?
G = C14.1042- (1+4) order 448 = 26·7
59th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1042- (1+4), (C7×D4)⋊18D4, C7⋊8(D4⋊6D4), D4⋊9(C7⋊D4), (D4×Dic7)⋊40C2, C28.264(C2×D4), D14⋊3Q8⋊43C2, Dic7⋊7(C4○D4), (C2×D4).234D14, (C2×Q8).192D14, Dic7⋊D4⋊43C2, Dic7⋊Q8⋊31C2, C28.48D4⋊39C2, (C2×C14).310C24, (C2×C28).559C23, D14⋊C4.90C22, (C22×C4).284D14, C14.162(C22×D4), (D4×C14).313C22, Dic7⋊C4.91C22, C4⋊Dic7.259C22, (Q8×C14).239C22, C22.321(C23×D7), C23.207(C22×D7), C23.D7.75C22, C23.18D14⋊31C2, C23.23D14⋊30C2, (C22×C14).236C23, (C22×C28).441C22, (C4×Dic7).173C22, (C2×Dic7).160C23, (C22×D7).136C23, C2.68(D4.10D14), (C2×Dic14).203C22, (C22×Dic7).165C22, (C2×C4○D4)⋊5D7, (C14×C4○D4)⋊5C2, (C4×C7⋊D4)⋊28C2, C4.71(C2×C7⋊D4), C2.102(D7×C4○D4), (C2×C14).78(C2×D4), (C2×D4⋊2D7)⋊28C2, C22.4(C2×C7⋊D4), (C2×Dic7⋊C4)⋊50C2, C14.213(C2×C4○D4), (C2×C4×D7).167C22, C2.35(C22×C7⋊D4), (C2×C4).248(C22×D7), (C2×C7⋊D4).81C22, SmallGroup(448,1277)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1140 in 292 conjugacy classes, 113 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×4], C22 [×10], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×23], D4 [×4], D4 [×10], Q8 [×4], C23, C23 [×2], C23, D7, C14 [×3], C14 [×5], C42, C22⋊C4 [×8], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×8], Dic7 [×2], Dic7 [×6], C28 [×2], C28 [×3], D14 [×3], C2×C14, C2×C14 [×4], C2×C14 [×7], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4, C2×C4○D4, Dic14 [×2], C4×D7 [×2], C2×Dic7 [×3], C2×Dic7 [×4], C2×Dic7 [×8], C7⋊D4 [×6], C2×C28 [×2], C2×C28 [×2], C2×C28 [×6], C7×D4 [×4], C7×D4 [×4], C7×Q8 [×2], C22×D7, C22×C14, C22×C14 [×2], D4⋊6D4, C4×Dic7, Dic7⋊C4, Dic7⋊C4 [×8], C4⋊Dic7, D14⋊C4, D14⋊C4 [×2], C23.D7, C23.D7 [×4], C2×Dic14, C2×C4×D7, D4⋊2D7 [×4], C22×Dic7 [×4], C2×C7⋊D4, C2×C7⋊D4 [×2], C22×C28, C22×C28 [×2], D4×C14, D4×C14 [×2], Q8×C14, C7×C4○D4 [×4], C2×Dic7⋊C4 [×2], C28.48D4, C4×C7⋊D4, C23.23D14 [×2], D4×Dic7, C23.18D14 [×2], Dic7⋊D4 [×2], Dic7⋊Q8, D14⋊3Q8, C2×D4⋊2D7, C14×C4○D4, C14.1042- (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], D4⋊6D4, C2×C7⋊D4 [×6], C23×D7, D7×C4○D4, D4.10D14, C22×C7⋊D4, C14.1042- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=a7b-1, bd=db, be=eb, cd=dc, ece-1=a7c, 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)
(1 122 55 185)(2 121 56 184)(3 120 43 183)(4 119 44 196)(5 118 45 195)(6 117 46 194)(7 116 47 193)(8 115 48 192)(9 114 49 191)(10 113 50 190)(11 126 51 189)(12 125 52 188)(13 124 53 187)(14 123 54 186)(15 218 157 112)(16 217 158 111)(17 216 159 110)(18 215 160 109)(19 214 161 108)(20 213 162 107)(21 212 163 106)(22 211 164 105)(23 224 165 104)(24 223 166 103)(25 222 167 102)(26 221 168 101)(27 220 155 100)(28 219 156 99)(29 70 75 129)(30 69 76 128)(31 68 77 127)(32 67 78 140)(33 66 79 139)(34 65 80 138)(35 64 81 137)(36 63 82 136)(37 62 83 135)(38 61 84 134)(39 60 71 133)(40 59 72 132)(41 58 73 131)(42 57 74 130)(85 154 171 199)(86 153 172 198)(87 152 173 197)(88 151 174 210)(89 150 175 209)(90 149 176 208)(91 148 177 207)(92 147 178 206)(93 146 179 205)(94 145 180 204)(95 144 181 203)(96 143 182 202)(97 142 169 201)(98 141 170 200)
