?
G = C14.732- (1+4) order 448 = 26·7
28th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.732- (1+4), C7⋊D4⋊3D4, C28⋊Q8⋊20C2, C4⋊D4⋊14D7, C7⋊4(D4⋊6D4), C28⋊10(C4○D4), C28⋊2D4⋊20C2, C4⋊4(D4⋊2D7), C22.7(D4×D7), C4⋊C4.181D14, (D4×Dic7)⋊21C2, D14.18(C2×D4), C22⋊C4.9D14, (C2×D4).155D14, Dic7.23(C2×D4), C14.70(C22×D4), D14.D4⋊19C2, (C2×C28).174C23, (C2×C14).155C24, (C22×C4).224D14, D14⋊C4.105C22, C22⋊Dic14⋊20C2, (D4×C14).123C22, C4⋊Dic7.371C22, (C22×C14).22C23, (C4×Dic7).95C22, C23.183(C22×D7), C22.176(C23×D7), C23.18D14⋊10C2, Dic7⋊C4.117C22, (C22×C28).242C22, (C2×Dic7).228C23, (C22×D7).189C23, C23.D7.111C22, C2.31(D4.10D14), (C2×Dic14).155C22, (C22×Dic7).110C22, C2.43(C2×D4×D7), (D7×C4⋊C4)⋊22C2, (C4×C7⋊D4)⋊17C2, (C2×C14).7(C2×D4), (C7×C4⋊D4)⋊17C2, (C2×C4⋊Dic7)⋊40C2, C14.84(C2×C4○D4), (C2×D4⋊2D7)⋊15C2, (C2×C4×D7).84C22, C2.37(C2×D4⋊2D7), (C7×C4⋊C4).145C22, (C2×C4).587(C22×D7), (C2×C7⋊D4).29C22, (C7×C22⋊C4).14C22, SmallGroup(448,1064)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1260 in 292 conjugacy classes, 107 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×12], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×23], D4 [×14], Q8 [×4], C23, C23 [×2], C23, D7 [×2], C14 [×3], C14 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×7], C2×D4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C4○D4 [×8], Dic7 [×2], Dic7 [×6], C28 [×2], C28 [×3], D14 [×2], D14 [×2], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C4⋊C4 [×2], C4×D4 [×2], C4⋊D4, C4⋊D4, C22⋊Q8 [×2], C22.D4 [×4], C4⋊Q8, C2×C4○D4 [×2], Dic14 [×4], C4×D7 [×6], C2×Dic7 [×3], C2×Dic7 [×4], C2×Dic7 [×8], C7⋊D4 [×4], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×6], C22×D7, C22×C14, C22×C14 [×2], D4⋊6D4, C4×Dic7, Dic7⋊C4, Dic7⋊C4 [×4], C4⋊Dic7 [×2], C4⋊Dic7 [×2], D14⋊C4, C23.D7, C23.D7 [×4], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7, C2×C4×D7 [×2], D4⋊2D7 [×8], C22×Dic7 [×4], C2×C7⋊D4, C2×C7⋊D4 [×2], C22×C28, D4×C14, D4×C14 [×2], C22⋊Dic14 [×2], D14.D4 [×2], C28⋊Q8, D7×C4⋊C4, C2×C4⋊Dic7, C4×C7⋊D4, D4×Dic7, C23.18D14 [×2], C28⋊2D4, C7×C4⋊D4, C2×D4⋊2D7 [×2], C14.732- (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), C22×D7 [×7], D4⋊6D4, D4×D7 [×2], D4⋊2D7 [×2], C23×D7, C2×D4×D7, C2×D4⋊2D7, D4.10D14, C14.732- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=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)
(1 122 189 221)(2 121 190 220)(3 120 191 219)(4 119 192 218)(5 118 193 217)(6 117 194 216)(7 116 195 215)(8 115 196 214)(9 114 183 213)(10 113 184 212)(11 126 185 211)(12 125 186 224)(13 124 187 223)(14 123 188 222)(15 70 110 73)(16 69 111 72)(17 68 112 71)(18 67 99 84)(19 66 100 83)(20 65 101 82)(21 64 102 81)(22 63 103 80)(23 62 104 79)(24 61 105 78)(25 60 106 77)(26 59 107 76)(27 58 108 75)(28 57 109 74)(29 135 179 91)(30 134 180 90)(31 133 181 89)(32 132 182 88)(33 131 169 87)(34 130 170 86)(35 129 171 85)(36 128 172 98)(37 127 173 97)(38 140 174 96)(39 139 175 95)(40 138 176 94)(41 137 177 93)(42 136 178 92)(43 199 142 167)(44 198 143 166)(45 197 144 165)(46 210 145 164)(47 209 146 163)(48 208 147 162)(49 207 148 161)(50 206 149 160)(51 205 150 159)(52 204 151 158)(53 203 152 157)(54 202 153 156)(55 201 154 155)(56 200 141 168)
