?
G = C14.722- (1+4) order 448 = 26·7
27th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.722- (1+4), C14.392+ (1+4), (C4×D7)⋊2D4, C4⋊D4⋊11D7, D14.3(C2×D4), C4.184(D4×D7), C28⋊2D4⋊19C2, C4⋊C4.180D14, (C2×D4).92D14, C28.228(C2×D4), C22⋊C4.8D14, D14⋊D4⋊19C2, D14⋊2Q8⋊21C2, (C2×C28).39C23, Dic7.46(C2×D4), C14.67(C22×D4), Dic7⋊D4⋊13C2, C28.48D4⋊34C2, (C2×C14).152C24, D14⋊C4.15C22, (C22×C4).223D14, C2.41(D4⋊6D14), C23.16(C22×D7), C22⋊Dic14⋊19C2, (D4×C14).122C22, (C2×D28).220C22, Dic7⋊C4.18C22, C4⋊Dic7.207C22, (C2×Dic7).73C23, C22.173(C23×D7), C23.D7.25C22, (C22×C28).241C22, (C22×C14).187C23, C7⋊3(C22.31C24), (C22×D7).187C23, C2.30(D4.10D14), (C2×Dic14).154C22, (C22×Dic7).109C22, C2.40(C2×D4×D7), (D7×C4⋊C4)⋊21C2, (C2×C4○D28)⋊21C2, (C7×C4⋊D4)⋊14C2, (C2×D4⋊2D7)⋊13C2, (C2×C4×D7).83C22, (C7×C4⋊C4).144C22, (C2×C4).586(C22×D7), (C2×C7⋊D4).28C22, (C7×C22⋊C4).13C22, SmallGroup(448,1061)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1420 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×10], C22, C22 [×16], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×20], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×2], D7 [×3], C14 [×3], C14 [×3], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×7], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic7 [×2], Dic7 [×5], C28 [×2], C28 [×3], D14 [×2], D14 [×5], C2×C14, C2×C14 [×9], C2×C4⋊C4, C4⋊D4, C4⋊D4 [×7], C22⋊Q8 [×4], C2×C4○D4 [×2], Dic14 [×4], C4×D7 [×4], C4×D7 [×4], D28 [×2], C2×Dic7 [×2], C2×Dic7 [×4], C2×Dic7 [×4], C7⋊D4 [×10], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×4], C22×D7 [×2], C22×C14, C22×C14 [×2], C22.31C24, Dic7⋊C4 [×4], C4⋊Dic7, C4⋊Dic7 [×2], D14⋊C4 [×2], C23.D7 [×4], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7 [×2], C2×C4×D7 [×2], C2×D28, C4○D28 [×4], D4⋊2D7 [×4], C22×Dic7 [×2], C2×C7⋊D4 [×2], C2×C7⋊D4 [×4], C22×C28, D4×C14, D4×C14 [×2], C22⋊Dic14 [×2], D14⋊D4 [×2], D7×C4⋊C4, D14⋊2Q8, C28.48D4, C28⋊2D4, C28⋊2D4 [×2], Dic7⋊D4 [×2], C7×C4⋊D4, C2×C4○D28, C2×D4⋊2D7, C14.722- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4), 2- (1+4), C22×D7 [×7], C22.31C24, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊6D14, D4.10D14, C14.722- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=a7b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, dbd-1=ebe-1=a7b, dcd-1=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)
(1 148 37 213)(2 147 38 212)(3 146 39 211)(4 145 40 224)(5 144 41 223)(6 143 42 222)(7 142 29 221)(8 141 30 220)(9 154 31 219)(10 153 32 218)(11 152 33 217)(12 151 34 216)(13 150 35 215)(14 149 36 214)(15 197 84 93)(16 210 71 92)(17 209 72 91)(18 208 73 90)(19 207 74 89)(20 206 75 88)(21 205 76 87)(22 204 77 86)(23 203 78 85)(24 202 79 98)(25 201 80 97)(26 200 81 96)(27 199 82 95)(28 198 83 94)(43 156 112 65)(44 155 99 64)(45 168 100 63)(46 167 101 62)(47 166 102 61)(48 165 103 60)(49 164 104 59)(50 163 105 58)(51 162 106 57)(52 161 107 70)(53 160 108 69)(54 159 109 68)(55 158 110 67)(56 157 111 66)(113 170 194 135)(114 169 195 134)(115 182 196 133)(116 181 183 132)(117 180 184 131)(118 179 185 130)(119 178 186 129)(120 177 187 128)(121 176 188 127)(122 175 189 140)(123 174 190 139)(124 173 191 138)(125 172 192 137)(126 171 193 136)