(1 70 8 63)(2 69 9 62)(3 68 10 61)(4 67 11 60)(5 66 12 59)(6 65 13 58)(7 64 14 57)(15 87 22 94)(16 86 23 93)(17 85 24 92)(18 98 25 91)(19 97 26 90)(20 96 27 89)(21 95 28 88)(29 122 36 115)(30 121 37 114)(31 120 38 113)(32 119 39 126)(33 118 40 125)(34 117 41 124)(35 116 42 123)(43 127 50 134)(44 140 51 133)(45 139 52 132)(46 138 53 131)(47 137 54 130)(48 136 55 129)(49 135 56 128)(71 189 78 196)(72 188 79 195)(73 187 80 194)(74 186 81 193)(75 185 82 192)(76 184 83 191)(77 183 84 190)(99 144 106 151)(100 143 107 150)(101 142 108 149)(102 141 109 148)(103 154 110 147)(104 153 111 146)(105 152 112 145)(155 175 162 182)(156 174 163 181)(157 173 164 180)(158 172 165 179)(159 171 166 178)(160 170 167 177)(161 169 168 176)(197 218 204 211)(198 217 205 224)(199 216 206 223)(200 215 207 222)(201 214 208 221)(202 213 209 220)(203 212 210 219)
(1 198 55 153)(2 199 56 154)(3 200 43 141)(4 201 44 142)(5 202 45 143)(6 203 46 144)(7 204 47 145)(8 205 48 146)(9 206 49 147)(10 207 50 148)(11 208 51 149)(12 209 52 150)(13 210 53 151)(14 197 54 152)(15 74 157 42)(16 75 158 29)(17 76 159 30)(18 77 160 31)(19 78 161 32)(20 79 162 33)(21 80 163 34)(22 81 164 35)(23 82 165 36)(24 83 166 37)(25 84 167 38)(26 71 168 39)(27 72 155 40)(28 73 156 41)(57 218 130 112)(58 219 131 99)(59 220 132 100)(60 221 133 101)(61 222 134 102)(62 223 135 103)(63 224 136 104)(64 211 137 105)(65 212 138 106)(66 213 139 107)(67 214 140 108)(68 215 127 109)(69 216 128 110)(70 217 129 111)(85 184 171 121)(86 185 172 122)(87 186 173 123)(88 187 174 124)(89 188 175 125)(90 189 176 126)(91 190 177 113)(92 191 178 114)(93 192 179 115)(94 193 180 116)(95 194 181 117)(96 195 182 118)(97 196 169 119)(98 183 170 120)
(1 86 55 172)(2 85 56 171)(3 98 43 170)(4 97 44 169)(5 96 45 182)(6 95 46 181)(7 94 47 180)(8 93 48 179)(9 92 49 178)(10 91 50 177)(11 90 51 176)(12 89 52 175)(13 88 53 174)(14 87 54 173)(15 130 157 57)(16 129 158 70)(17 128 159 69)(18 127 160 68)(19 140 161 67)(20 139 162 66)(21 138 163 65)(22 137 164 64)(23 136 165 63)(24 135 166 62)(25 134 167 61)(26 133 168 60)(27 132 155 59)(28 131 156 58)(29 111 75 217)(30 110 76 216)(31 109 77 215)(32 108 78 214)(33 107 79 213)(34 106 80 212)(35 105 81 211)(36 104 82 224)(37 103 83 223)(38 102 84 222)(39 101 71 221)(40 100 72 220)(41 99 73 219)(42 112 74 218)(113 148 190 207)(114 147 191 206)(115 146 192 205)(116 145 193 204)(117 144 194 203)(118 143 195 202)(119 142 196 201)(120 141 183 200)(121 154 184 199)(122 153 185 198)(123 152 186 197)(124 151 187 210)(125 150 188 209)(126 149 189 208)