(1 214)(2 215)(3 216)(4 217)(5 218)(6 219)(7 220)(8 221)(9 222)(10 223)(11 224)(12 211)(13 212)(14 213)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 132)(30 133)(31 134)(32 135)(33 136)(34 137)(35 138)(36 139)(37 140)(38 127)(39 128)(40 129)(41 130)(42 131)(43 202)(44 203)(45 204)(46 205)(47 206)(48 207)(49 208)(50 209)(51 210)(52 197)(53 198)(54 199)(55 200)(56 201)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 99)(84 100)(85 176)(86 177)(87 178)(88 179)(89 180)(90 181)(91 182)(92 169)(93 170)(94 171)(95 172)(96 173)(97 174)(98 175)(113 187)(114 188)(115 189)(116 190)(117 191)(118 192)(119 193)(120 194)(121 195)(122 196)(123 183)(124 184)(125 185)(126 186)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)
(1 197 189 165)(2 210 190 164)(3 209 191 163)(4 208 192 162)(5 207 193 161)(6 206 194 160)(7 205 195 159)(8 204 196 158)(9 203 183 157)(10 202 184 156)(11 201 185 155)(12 200 186 168)(13 199 187 167)(14 198 188 166)(15 41 110 177)(16 40 111 176)(17 39 112 175)(18 38 99 174)(19 37 100 173)(20 36 101 172)(21 35 102 171)(22 34 103 170)(23 33 104 169)(24 32 105 182)(25 31 106 181)(26 30 107 180)(27 29 108 179)(28 42 109 178)(43 124 142 223)(44 123 143 222)(45 122 144 221)(46 121 145 220)(47 120 146 219)(48 119 147 218)(49 118 148 217)(50 117 149 216)(51 116 150 215)(52 115 151 214)(53 114 152 213)(54 113 153 212)(55 126 154 211)(56 125 141 224)(57 136 74 92)(58 135 75 91)(59 134 76 90)(60 133 77 89)(61 132 78 88)(62 131 79 87)(63 130 80 86)(64 129 81 85)(65 128 82 98)(66 127 83 97)(67 140 84 96)(68 139 71 95)(69 138 72 94)(70 137 73 93)
(1 137 196 86)(2 138 183 87)(3 139 184 88)(4 140 185 89)(5 127 186 90)(6 128 187 91)(7 129 188 92)(8 130 189 93)(9 131 190 94)(10 132 191 95)(11 133 192 96)(12 134 193 97)(13 135 194 98)(14 136 195 85)(15 144 103 52)(16 145 104 53)(17 146 105 54)(18 147 106 55)(19 148 107 56)(20 149 108 43)(21 150 109 44)(22 151 110 45)(23 152 111 46)(24 153 112 47)(25 154 99 48)(26 141 100 49)(27 142 101 50)(28 143 102 51)(29 117 172 223)(30 118 173 224)(31 119 174 211)(32 120 175 212)(33 121 176 213)(34 122 177 214)(35 123 178 215)(36 124 179 216)(37 125 180 217)(38 126 181 218)(39 113 182 219)(40 114 169 220)(41 115 170 221)(42 116 171 222)(57 166 81 205)(58 167 82 206)(59 168 83 207)(60 155 84 208)(61 156 71 209)(62 157 72 210)(63 158 73 197)(64 159 74 198)(65 160 75 199)(66 161 76 200)(67 162 77 201)(68 163 78 202)(69 164 79 203)(70 165 80 204)
