5,87,20,103)(6,88,21,104)(7,89,22,105)(8,90,23,106)(9,91,24,107)(10,92,25,108)(11,93,26,109)(12,94,27,110)(13,95,28,111)(14,96,15,112)(29,71,51,60)(30,72,52,61)(31,73,53,62)(32,74,54,63)(33,75,55,64)(34,76,56,65)(35,77,43,66)(36,78,44,67)(37,79,45,68)(38,80,46,69)(39,81,47,70)(40,82,48,57)(41,83,49,58)(42,84,50,59)(113,220,130,208)(114,221,131,209)(115,222,132,210)(116,223,133,197)(117,224,134,198)(118,211,135,199)(119,212,136,200)(120,213,137,201)(121,214,138,202)(122,215,139,203)(123,216,140,204)(124,217,127,205)(125,218,128,206)(126,219,129,207)(141,180,155,195)(142,181,156,196)(143,182,157,183)(144,169,158,184)(145,170,159,185)(146,171,160,186)(147,172,161,187)(148,173,162,188)(149,174,163,189)(150,175,164,190)(151,176,165,191)(152,177,166,192)(153,178,167,193)(154,179,168,194), (1,119)(2,120)(3,121)(4,122)(5,123)(6,124)(7,125)(8,126)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,127)(22,128)(23,129)(24,130)(25,131)(26,132)(27,133)(28,134)(29,146)(30,147)(31,148)(32,149)(33,150)(34,151)(35,152)(36,153)(37,154)(38,141)(39,142)(40,143)(41,144)(42,145)(43,166)(44,167)(45,168)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,165)(57,182)(58,169)(59,170)(60,171)(61,172)(62,173)(63,174)(64,175)(65,176)(66,177)(67,178)(68,179)(69,180)(70,181)(71,186)(72,187)(73,188)(74,189)(75,190)(76,191)(77,192)(78,193)(79,194)(80,195)(81,196)(82,183)(83,184)(84,185)(85,202)(86,203)(87,204)(88,205)(89,206)(90,207)(91,208)(92,209)(93,210)(94,197)(95,198)(96,199)(97,200)(98,201)(99,212)(100,213)(101,214)(102,215)(103,216)(104,217)(105,218)(106,219)(107,220)(108,221)(109,222)(110,223)(111,224)(112,211), (1,52,16,30)(2,53,17,31)(3,54,18,32)(4,55,19,33)(5,56,20,34)(6,43,21,35)(7,44,22,36)(8,45,23,37)(9,46,24,38)(10,47,25,39)(11,48,26,40)(12,49,27,41)(13,50,28,42)(14,51,15,29)(57,109,82,93)(58,110,83,94)(59,111,84,95)(60,112,71,96)(61,99,72,97)(62,100,73,98)(63,101,74,85)(64,102,75,86)(65,103,76,87)(66,104,77,88)(67,105,78,89)(68,106,79,90)(69,107,80,91)(70,108,81,92)(113,155,130,141)(114,156,131,142)(115,157,132,143)(116,158,133,144)(117,159,134,145)(118,160,135,146)(119,161,136,147)(120,162,137,148)(121,163,138,149)(122,164,139,150)(123,165,140,151)(124,166,127,152)(125,167,128,153)(126,168,129,154)(169,223,184,197)(170,224,185,198)(171,211,186,199)(172,212,187,200)(173,213,188,201)(174,214,189,202)(175,215,190,203)(176,216,191,204)(177,217,192,205)(178,218,193,206)(179,219,194,207)(180,220,195,208)(181,221,196,209)(182,222,183,210), (1,154)(2,153)(3,152)(4,151)(5,150)(6,149)(7,148)(8,147)(9,146)(10,145)(11,144)(12,143)(13,142)(14,141)(15,155)(16,168)(17,167)(18,166)(19,165)(20,164)(21,163)(22,162)(23,161)(24,160)(25,159)(26,158)(27,157)(28,156)(29,120)(30,119)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,131)(44,130)(45,129)(46,128)(47,127)(48,140)(49,139)(50,138)(51,137)(52,136)(53,135)(54,134)(55,133)(56,132)(57,204)(58,203)(59,202)(60,201)(61,200)(62,199)(63,198)(64,197)(65,210)(66,209)(67,208)(68,207)(69,206)(70,205)(71,213)(72,212)(73,211)(74,224)(75,223)(76,222)(77,221)(78,220)(79,219)(80,218)(81,217)(82,216)(83,215)(84,214)(85,177)(86,176)(87,175)(88,174)(89,173)(90,172)(91,171)(92,170)(93,169)(94,182)(95,181)(96,180)(97,179)(98,178)(99,194)(100,193)(101,192)(102,191)(103,190)(104,189)(105,188)(106,187)(107,186)(108,185)(109,184)(110,183)(111,196)(112,195) );
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)], [(1,97,16,99),(2,98,17,100),(3,85,18,101),(4,86,19,102),(5,87,20,103),(6,88,21,104),(7,89,22,105),(8,90,23,106),(9,91,24,107),(10,92,25,108),(11,93,26,109),(12,94,27,110),(13,95,28,111),(14,96,15,112),(29,71,51,60),(30,72,52,61),(31,73,53,62),(32,74,54,63),(33,75,55,64),(34,76,56,65),(35,77,43,66),(36,78,44,67),(37,79,45,68),(38,80,46,69),(39,81,47,70),(40,82,48,57),(41,83,49,58),(42,84,50,59),(113,220,130,208),(114,221,131,209),(115,222,132,210),(116,223,133,197),(117,224,134,198),(118,211,135,199),(119,212,136,200),(120,213,137,201),(121,214,138,202),(122,215,139,203),(123,216,140,204),(124,217,127,205),(125,218,128,206),(126,219,129,207),(141,180,155,195),(142,181,156,196),(143,182,157,183),(144,169,158,184),(145,170,159,185),(146,171,160,186),(147,172,161,187),(148,173,162,188),(149,174,163,189