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 204)(16 205)(17 206)(18 207)(19 208)(20 209)(21 210)(22 197)(23 198)(24 199)(25 200)(26 201)(27 202)(28 203)(29 212)(30 213)(31 214)(32 215)(33 216)(34 217)(35 218)(36 219)(37 220)(38 221)(39 222)(40 223)(41 224)(42 211)(43 66)(44 67)(45 68)(46 69)(47 70)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 65)(71 87)(72 88)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 97)(82 98)(83 85)(84 86)(99 158)(100 159)(101 160)(102 161)(103 162)(104 163)(105 164)(106 165)(107 166)(108 167)(109 168)(110 155)(111 156)(112 157)(113 130)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)(120 137)(121 138)(122 139)(123 140)(124 127)(125 128)(126 129)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 190)(176 191)(177 192)(178 193)(179 194)(180 195)(181 196)(182 183)
(1 62 30 160)(2 61 31 159)(3 60 32 158)(4 59 33 157)(5 58 34 156)(6 57 35 155)(7 70 36 168)(8 69 37 167)(9 68 38 166)(10 67 39 165)(11 66 40 164)(12 65 41 163)(13 64 42 162)(14 63 29 161)(15 136 77 178)(16 135 78 177)(17 134 79 176)(18 133 80 175)(19 132 81 174)(20 131 82 173)(21 130 83 172)(22 129 84 171)(23 128 71 170)(24 127 72 169)(25 140 73 182)(26 139 74 181)(27 138 75 180)(28 137 76 179)(43 216 105 144)(44 215 106 143)(45 214 107 142)(46 213 108 141)(47 212 109 154)(48 211 110 153)(49 224 111 152)(50 223 112 151)(51 222 99 150)(52 221 100 149)(53 220 101 148)(54 219 102 147)(55 218 103 146)(56 217 104 145)(85 194 210 120)(86 193 197 119)(87 192 198 118)(88 191 199 117)(89 190 200 116)(90 189 201 115)(91 188 202 114)(92 187 203 113)(93 186 204 126)(94 185 205 125)(95 184 206 124)(96 183 207 123)(97 196 208 122)(98 195 209 121)
(1 84 30 22)(2 71 31 23)(3 72 32 24)(4 73 33 25)(5 74 34 26)(6 75 35 27)(7 76 36 28)(8 77 37 15)(9 78 38 16)(10 79 39 17)(11 80 40 18)(12 81 41 19)(13 82 42 20)(14 83 29 21)(43 116 105 190)(44 117 106 191)(45 118 107 192)(46 119 108 193)(47 120 109 194)(48 121 110 195)(49 122 111 196)(50 123 112 183)(51 124 99 184)(52 125 100 185)(53 126 101 186)(54 113 102 187)(55 114 103 188)(56 115 104 189)(57 131 155 173)(58 132 156 174)(59 133 157 175)(60 134 158 176)(61 135 159 177)(62 136 160 178)(63 137 161 179)(64 138 162 180)(65 139 163 181)(66 140 164 182)(67 127 165 169)(68 128 166 170)(69 129 167 171)(70 130 168 172)(85 219 210 147)(86 220 197 148)(87 221 198 149)(88 222 199 150)(89 223 200 151)(90 224 201 152)(91 211 202 153)(92 212 203 154)(93 213 204 141)(94 214 205 142)(95 215 206 143)(96 216 207 144)(97 217 208 145)(98 218 209 146)