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), (1,122,55,185)(2,121,56,184)(3,120,43,183)(4,119,44,196)(5,118,45,195)(6,117,46,194)(7,116,47,193)(8,115,48,192)(9,114,49,191)(10,113,50,190)(11,126,51,189)(12,125,52,188)(13,124,53,187)(14,123,54,186)(15,218,157,112)(16,217,158,111)(17,216,159,110)(18,215,160,109)(19,214,161,108)(20,213,162,107)(21,212,163,106)(22,211,164,105)(23,224,165,104)(24,223,166,103)(25,222,167,102)(26,221,168,101)(27,220,155,100)(28,219,156,99)(29,70,75,129)(30,69,76,128)(31,68,77,127)(32,67,78,140)(33,66,79,139)(34,65,80,138)(35,64,81,137)(36,63,82,136)(37,62,83,135)(38,61,84,134)(39,60,71,133)(40,59,72,132)(41,58,73,131)(42,57,74,130)(85,154,171,199)(86,153,172,198)(87,152,173,197)(88,151,174,210)(89,150,175,209)(90,149,176,208)(91,148,177,207)(92,147,178,206)(93,146,179,205)(94,145,180,204)(95,144,181,203)(96,143,182,202)(97,142,169,201)(98,141,170,200), (1,70,8,63)(2,69,9,62)(3,68,10,61)(4,67,11,60)(5,66,12,59)(6,65,13,58)(7,64,14,57)(15,87,22,94)(16,86,23,93)(17,85,24,92)(18,98,25,91)(19,97,26,90)(20,96,27,89)(21,95,28,88)(29,122,36,115)(30,121,37,114)(31,120,38,113)(32,119,39,126)(33,118,40,125)(34,117,41,124)(35,116,42,123)(43,127,50,134)(44,140,51,133)(45,139,52,132)(46,138,53,131)(47,137,54,130)(48,136,55,129)(49,135,56,128)(71,189,78,196)(72,188,79,195)(73,187,80,194)(74,186,81,193)(75,185,82,192)(76,184,83,191)(77,183,84,190)(99,144,106,151)(100,143,107,150)(101,142,108,149)(102,141,109,148)(103,154,110,147)(104,153,111,146)(105,152,112,145)(155,175,162,182)(156,174,163,181)(157,173,164,180)(158,172,165,179)(159,171,166,178)(160,170,167,177)(161,169,168,176)(197,218,204,211)(198,217,205,224)(199,216,206,223)(200,215,207,222)(201,214,208,221)(202,213,209,220)(203,212,210,219), (1,198,55,153)(2,199,56,154)(3,200,43,141)(4,201,44,142)(5,202,45,143)(6,203,46,144)(7,204,47,145)(8,205,48,146)(9,206,49,147)(10,207,50,148)(11,208,51,149)(12,209,52,150)(13,210,53,151)(14,197,54,152)(15,74,157,42)(16,75,158,29)(17,76,159,30)(18,77,160,31)(19,78,161,32)(20,79,162,33)(21,80,163,34)(22,81,164,35)(23,82,165,36)(24,83,166,37)(25,84,167,38)(26,71,168,39)(27,72,155,40)(28,73,156,41)(57,218,130,112)(58,219,131,99)(59,220,132,100)(60,221,133,101)(61,222,134,102)(62,223,135,103)(63,224,136,104)(64,211,137,105)(65,212,138,106)(66,213,139,107)(67,214,140,108)(68,215,127,109)(69,216,128,110)(70,217,129,111)(85,184,171,121)(86,185,172,122)(87,186,173,123)(88,187,174,124)(89,188,175,125)(90,189,176,126)(91,190,177,113)(92,191,178,114)(93,192,179,115)(94,193,180,116)(95,194,181,117)(96,195,182,118)(97,196,169,119)(98,183,170,120), (1,86,55,172)(2,85,56,171)(3,98,43,170)(4,97,44,169)(5,96,45,182)(6,95,46,181)(7,94,47,180)(8,93,48,179)(9,92,49,178)(10,91,50,177)(11,90,51,176)(12,89,52,175)(13,88,53,174)(14,87,54,173)(15,130,157,57)(16,129,158,70)(17,128,159,69)(18,127,160,68)(19,140,161,67)(20,139,162,66)(21,138,163,65)(22,137,164,64)(23,136,165,63)(24,135,166,62)(25,134,167,61)(26,133,168,60)(27,132,155,59)(28,131,156,58)(29,111,75,217)(30,110,76,216)(31,109,77,215)(32,108,78,214)(33,107,79,213)(34,106,80,212)(35,105,81,211)(36,104,82,224)(37,103,83,223)(38,102,84,222)(39,101,71,221)(40,100,72,220)(41,99,73,219)(42,112,74,218)(113,148,190,207)(114,147,191,206)(115,146,192,205)(116,145,193,204)(117,144,194,203)(118,143,195,202)(119,142,196,201)(120,141,183,200)(121,154,184,199)(122,153,185,198)(123,152,186,197)(124,151,187,210)(125,150,188,209)(126,149,189,208)>;