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), (1,122,189,221)(2,121,190,220)(3,120,191,219)(4,119,192,218)(5,118,193,217)(6,117,194,216)(7,116,195,215)(8,115,196,214)(9,114,183,213)(10,113,184,212)(11,126,185,211)(12,125,186,224)(13,124,187,223)(14,123,188,222)(15,70,110,73)(16,69,111,72)(17,68,112,71)(18,67,99,84)(19,66,100,83)(20,65,101,82)(21,64,102,81)(22,63,103,80)(23,62,104,79)(24,61,105,78)(25,60,106,77)(26,59,107,76)(27,58,108,75)(28,57,109,74)(29,135,179,91)(30,134,180,90)(31,133,181,89)(32,132,182,88)(33,131,169,87)(34,130,170,86)(35,129,171,85)(36,128,172,98)(37,127,173,97)(38,140,174,96)(39,139,175,95)(40,138,176,94)(41,137,177,93)(42,136,178,92)(43,199,142,167)(44,198,143,166)(45,197,144,165)(46,210,145,164)(47,209,146,163)(48,208,147,162)(49,207,148,161)(50,206,149,160)(51,205,150,159)(52,204,151,158)(53,203,152,157)(54,202,153,156)(55,201,154,155)(56,200,141,168), (1,214)(2,215)(3,216)(4,217)(5,218)(6,219)(7,220)(8,221)(9,222)(10,223)(11,224)(12,211)(13,212)(14,213)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,132)(30,133)(31,134)(32,135)(33,136)(34,137)(35,138)(36,139)(37,140)(38,127)(39,128)(40,129)(41,130)(42,131)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,197)(53,198)(54,199)(55,200)(56,201)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,99)(84,100)(85,176)(86,177)(87,178)(88,179)(89,180)(90,181)(91,182)(92,169)(93,170)(94,171)(95,172)(96,173)(97,174)(98,175)(113,187)(114,188)(115,189)(116,190)(117,191)(118,192)(119,193)(120,194)(121,195)(122,196)(123,183)(124,184)(125,185)(126,186)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168), (1,197,189,165)(2,210,190,164)(3,209,191,163)(4,208,192,162)(5,207,193,161)(6,206,194,160)(7,205,195,159)(8,204,196,158)(9,203,183,157)(10,202,184,156)(11,201,185,155)(12,200,186,168)(13,199,187,167)(14,198,188,166)(15,41,110,177)(16,40,111,176)(17,39,112,175)(18,38,99,174)(19,37,100,173)(20,36,101,172)(21,35,102,171)(22,34,103,170)(23,33,104,169)(24,32,105,182)(25,31,106,181)(26,30,107,180)(27,29,108,179)(28,42,109,178)(43,124,142,223)(44,123,143,222)(45,122,144,221)(46,121,145,220)(47,120,146,219)(48,119,147,218)(49,118,148,217)(50,117,149,216)(51,116,150,215)(52,115,151,214)(53,114,152,213)(54,113,153,212)(55,126,154,211)(56,125,141,224)(57,136,74,92)(58,135,75,91)(59,134,76,90)(60,133,77,89)(61,132,78,88)(62,131,79,87)(63,130,80,86)(64,129,81,85)(65,128,82,98)(66,127,83,97)(67,140,84,96)(68,139,71,95)(69,138,72,94)(70,137,73,93), (1,137,196,86)(2,138,183,87)(3,139,184,88)(4,140,185,89)(5,127,186,90)(6,128,187,91)(7,129,188,92)(8,130,189,93)(9,131,190,94)(10,132,191,95)(11,133,192,96)(12,134,193,97)(13,135,194,98)(14,136,195,85)(15,144,103,52)(16,145,104,53)(17,146,105,54)(18,147,106,55)(19,148,107,56)(20,149,108,43)(21,150,109,44)(22,151,110,45)(23,152,111,46)(24,153,112,47)(25,154,99,48)(26,141,100,49)(27,142,101,50)(28,143,102,51)(29,117,172,223)(30,118,173,224)(31,119,174,211)(32,120,175,212)(33,121,176,213)(34,122,177,214)(35,123,178,215)(36,124,179,216)(37,125,180,217)(38,126,181,218)(39,113,182,219)(40,114,169,220)(41,115,170,221)(42,116,171,222)(57,166,81,205)(58,167,82,206)(59,168,83,207)(60,155,84,208)(61,156,71,209)(62,157,72,210)(63,158,73,197)(64,159,74,198)(65,160,75,199)(66,161,76,200)(67,162,77,201)(68,163,78,202)(69,164,79,203)(70,165,80,204)>;
