?
G = C14.852- (1+4) order 448 = 26·7
40th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.852- (1+4), C14.1232+ (1+4), C4⋊C4.108D14, (D4×Dic7)⋊29C2, D14⋊2Q8⋊33C2, (C2×D4).167D14, Dic7.Q8⋊30C2, D14⋊C4.9C22, C22⋊C4.69D14, C4.Dic14⋊30C2, Dic7⋊4D4⋊26C2, D14.D4⋊37C2, (C2×C28).183C23, (C2×C14).209C24, Dic7⋊D4.3C2, (C22×C4).262D14, C22.D4⋊14D7, C2.45(D4⋊8D14), C23.33(C22×D7), C22⋊Dic14⋊35C2, (D4×C14).147C22, C23.D14⋊35C2, C22.D28⋊23C2, C23.23D14⋊8C2, Dic7⋊C4.10C22, C4⋊Dic7.313C22, (C22×C14).41C23, (C22×C28).87C22, (C22×D7).90C23, C22.230(C23×D7), C23.D7.47C22, C22.12(D4⋊2D7), C7⋊7(C22.33C24), (C2×Dic7).108C23, (C4×Dic7).128C22, C2.46(D4.10D14), (C2×Dic14).171C22, (C22×Dic7).135C22, C4⋊C4⋊D7⋊31C2, (C2×C4⋊Dic7)⋊27C2, C14.92(C2×C4○D4), C2.54(C2×D4⋊2D7), (C2×C4×D7).116C22, (C2×C14).48(C4○D4), (C7×C4⋊C4).182C22, (C2×C4).190(C22×D7), (C2×C7⋊D4).53C22, (C7×C22.D4)⋊17C2, (C7×C22⋊C4).57C22, SmallGroup(448,1118)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 940 in 218 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C7, C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23, D7, C14 [×3], C14 [×3], C42 [×2], C22⋊C4 [×3], C22⋊C4 [×7], C4⋊C4 [×2], C4⋊C4 [×12], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, Dic7 [×7], C28 [×5], D14 [×3], C2×C14, C2×C14 [×2], C2×C14 [×5], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8 [×3], C22.D4, C22.D4 [×3], C42.C2 [×2], C42⋊2C2 [×2], Dic14, C4×D7, C2×Dic7 [×7], C2×Dic7 [×3], C7⋊D4 [×3], C2×C28 [×5], C2×C28 [×2], C7×D4 [×2], C22×D7, C22×C14 [×2], C22.33C24, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×6], D14⋊C4 [×4], C23.D7 [×3], C7×C22⋊C4 [×3], C7×C4⋊C4 [×2], C2×Dic14, C2×C4×D7, C22×Dic7 [×3], C2×C7⋊D4 [×2], C22×C28, D4×C14, C22⋊Dic14 [×2], C23.D14, Dic7⋊4D4, D14.D4, C22.D28, Dic7.Q8, C4.Dic14, D14⋊2Q8, C4⋊C4⋊D7, C2×C4⋊Dic7, C23.23D14, D4×Dic7, Dic7⋊D4, C7×C22.D4, C14.852- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D7 [×7], C22.33C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D4⋊8D14, D4.10D14, C14.852- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=a7b2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a7b, cd=dc, ce=ec, 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)
(1 121 49 35)(2 122 50 36)(3 123 51 37)(4 124 52 38)(5 125 53 39)(6 126 54 40)(7 113 55 41)(8 114 56 42)(9 115 43 29)(10 116 44 30)(11 117 45 31)(12 118 46 32)(13 119 47 33)(14 120 48 34)(15 133 169 108)(16 134 170 109)(17 135 171 110)(18 136 172 111)(19 137 173 112)(20 138 174 99)(21 139 175 100)(22 140 176 101)(23 127 177 102)(24 128 178 103)(25 129 179 104)(26 130 180 105)(27 131 181 106)(28 132 182 107)(57 77 200 221)(58 78 201 222)(59 79 202 223)(60 80 203 224)(61 81 204 211)(62 82 205 212)(63 83 206 213)(64 84 207 214)(65 71 208 215)(66 72 209 216)(67 73 210 217)(68 74 197 218)(69 75 198 219)(70 76 199 220)(85 192 149 162)(86 193 150 163)(87 194 151 164)(88 195 152 165)(89 196 153 166)(90 183 154 167)(91 184 141 168)(92 185 142 155)(93 186 143 156)(94 187 144 157)(95 188 145 158)(96 189 146 159)(97 190 147 160)(98 191 148 161)
(29 115)(30 116)(31 117)(32 118)(33 119)(34 120)(35 121)(36 122)(37 123)(38 124)(39 125)(40 126)(41 113)(42 114)(57 200)(58 201)(59 202)(60 203)(61 204)(62 205)(63 206)(64 207)(65 208)(66 209)(67 210)(68 197)(69 198)(70 199)(85 149)(86 150)(87 151)(88 152)(89 153)(90 154)(91 141)(92 142)(93 143)(94 144)(95 145)(96 146)(97 147)(98 148)(99 138)(100 139)(101 140)(102 127)(103 128)(104 129)(105 130)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 137)