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), (1,122,55,185)(2,121,56,184)(3,120,43,183)(4,119,44,196)(5,118,45,195)(6,117,46,194)(7,116,47,193)(8,115,48,192)(9,114,49,191)(10,113,50,190)(11,126,51,189)(12,125,52,188)(13,124,53,187)(14,123,54,186)(15,218,157,112)(16,217,158,111)(17,216,159,110)(18,215,160,109)(19,214,161,108)(20,213,162,107)(21,212,163,106)(22,211,164,105)(23,224,165,104)(24,223,166,103)(25,222,167,102)(26,221,168,101)(27,220,155,100)(28,219,156,99)(29,70,75,129)(30,69,76,128)(31,68,77,127)(32,67,78,140)(33,66,79,139)(34,65,80,138)(35,64,81,137)(36,63,82,136)(37,62,83,135)(38,61,84,134)(39,60,71,133)(40,59,72,132)(41,58,73,131)(42,57,74,130)(85,154,171,199)(86,153,172,198)(87,152,173,197)(88,151,174,210)(89,150,175,209)(90,149,176,208)(91,148,177,207)(92,147,178,206)(93,146,179,205)(94,145,180,204)(95,144,181,203)(96,143,182,202)(97,142,169,201)(98,141,170,200), (1,70,8,63)(2,69,9,62)(3,68,10,61)(4,67,11,60)(5,66,12,59)(6,65,13,58)(7,64,14,57)(15,87,22,94)(16,86,23,93)(17,85,24,92)(18,98,25,91)(19,97,26,90)(20,96,27,89)(21,95,28,88)(29,122,36,115)(30,121,37,114)(31,120,38,113)(32,119,39,126)(33,118,40,125)(34,117,41,124)(35,116,42,123)(43,127,50,134)(44,140,51,133)(45,139,52,132)(46,138,53,131)(47,137,54,130)(48,136,55,129)(49,135,56,128)(71,189,78,196)(72,188,79,195)(73,187,80,194)(74,186,81,193)(75,185,82,192)(76,184,83,191)(77,183,84,190)(99,144,106,151)(100,143,107,150)(101,142,108,149)(102,141,109,148)(103,154,110,147)(104,153,111,146)(105,152,112,145)(155,175,162,182)(156,174,163,181)(157,173,164,180)(158,172,165,179)(159,171,166,178)(160,170,167,177)(161,169,168,176)(197,218,204,211)(198,217,205,224)(199,216,206,223)(200,215,207,222)(201,214,208,221)(202,213,209,220)(203,212,210,219), (1,198,55,153)(2,199,56,154)(3,200,43,141)(4,201,44,142)(5,202,45,143)(6,203,46,144)(7,204,47,145)(8,205,48,146)(9,206,49,147)(10,207,50,148)(11,208,51,149)(12,209,52,150)(13,210,53,151)(14,197,54,152)(15,74,157,42)(16,75,158,29)(17,76,159,30)(18,77,160,31)(19,78,161,32)(20,79,162,33)(21,80,163,34)(22,81,164,35)(23,82,165,36)(24,83,166,37)(25,84,167,38)(26,71,168,39)(27,72,155,40)(28,73,156,41)(57,218,130,112)(58,219,131,99)(59,220,132,100)(60,221,133,101)(61,222,134,102)(62,223,135,103)(63,224,136,104)(64,211,137,105)(65,212,138,106)(66,213,139,107)(67,214,140,108)(68,215,127,109)(69,216,128,110)(70,217,129,111)(85,184,171,121)(86,185,172,122)(87,186,173,123)(88,187,174,124)(89,188,175,125)(90,189,176,126)(91,190,177,113)(92,191,178,114)(93,192,179,115)(94,193,180,116)(95,194,181,117)(96,195,182,118)(97,196,169,119)(98,183,170,120), (1,86,55,172)(2,85,56,171)(3,98,43,170)(4,97,44,169)(5,96,45,182)(6,95,46,181)(7,94,47,180)(8,93,48,179)(9,92,49,178)(10,91,50,177)(11,90,51,176)(12,89,52,175)(13,88,53,174)(14,87,54,173)(15,130,157,57)(16,129,158,70)(17,128,159,69)(18,127,160,68)(19,140,161,67)(20,139,162,66)(21,138,163,65)(22,137,164,64)(23,136,165,63)(24,135,166,62)(25,134,167,61)(26,133,168,60)(27,132,155,59)(28,131,156,58)(29,111,75,217)(30,110,76,216)(31,109,77,215)(32,108,78,214)(33,107,79,213)(34,106,80,212)(35,105,81,211)(36,104,82,224)(37,103,83,223)(38,102,84,222)(39,101,71,221)(40,100,72,220)(41,99,73,219)(42,112,74,218)(113,148,190,207)(114,147,191,206)(115,146,192,205)(116,145,193,204)(117,144,194,203)(118,143,195,202)(119,142,196,201)(120,141,183,200)(121,154,184,199)(122,153,185,198)(123,152,186,197)(124,151,187,210)(125,150,188,209)(126,149,189,208) );