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), (1,148,37,213)(2,147,38,212)(3,146,39,211)(4,145,40,224)(5,144,41,223)(6,143,42,222)(7,142,29,221)(8,141,30,220)(9,154,31,219)(10,153,32,218)(11,152,33,217)(12,151,34,216)(13,150,35,215)(14,149,36,214)(15,197,84,93)(16,210,71,92)(17,209,72,91)(18,208,73,90)(19,207,74,89)(20,206,75,88)(21,205,76,87)(22,204,77,86)(23,203,78,85)(24,202,79,98)(25,201,80,97)(26,200,81,96)(27,199,82,95)(28,198,83,94)(43,156,112,65)(44,155,99,64)(45,168,100,63)(46,167,101,62)(47,166,102,61)(48,165,103,60)(49,164,104,59)(50,163,105,58)(51,162,106,57)(52,161,107,70)(53,160,108,69)(54,159,109,68)(55,158,110,67)(56,157,111,66)(113,170,194,135)(114,169,195,134)(115,182,196,133)(116,181,183,132)(117,180,184,131)(118,179,185,130)(119,178,186,129)(120,177,187,128)(121,176,188,127)(122,175,189,140)(123,174,190,139)(124,173,191,138)(125,172,192,137)(126,171,193,136), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,204)(16,205)(17,206)(18,207)(19,208)(20,209)(21,210)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,212)(30,213)(31,214)(32,215)(33,216)(34,217)(35,218)(36,219)(37,220)(38,221)(39,222)(40,223)(41,224)(42,211)(43,66)(44,67)(45,68)(46,69)(47,70)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(71,87)(72,88)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,97)(82,98)(83,85)(84,86)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,155)(111,156)(112,157)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136)(120,137)(121,138)(122,139)(123,140)(124,127)(125,128)(126,129)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195)(181,196)(182,183), (1,62,30,160)(2,61,31,159)(3,60,32,158)(4,59,33,157)(5,58,34,156)(6,57,35,155)(7,70,36,168)(8,69,37,167)(9,68,38,166)(10,67,39,165)(11,66,40,164)(12,65,41,163)(13,64,42,162)(14,63,29,161)(15,136,77,178)(16,135,78,177)(17,134,79,176)(18,133,80,175)(19,132,81,174)(20,131,82,173)(21,130,83,172)(22,129,84,171)(23,128,71,170)(24,127,72,169)(25,140,73,182)(26,139,74,181)(27,138,75,180)(28,137,76,179)(43,216,105,144)(44,215,106,143)(45,214,107,142)(46,213,108,141)(47,212,109,154)(48,211,110,153)(49,224,111,152)(50,223,112,151)(51,222,99,150)(52,221,100,149)(53,220,101,148)(54,219,102,147)(55,218,103,146)(56,217,104,145)(85,194,210,120)(86,193,197,119)(87,192,198,118)(88,191,199,117)(89,190,200,116)(90,189,201,115)(91,188,202,114)(92,187,203,113)(93,186,204,126)(94,185,205,125)(95,184,206,124)(96,183,207,123)(97,196,208,122)(98,195,209,121), (1,84,30,22)(2,71,31,23)(3,72,32,24)(4,73,33,25)(5,74,34,26)(6,75,35,27)(7,76,36,28)(8,77,37,15)(9,78,38,16)(10,79,39,17)(11,80,40,18)(12,81,41,19)(13,82,42,20)(14,83,29,21)(43,116,105,190)(44,117,106,191)(45,118,107,192)(46,119,108,193)(47,120,109,194)(48,121,110,195)(49,122,111,196)(50,123,112,183)(51,124,99,184)(52,125,100,185)(53,126,101,186)(54,113,102,187)(55,114,103,188)(56,115,104,189)(57,131,155,173)(58,132,156,174)(59,133,157,175)(60,134,158,176)(61,135,159,177)(62,136,160,178)(63,137,161,179)(64,138,162,180)(65,139,163,181)(66,140,164,182)(67,127,165,169)(68,128,166,170)(69,129,167,171)(70,130,168,172)(85,219,210,147)(86,220,197,148)(87,221,198,149)(88,222,199,150)(89,223,200,151)(90,224,201,152)(91,211,202,153)(92,212,203,154)(93,213,204,141)(94,214,205,142)(95,215,206,143)(96,216,207,144)(97,217,208,145)(98,218,209,146)>;