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), (1,122,189,221)(2,121,190,220)(3,120,191,219)(4,119,192,218)(5,118,193,217)(6,117,194,216)(7,116,195,215)(8,115,196,214)(9,114,183,213)(10,113,184,212)(11,126,185,211)(12,125,186,224)(13,124,187,223)(14,123,188,222)(15,70,110,73)(16,69,111,72)(17,68,112,71)(18,67,99,84)(19,66,100,83)(20,65,101,82)(21,64,102,81)(22,63,103,80)(23,62,104,79)(24,61,105,78)(25,60,106,77)(26,59,107,76)(27,58,108,75)(28,57,109,74)(29,135,179,91)(30,134,180,90)(31,133,181,89)(32,132,182,88)(33,131,169,87)(34,130,170,86)(35,129,171,85)(36,128,172,98)(37,127,173,97)(38,140,174,96)(39,139,175,95)(40,138,176,94)(41,137,177,93)(42,136,178,92)(43,199,142,167)(44,198,143,166)(45,197,144,165)(46,210,145,164)(47,209,146,163)(48,208,147,162)(49,207,148,161)(50,206,149,160)(51,205,150,159)(52,204,151,158)(53,203,152,157)(54,202,153,156)(55,201,154,155)(56,200,141,168), (1,214)(2,215)(3,216)(4,217)(5,218)(6,219)(7,220)(8,221)(9,222)(10,223)(11,224)(12,211)(13,212)(14,213)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,132)(30,133)(31,134)(32,135)(33,136)(34,137)(35,138)(36,139)(37,140)(38,127)(39,128)(40,129)(41,130)(42,131)(43,202)(44,203)(45,204)(46,205)(47,206)(48,207)(49,208)(50,209)(51,210)(52,197)(53,198)(54,199)(55,200)(56,201)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,99)(84,100)(85,176)(86,177)(87,178)(88,179)(89,180)(90,181)(91,182)(92,169)(93,170)(94,171)(95,172)(96,173)(97,174)(98,175)(113,187)(114,188)(115,189)(116,190)(117,191)(118,192)(119,193)(120,194)(121,195)(122,196)(123,183)(124,184)(125,185)(126,186)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168), (1,197,189,165)(2,210,190,164)(3,209,191,163)(4,208,192,162)(5,207,193,161)(6,206,194,160)(7,205,195,159)(8,204,196,158)(9,203,183,157)(10,202,184,156)(11,201,185,155)(12,200,186,168)(13,199,187,167)(14,198,188,166)(15,41,110,177)(16,40,111,176)(17,39,112,175)(18,38,99,174)(19,37,100,173)(20,36,101,172)(21,35,102,171)(22,34,103,170)(23,33,104,169)(24,32,105,182)(25,31,106,181)(26,30,107,180)(27,29,108,179)(28,42,109,178)(43,124,142,223)(44,123,143,222)(45,122,144,221)(46,121,145,220)(47,120,146,219)(48,119,147,218)(49,118,148,217)(50,117,149,216)(51,116,150,215)(52,115,151,214)(53,114,152,213)(54,113,153,212)(55,126,154,211)(56,125,141,224)(57,136,74,92)(58,135,75,91)(59,134,76,90)(60,133,77,89)(61,132,78,88)(62,131,79,87)(63,130,80,86)(64,129,81,85)(65,128,82,98)(66,127,83,97)(67,140,84,96)(68,139,71,95)(69,138,72,94)(70,137,73,93), (1,137,196,86)(2,138,183,87)(3,139,184,88)(4,140,185,89)(5,127,186,90)(6,128,187,91)(7,129,188,92)(8,130,189,93)(9,131,190,94)(10,132,191,95)(11,133,192,96)(12,134,193,97)(13,135,194,98)(14,136,195,85)(15,144,103,52)(16,145,104,53)(17,146,105,54)(18,147,106,55)(19,148,107,56