),(150,175,164,190),(151,176,165,191),(152,177,166,192),(153,178,167,193),(154,179,168,194)], [(1,119),(2,120),(3,121),(4,122),(5,123),(6,124),(7,125),(8,126),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,127),(22,128),(23,129),(24,130),(25,131),(26,132),(27,133),(28,134),(29,146),(30,147),(31,148),(32,149),(33,150),(34,151),(35,152),(36,153),(37,154),(38,141),(39,142),(40,143),(41,144),(42,145),(43,166),(44,167),(45,168),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,161),(53,162),(54,163),(55,164),(56,165),(57,182),(58,169),(59,170),(60,171),(61,172),(62,173),(63,174),(64,175),(65,176),(66,177),(67,178),(68,179),(69,180),(70,181),(71,186),(72,187),(73,188),(74,189),(75,190),(76,191),(77,192),(78,193),(79,194),(80,195),(81,196),(82,183),(83,184),(84,185),(85,202),(86,203),(87,204),(88,205),(89,206),(90,207),(91,208),(92,209),(93,210),(94,197),(95,198),(96,199),(97,200),(98,201),(99,212),(100,213),(101,214),(102,215),(103,216),(104,217),(105,218),(106,219),(107,220),(108,221),(109,222),(110,223),(111,224),(112,211)], [(1,52,16,30),(2,53,17,31),(3,54,18,32),(4,55,19,33),(5,56,20,34),(6,43,21,35),(7,44,22,36),(8,45,23,37),(9,46,24,38),(10,47,25,39),(11,48,26,40),(12,49,27,41),(13,50,28,42),(14,51,15,29),(57,109,82,93),(58,110,83,94),(59,111,84,95),(60,112,71,96),(61,99,72,97),(62,100,73,98),(63,101,74,85),(64,102,75,86),(65,103,76,87),(66,104,77,88),(67,105,78,89),(68,106,79,90),(69,107,80,91),(70,108,81,92),(113,155,130,141),(114,156,131,142),(115,157,132,143),(116,158,133,144),(117,159,134,145),(118,160,135,146),(119,161,136,147),(120,162,137,148),(121,163,138,149),(122,164,139,150),(123,165,140,151),(124,166,127,152),(125,167,128,153),(126,168,129,154),(169,223,184,197),(170,224,185,198),(171,211,186,199),(172,212,187,200),(173,213,188,201),(174,214,189,202),(175,215,190,203),(176,216,191,204),(177,217,192,205),(178,218,193,206),(179,219,194,207),(180,220,195,208),(181,221,196,209),(182,222,183,210)], [(1,154),(2,153),(3,152),(4,151),(5,150),(6,149),(7,148),(8,147),(9,146),(10,145),(11,144),(12,143),(13,142),(14,141),(15,155),(16,168),(17,167),(18,166),(19,165),(20,164),(21,163),(22,162),(23,161),(24,160),(25,159),(26,158),(27,157),(28,156),(29,120),(30,119),(31,118),(32,117),(33,116),(34,115),(35,114),(36,113),(37,126),(38,125),(39,124),(40,123),(41,122),(42,121),(43,131),(44,130),(45,129),(46,128),(47,127),(48,140),(49,139),(50,138),(51,137),(52,136),(53,135),(54,134),(55,133),(56,132),(57,204),(58,203),(59,202),(60,201),(61,200),(62,199),(63,198),(64,197),(65,210),(66,209),(67,208),(68,207),(69,206),(70,205),(71,213),(72,212),(73,211),(74,224),(75,223),(76,222),(77,221),(78,220),(79,219),(80,218),(81,217),(82,216),(83,215),(84,214),(85,177),(86,176),(87,175),(88,174),(89,173),(90,172),(91,171),(92,170),(93,169),(94,182),(95,181),(96,180),(97,179),(98,178),(99,194),(100,193),(101,192),(102,191),(103,190),(104,189),(105,188),(106,187),(107,186),(108,185),(109,184),(110,183),(111,196),(112,195)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 19 | 19 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 5 |
| 0 | 0 | 17 | 1 |
| 9 | 14 | 0 | 0 |
| 15 | 20 | 0 | 0 |
| 0 | 0 | 1 | 24 |
| 0 | 0 | 0 | 28 |
| 9 | 14 | 0 | 0 |
| 15 | 20 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 7 | 28 | 0 | 0 |
| 0 | 0 | 12 | 27 |
| 0 | 0 | 28 | 17 |
G:=sub<GL(4,GF(29))| [19,10,0,0,19,7,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,17,0,0,5,1],[9,15,0,0,14,20,0,0,0,0,1,0,0,0,24,28],[9,15,0,0,14,20,0,0,0,0,12,0,0,0,0,12],[1,7,0,0,0,28,0,0,0,0,12,28,0,0,27,17] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 4 | 4 | 28 | 28 | 28 | 2 | ··· | 2 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | 2+ (1+4) | D7×C4○D4 | D4⋊8D14 |
| kernel | C14.1482+ (1+4) | C4×C7⋊D4 | C28⋊7D4 | C28⋊2D4 | C28⋊D4 | Q8×Dic7 | C2×Q8⋊2D7 | C14×C4○D4 | C7×Q8 | C2×C4○D4 | Dic7 | C22×C4 | C2×D4 | C2×Q8 | Q8 | C14 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{148}2_+^{(1+4)} % in TeX
G:=Group("C14.148ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1287);
// by ID
G=gap.SmallGroup(448,1287);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,758,219,1571,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