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), (1,148,37,213)(2,147,38,212)(3,146,39,211)(4,145,40,224)(5,144,41,223)(6,143,42,222)(7,142,29,221)(8,141,30,220)(9,154,31,219)(10,153,32,218)(11,152,33,217)(12,151,34,216)(13,150,35,215)(14,149,36,214)(15,197,84,93)(16,210,71,92)(17,209,72,91)(18,208,73,90)(19,207,74,89)(20,206,75,88)(21,205,76,87)(22,204,77,86)(23,203,78,85)(24,202,79,98)(25,201,80,97)(26,200,81,96)(27,199,82,95)(28,198,83,94)(43,156,112,65)(44,155,99,64)(45,168,100,63)(46,167,101,62)(47,166,102,61)(48,165,103,60)(49,164,104,59)(50,163,105,58)(51,162,106,57)(52,161,107,70)(53,160,108,69)(54,159,109,68)(55,158,110,67)(56,157,111,66)(113,170,194,135)(114,169,195,134)(115,182,196,133)(116,181,183,132)(117,180,184,131)(118,179,185,130)(119,178,186,129)(120,177,187,128)(121,176,188,127)(122,175,189,140)(123,174,190,139)(124,173,191,138)(125,172,192,137)(126,171,193,136), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,204)(16,205)(17,206)(18,207)(19,208)(20,209)(21,210)(22,197)(23,198)(24,199)(25,200)(26,201)(27,202)(28,203)(29,212)(30,213)(31,214)(32,215)(33,216)(34,217)(35,218)(36,219)(37,220)(38,221)(39,222)(40,223)(41,224)(42,211)(43,66)(44,67)(45,68)(46,69)(47,70)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(71,87)(72,88)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,97)(82,98)(83,85)(84,86)(99,158)(100,159)(101,160)(102,161)(103,162)(104,163)(105,164)(106,165)(107,166)(108,167)(109,168)(110,155)(111,156)(112,157)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136)(120,137)(121,138)(122,139)(123,140)(124,127)(125,128)(126,129)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,190)(176,191)(177,192)(178,193)(179,194)(180,195)(181,196)(182,183), (1,62,30,160)(2,61,31,159)(3,60,32,158)(4,59,33,157)(5,58,34,156)(6,57,35,155)(7,70,36,168)(8,69,37,167)(9,68,38,166)(10,67,39,165)(11,66,40,164)(12,65,41,163)(13,64,42,162)(14,63,29,161)(15,136,77,178)(16,135,78,177)(17,134,79,176)(18,133,80,175)(19,132,81,174)(20,131,82,173)(21,130,83,172)(22,129,84,171)(23,128,71,170)(24,127,72,169)(25,140,73,182)(26,139,74,181)(27,138,75,180)(28,137,76,179)(43,216,105,144)(44,215,106,143)(45,214,107,142)(46,213,108,141)(47,212,109,154)(48,211,110,153)(49,224,111,152)(50,223,112,151)(51,222,99,150)(52,221,100,149)(53,220,101,148)(54,219,102,147)(55,218,103,146)(56,217,104,145)(85,194,210,120)(86,193,197,119)(87,192,198,118)(88,191,199,117)(89,190,200,116)(90,189,201,115)(91,188,202,114)(92,187,203,113)(93,186,204,126)(94,185,205,125)(95,184,206,124)(96,183,207,123)(97,196,208,122)(98,195,209,121), (1,84,30,22)(2,71,31,23)(3,72,32,24)(4,73,33,25)(5,74,34,26)(6,75,35,27)(7,76,36,28)(8,77,37,15)(9,78,38,16)(10,79,39,17)(11,80,40,18)(12,81,41,19)(13,82,42,20)(14,83,29,21)(43,116,105,190)(44,117,106,191)(45,118,107,192)(46,119,108,193)(47,120,109,194)(48,121,110,195)(49,122,111,196)(50,123,112,183)(51,124,99,184)(52,125,100,185)(53,126,101,186)(54,113,102,187)(55,114,103,188)(56,115,104,189)(57,131,155,173)(58,132,156,174)(59,133,157,175)(60,134,158,176)(61,135,159,177)(62,136,160,178)(63,137,161,179)(64,138,162,180)(65,139,163,181)(66,140,164,182)(67,127,165,169)(68,128,166,170)(69,129,167,171)(70,130,168,172)(85,219,210,147)(86,220,197,148)(87,221,198,149)(88,222,199,150)(89,223,200,151)(90,224,201,152)(91,211,202,153)(92,212,203,154)(93,213,204,141)(94,214,205,142)(95,215,206,143)(96,216,207,144)(97,217,208,145)(98,218,209,146) );