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)], [(1,122,55,185),(2,121,56,184),(3,120,43,183),(4,119,44,196),(5,118,45,195),(6,117,46,194),(7,116,47,193),(8,115,48,192),(9,114,49,191),(10,113,50,190),(11,126,51,189),(12,125,52,188),(13,124,53,187),(14,123,54,186),(15,218,157,112),(16,217,158,111),(17,216,159,110),(18,215,160,109),(19,214,161,108),(20,213,162,107),(21,212,163,106),(22,211,164,105),(23,224,165,104),(24,223,166,103),(25,222,167,102),(26,221,168,101),(27,220,155,100),(28,219,156,99),(29,70,75,129),(30,69,76,128),(31,68,77,127),(32,67,78,140),(33,66,79,139),(34,65,80,138),(35,64,81,137),(36,63,82,136),(37,62,83,135),(38,61,84,134),(39,60,71,133),(40,59,72,132),(41,58,73,131),(42,57,74,130),(85,154,171,199),(86,153,172,198),(87,152,173,197),(88,151,174,210),(89,150,175,209),(90,149,176,208),(91,148,177,207),(92,147,178,206),(93,146,179,205),(94,145,180,204),(95,144,181,203),(96,143,182,202),(97,142,169,201),(98,141,170,200)], [(1,70,8,63),(2,69,9,62),(3,68,10,61),(4,67,11,60),(5,66,12,59),(6,65,13,58),(7,64,14,57),(15,87,22,94),(16,86,23,93),(17,85,24,92),(18,98,25,91),(19,97,26,90),(20,96,27,89),(21,95,28,88),(29,122,36,115),(30,121,37,114),(31,120,38,113),(32,119,39,126),(33,118,40,125),(34,117,41,124),(35,116,42,123),(43,127,50,134),(44,140,51,133),(45,139,52,132),(46,138,53,131),(47,137,54,130),(48,136,55,129),(49,135,56,128),(71,189,78,196),(72,188,79,195),(73,187,80,194),(74,186,81,193),(75,185,82,192),(76,184,83,191),(77,183,84,190),(99,144,106,151),(100,143,107,150),(101,142,108,149),(102,141,109,148),(103,154,110,147),(104,153,111,146),(105,152,112,145),(155,175,162,182),(156,174,163,181),(157,173,164,180),(158,172,165,179),(159,171,166,178),(160,170,167,177),(161,169,168,176),(197,218,204,211),(198,217,205,224),(199,216,206,223),(200,215,207,222),(201,214,208,221),(202,213,209,220),(203,212,210,219)], [(1,198,55,153),(2,199,56,154),(3,200,43,141),(4,201,44,142),(5,202,45,143),(6,203,46,144),(7,204,47,145),(8,205,48,146),(9,206,49,147),(10,207,50,148),(11,208,51,149),(12,209,52,150),(13,210,53,151),(14,197,54,152),(15,74,157,42),(16,75,158,29),(17,76,159,30),(18,77,160,31),(19,78,161,32),(20,79,162,33),(21,80,163,34),(22,81,164,35),(23,82,165,36),(24,83,166,37),(25,84,167,38),(26,71,168,39),(27,72,155,40),(28,73,156,41),(57,218,130,112),(58,219,131,99),(59,220,132,100),(60,221,133,101),(61,222,134,102),(62,223,135,103),(63,224,136,104),(64,211,137,105),(65,212,138,106),(66,213,139,107),(67,214,140,108),(68,215,127,109),(69,216,128,110),(70,217,129,111),(85,184,171,121),(86,185,172,122),(87,186,173,123),(88,187,174,124),(89,188,175,125),(90,189,176,126),(91,190,177,113),(92,191,178,114),(93,192,179,115),(94,193,180,116),(95,194,181,117