)(20,149,108,43)(21,150,109,44)(22,151,110,45)(23,152,111,46)(24,153,112,47)(25,154,99,48)(26,141,100,49)(27,142,101,50)(28,143,102,51)(29,117,172,223)(30,118,173,224)(31,119,174,211)(32,120,175,212)(33,121,176,213)(34,122,177,214)(35,123,178,215)(36,124,179,216)(37,125,180,217)(38,126,181,218)(39,113,182,219)(40,114,169,220)(41,115,170,221)(42,116,171,222)(57,166,81,205)(58,167,82,206)(59,168,83,207)(60,155,84,208)(61,156,71,209)(62,157,72,210)(63,158,73,197)(64,159,74,198)(65,160,75,199)(66,161,76,200)(67,162,77,201)(68,163,78,202)(69,164,79,203)(70,165,80,204) );
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)], [(1,122,189,221),(2,121,190,220),(3,120,191,219),(4,119,192,218),(5,118,193,217),(6,117,194,216),(7,116,195,215),(8,115,196,214),(9,114,183,213),(10,113,184,212),(11,126,185,211),(12,125,186,224),(13,124,187,223),(14,123,188,222),(15,70,110,73),(16,69,111,72),(17,68,112,71),(18,67,99,84),(19,66,100,83),(20,65,101,82),(21,64,102,81),(22,63,103,80),(23,62,104,79),(24,61,105,78),(25,60,106,77),(26,59,107,76),(27,58,108,75),(28,57,109,74),(29,135,179,91),(30,134,180,90),(31,133,181,89),(32,132,182,88),(33,131,169,87),(34,130,170,86),(35,129,171,85),(36,128,172,98),(37,127,173,97),(38,140,174,96),(39,139,175,95),(40,138,176,94),(41,137,177,93),(42,136,178,92),(43,199,142,167),(44,198,143,166),(45,197,144,165),(46,210,145,164),(47,209,146,163),(48,208,147,162),(49,207,148,161),(50,206,149,160),(51,205,150,159),(52,204,151,158),(53,203,152,157),(54,202,153,156),(55,201,154,155),(56,200,141,168)], [(1,214),(2,215),(3,216),(4,217),(5,218),(6,219),(7,220),(8,221),(9,222),(10,223),(11,224),(12,211),(13,212),(14,213),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,57),(24,58),(25,59),(26,60),(27,61),(28,62),(29,132),(30,133),(31,134),(32,135),(33,136),(34,137),(35,138),(36,139),(37,140),(38,127),(39,128),(40,129),(41,130),(42,131),(43,202),(44,203),(45,204),(46,205),(47,206),(48,207),(49,208),(50,209),(51,210),(52,197),(53,198),(54,199),(55,200),(56,201),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,99),(84,100),(85,176),(86,177),(87,178),(88,179),(89,180),(90,181),(91,182),(92,169),(93,170),(94,171),(95,172),(96,173),(97,174),(98,175),(113,187),(114,188),(115,189),(116,190),(117,191),(118,192),(119,193),(120,194),(121,195),(122,196),(123,183),(124,184),(125,185),(126,186),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168)], [(1,197,189,165),(2,210,190,164),(3,209,191,163),(4,208,192,162),(5,207,193,161),(6,206,194,160),(7,205,195,159),(8,204,196,158),(9,203,183,157),(10,202,184,156),(11,201,185,155),(12,200,186,168),(13,199,187,167),(14,198,188,166),(15,41,110,177),(16,40,111,176),(17,39,112,175),(18,38,99,174),(19,37,100,173),(20,36,101,172),(21,35,102,171),(22,34,103,170),(23,33,104,169),(24,32,105,182),(25,31,106,181),(26,30,107,180),(27,29,108,179),(28,42,109,178),(43,124,142,223),(44,123,143,222),(45,122,144,221),(46,121,145,220),(47,120,146