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)], [(1,148,37,213),(2,147,38,212),(3,146,39,211),(4,145,40,224),(5,144,41,223),(6,143,42,222),(7,142,29,221),(8,141,30,220),(9,154,31,219),(10,153,32,218),(11,152,33,217),(12,151,34,216),(13,150,35,215),(14,149,36,214),(15,197,84,93),(16,210,71,92),(17,209,72,91),(18,208,73,90),(19,207,74,89),(20,206,75,88),(21,205,76,87),(22,204,77,86),(23,203,78,85),(24,202,79,98),(25,201,80,97),(26,200,81,96),(27,199,82,95),(28,198,83,94),(43,156,112,65),(44,155,99,64),(45,168,100,63),(46,167,101,62),(47,166,102,61),(48,165,103,60),(49,164,104,59),(50,163,105,58),(51,162,106,57),(52,161,107,70),(53,160,108,69),(54,159,109,68),(55,158,110,67),(56,157,111,66),(113,170,194,135),(114,169,195,134),(115,182,196,133),(116,181,183,132),(117,180,184,131),(118,179,185,130),(119,178,186,129),(120,177,187,128),(121,176,188,127),(122,175,189,140),(123,174,190,139),(124,173,191,138),(125,172,192,137),(126,171,193,136)], [(1,141),(2,142),(3,143),(4,144),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,204),(16,205),(17,206),(18,207),(19,208),(20,209),(21,210),(22,197),(23,198),(24,199),(25,200),(26,201),(27,202),(28,203),(29,212),(30,213),(31,214),(32,215),(33,216),(34,217),(35,218),(36,219),(37,220),(38,221),(39,222),(40,223),(41,224),(42,211),(43,66),(44,67),(45,68),(46,69),(47,70),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,65),(71,87),(72,88),(73,89),(74,90),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96),(81,97),(82,98),(83,85),(84,86),(99,158),(100,159),(101,160),(102,161),(103,162),(104,163),(105,164),(106,165),(107,166),(108,167),(109,168),(110,155),(111,156),(112,157),(113,130),(114,131),(115,132),(116,133),(117,134),(118,135),(119,136),(120,137),(121,138),(122,139),(123,140),(124,127),(125,128),(126,129),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,190),(176,191),(177,192),(178,193),(179,194),(180,195),(181,196),(182,183)], [(1,62,30,160),(2,61,31,159),(3,60,32,158),(4,59,33,157),(5,58,34,156),(6,57,35,155),(7,70,36,168),(8,69,37,167),(9,68,38,166),(10,67,39,165),(11,66,40,164),(12,65,41,163),(13,64,42,162),(14,63,29,161),(15,136,77,178),(16,135,78,177),(17,134,79,176),(18,133,80,175),(19,132,81,174),(20,131,82,173),(21,130,83,172),(22,129,84,171),(23,128,71,170),(24,127,72,169),(25,140,73,182),(26,139,74,181),(27,138,75,180),(28,137,76,179),(43,216,105,144),(44,215,106,143),(45,214,107,142),(46,213,108,141),(47,212,109,154),(48,211,110,153),(49,224,111,152),(50,223,112,151),(51,222,99,150),(52,221,100,149),(53,220,101,148),(54,219,102,147),(55,218,103,146),(56,217,104,145),(85,194,210,120),(86,193,197,119),(87,192,198,118),(88,191,199,117),(89,190,200,116