),(96,195,182,118),(97,196,169,119),(98,183,170,120)], [(1,86,55,172),(2,85,56,171),(3,98,43,170),(4,97,44,169),(5,96,45,182),(6,95,46,181),(7,94,47,180),(8,93,48,179),(9,92,49,178),(10,91,50,177),(11,90,51,176),(12,89,52,175),(13,88,53,174),(14,87,54,173),(15,130,157,57),(16,129,158,70),(17,128,159,69),(18,127,160,68),(19,140,161,67),(20,139,162,66),(21,138,163,65),(22,137,164,64),(23,136,165,63),(24,135,166,62),(25,134,167,61),(26,133,168,60),(27,132,155,59),(28,131,156,58),(29,111,75,217),(30,110,76,216),(31,109,77,215),(32,108,78,214),(33,107,79,213),(34,106,80,212),(35,105,81,211),(36,104,82,224),(37,103,83,223),(38,102,84,222),(39,101,71,221),(40,100,72,220),(41,99,73,219),(42,112,74,218),(113,148,190,207),(114,147,191,206),(115,146,192,205),(116,145,193,204),(117,144,194,203),(118,143,195,202),(119,142,196,201),(120,141,183,200),(121,154,184,199),(122,153,185,198),(123,152,186,197),(124,151,187,210),(125,150,188,209),(126,149,189,208)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 21 | 0 | 0 |
| 0 | 0 | 11 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 21 | 11 | 0 | 0 | 0 | 0 |
| 18 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 17 | 0 | 0 |
| 0 | 0 | 4 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 11 | 21 | 0 | 0 | 0 | 0 |
| 8 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 17 | 0 | 0 |
| 0 | 0 | 4 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 0 | 1 | 28 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 0 | 1 | 28 |
| 8 | 18 | 0 | 0 | 0 | 0 |
| 11 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 17 | 0 | 0 |
| 0 | 0 | 4 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 24 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,11,0,0,0,0,21,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[21,18,0,0,0,0,11,8,0,0,0,0,0,0,7,4,0,0,0,0,17,22,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[11,8,0,0,0,0,21,18,0,0,0,0,0,0,7,4,0,0,0,0,17,22,0,0,0,0,0,0,1,1,0,0,0,0,27,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,27,28],[8,11,0,0,0,0,18,21,0,0,0,0,0,0,7,4,0,0,0,0,17,22,0,0,0,0,0,0,17,0,0,0,0,0,24,12] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 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 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | 2- (1+4) | D7×C4○D4 | D4.10D14 |
| kernel | C14.1042- (1+4) | C2×Dic7⋊C4 | C28.48D4 | C4×C7⋊D4 | C23.23D14 | D4×Dic7 | C23.18D14 | Dic7⋊D4 | Dic7⋊Q8 | D14⋊3Q8 | C2×D4⋊2D7 | C14×C4○D4 | C7×D4 | C2×C4○D4 | Dic7 | C22×C4 | C2×D4 | C2×Q8 | D4 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{104}2_-^{(1+4)} % in TeX
G:=Group("C14.104ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1277);
// by ID
G=gap.SmallGroup(448,1277);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,758,100,346,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=e^2=b^2,b*a*b^-1=c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=a^7*b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^7*c,e*d*e^-1=b^2*d>;
// generators/relations
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