,219),(48,119,147,218),(49,118,148,217),(50,117,149,216),(51,116,150,215),(52,115,151,214),(53,114,152,213),(54,113,153,212),(55,126,154,211),(56,125,141,224),(57,136,74,92),(58,135,75,91),(59,134,76,90),(60,133,77,89),(61,132,78,88),(62,131,79,87),(63,130,80,86),(64,129,81,85),(65,128,82,98),(66,127,83,97),(67,140,84,96),(68,139,71,95),(69,138,72,94),(70,137,73,93)], [(1,137,196,86),(2,138,183,87),(3,139,184,88),(4,140,185,89),(5,127,186,90),(6,128,187,91),(7,129,188,92),(8,130,189,93),(9,131,190,94),(10,132,191,95),(11,133,192,96),(12,134,193,97),(13,135,194,98),(14,136,195,85),(15,144,103,52),(16,145,104,53),(17,146,105,54),(18,147,106,55),(19,148,107,56),(20,149,108,43),(21,150,109,44),(22,151,110,45),(23,152,111,46),(24,153,112,47),(25,154,99,48),(26,141,100,49),(27,142,101,50),(28,143,102,51),(29,117,172,223),(30,118,173,224),(31,119,174,211),(32,120,175,212),(33,121,176,213),(34,122,177,214),(35,123,178,215),(36,124,179,216),(37,125,180,217),(38,126,181,218),(39,113,182,219),(40,114,169,220),(41,115,170,221),(42,116,171,222),(57,166,81,205),(58,167,82,206),(59,168,83,207),(60,155,84,208),(61,156,71,209),(62,157,72,210),(63,158,73,197),(64,159,74,198),(65,160,75,199),(66,161,76,200),(67,162,77,201),(68,163,78,202),(69,164,79,203),(70,165,80,204)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 25 | 0 | 0 |
| 0 | 0 | 8 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 18 | 0 | 0 |
| 0 | 0 | 23 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 11 | 0 | 0 |
| 0 | 0 | 6 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 27 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 2 |
| 0 | 0 | 0 | 0 | 28 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,24,8,0,0,0,0,25,12,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,15,23,0,0,0,0,18,14,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,14,6,0,0,0,0,11,15,0,0,0,0,0,0,28,0,0,0,0,0,27,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,28,0,0,0,0,2,28] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4K | 4L | 4M | 4N | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 14 | 14 | 2 | 2 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
67 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2- (1+4) | D4⋊2D7 | D4×D7 | D4.10D14 |
| kernel | C14.732- (1+4) | C22⋊Dic14 | D14.D4 | C28⋊Q8 | D7×C4⋊C4 | C2×C4⋊Dic7 | C4×C7⋊D4 | D4×Dic7 | C23.18D14 | C28⋊2D4 | C7×C4⋊D4 | C2×D4⋊2D7 | C7⋊D4 | C4⋊D4 | C28 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 3 | 4 | 6 | 3 | 3 | 9 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{73}2_-^{(1+4)} % in TeX
G:=Group("C14.73ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1064);
// by ID
G=gap.SmallGroup(448,1064);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,100,675,185,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7*b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations
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