),(90,189,201,115),(91,188,202,114),(92,187,203,113),(93,186,204,126),(94,185,205,125),(95,184,206,124),(96,183,207,123),(97,196,208,122),(98,195,209,121)], [(1,84,30,22),(2,71,31,23),(3,72,32,24),(4,73,33,25),(5,74,34,26),(6,75,35,27),(7,76,36,28),(8,77,37,15),(9,78,38,16),(10,79,39,17),(11,80,40,18),(12,81,41,19),(13,82,42,20),(14,83,29,21),(43,116,105,190),(44,117,106,191),(45,118,107,192),(46,119,108,193),(47,120,109,194),(48,121,110,195),(49,122,111,196),(50,123,112,183),(51,124,99,184),(52,125,100,185),(53,126,101,186),(54,113,102,187),(55,114,103,188),(56,115,104,189),(57,131,155,173),(58,132,156,174),(59,133,157,175),(60,134,158,176),(61,135,159,177),(62,136,160,178),(63,137,161,179),(64,138,162,180),(65,139,163,181),(66,140,164,182),(67,127,165,169),(68,128,166,170),(69,129,167,171),(70,130,168,172),(85,219,210,147),(86,220,197,148),(87,221,198,149),(88,222,199,150),(89,223,200,151),(90,224,201,152),(91,211,202,153),(92,212,203,154),(93,213,204,141),(94,214,205,142),(95,215,206,143),(96,216,207,144),(97,217,208,145),(98,218,209,146)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 21 | 0 | 0 |
| 0 | 0 | 17 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 21 |
| 0 | 0 | 0 | 0 | 17 | 28 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 26 | 0 | 0 |
| 0 | 0 | 19 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 14 | 20 | 0 | 0 | 0 | 0 |
| 9 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 3 | 17 |
| 0 | 0 | 7 | 0 | 11 | 26 |
| 0 | 0 | 3 | 17 | 0 | 5 |
| 0 | 0 | 11 | 26 | 7 | 0 |
| 14 | 20 | 0 | 0 | 0 | 0 |
| 9 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 18 |
| 0 | 0 | 0 | 0 | 27 | 8 |
| 0 | 0 | 21 | 18 | 0 | 0 |
| 0 | 0 | 27 | 8 | 0 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,19,17,0,0,0,0,21,28,0,0,0,0,0,0,19,17,0,0,0,0,21,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,26,0,0,0,0,10,0,0,0,0,3,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[14,9,0,0,0,0,20,15,0,0,0,0,0,0,0,7,3,11,0,0,5,0,17,26,0,0,3,11,0,7,0,0,17,26,5,0],[14,9,0,0,0,0,20,15,0,0,0,0,0,0,0,0,21,27,0,0,0,0,18,8,0,0,21,27,0,0,0,0,18,8,0,0] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 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 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4×D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.722- (1+4) | C22⋊Dic14 | D14⋊D4 | D7×C4⋊C4 | D14⋊2Q8 | C28.48D4 | C28⋊2D4 | Dic7⋊D4 | C7×C4⋊D4 | C2×C4○D28 | C2×D4⋊2D7 | C4×D7 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 4 | 3 | 6 | 3 | 3 | 9 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{72}2_-^{(1+4)} % in TeX
G:=Group("C14.72ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1061);
// by ID
G=gap.SmallGroup(448,1061);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,1123,570,185,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^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,d*b*d^-1=e*b*e^-1=a^7*b,d*c*d^-1=e*c*e^-1=a^7*c,e*d*e^-1=b^2*d>;
// generators/relations
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