(1 27 56 174)(2 26 43 173)(3 25 44 172)(4 24 45 171)(5 23 46 170)(6 22 47 169)(7 21 48 182)(8 20 49 181)(9 19 50 180)(10 18 51 179)(11 17 52 178)(12 16 53 177)(13 15 54 176)(14 28 55 175)(29 105 122 137)(30 104 123 136)(31 103 124 135)(32 102 125 134)(33 101 126 133)(34 100 113 132)(35 99 114 131)(36 112 115 130)(37 111 116 129)(38 110 117 128)(39 109 118 127)(40 108 119 140)(41 107 120 139)(42 106 121 138)(57 154 207 97)(58 153 208 96)(59 152 209 95)(60 151 210 94)(61 150 197 93)(62 149 198 92)(63 148 199 91)(64 147 200 90)(65 146 201 89)(66 145 202 88)(67 144 203 87)(68 143 204 86)(69 142 205 85)(70 141 206 98)(71 166 222 189)(72 165 223 188)(73 164 224 187)(74 163 211 186)(75 162 212 185)(76 161 213 184)(77 160 214 183)(78 159 215 196)(79 158 216 195)(80 157 217 194)(81 156 218 193)(82 155 219 192)(83 168 220 191)(84 167 221 190)
(1 223 49 79)(2 224 50 80)(3 211 51 81)(4 212 52 82)(5 213 53 83)(6 214 54 84)(7 215 55 71)(8 216 56 72)(9 217 43 73)(10 218 44 74)(11 219 45 75)(12 220 46 76)(13 221 47 77)(14 222 48 78)(15 160 169 190)(16 161 170 191)(17 162 171 192)(18 163 172 193)(19 164 173 194)(20 165 174 195)(21 166 175 196)(22 167 176 183)(23 168 177 184)(24 155 178 185)(25 156 179 186)(26 157 180 187)(27 158 181 188)(28 159 182 189)(29 203 115 60)(30 204 116 61)(31 205 117 62)(32 206 118 63)(33 207 119 64)(34 208 120 65)(35 209 121 66)(36 210 122 67)(37 197 123 68)(38 198 124 69)(39 199 125 70)(40 200 126 57)(41 201 113 58)(42 202 114 59)(85 103 149 128)(86 104 150 129)(87 105 151 130)(88 106 152 131)(89 107 153 132)(90 108 154 133)(91 109 141 134)(92 110 142 135)(93 111 143 136)(94 112 144 137)(95 99 145 138)(96 100 146 139)(97 101 147 140)(98 102 148 127)
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), (1,121,49,35)(2,122,50,36)(3,123,51,37)(4,124,52,38)(5,125,53,39)(6,126,54,40)(7,113,55,41)(8,114,56,42)(9,115,43,29)(10,116,44,30)(11,117,45,31)(12,118,46,32)(13,119,47,33)(14,120,48,34)(15,133,169,108)(16,134,170,109)(17,135,171,110)(18,136,172,111)(19,137,173,112)(20,138,174,99)(21,139,175,100)(22,140,176,101)(23,127,177,102)(24,128,178,103)(25,129,179,104)(26,130,180,105)(27,131,181,106)(28,132,182,107)(57,77,200,221)(58,78,201,222)(59,79,202,223)(60,80,203,224)(61,81,204,211)(62,82,205,212)(63,83,206,213)(64,84,207,214)(65,71,208,215)(66,72,209,216)(67,73,210,217)(68,74,197,218)(69,75,198,219)(70,76,199,220)(85,192,149,162)(86,193,150,163)(87,194,151,164)(88,195,152,165)(89,196,153,166)(90,183,154,167)(91,184,141,168)(92,185,142,155)(93,186,143,156)(94,187,144,157)(95,188,145,158)(96,189,146,159)(97,190,147,160)(98,191,148,161), (29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,113)(42,114)(57,200)(58,201)(59,202)(60,203)(61,204)(62,205)(63,206)(64,207)(65,208)(66,209)(67,210)(68,197)(69,198)(70,199)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,141)(92,142)(93,143)(94,144)(95,145)(96,146)(97,147)(98,148)(99,138)(100,139)(101,140)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137), (1,27,56,174)(2,26,43,173)(3,25,44,172)(4,24,45,171)(5,23,46,170)(6,22,47,169)(7,21,48,182)(8,20,49,181)(9,19,50,180)(10,18,51,179)(11,17,52,178)(12,16,53,177)(13,15,54,176)(14,28,55,175)(29,105,122,137)(30,104,123,136)(31,103,124,135)(32,102,125,134)(33,101,126,133)(34,100,113,132)(35,99,114,131)(36,112,115,130)(37,111,116,129)(38,110,117,128)(39,109,118,127)(40,108,119,140)(41,107,120,139)(42,106,121,138)(57,154,207,97)(58,153,208,96)(59,152,209,95)(60,151,210,94)(61,150,197,93)(62,149,198,92)(63,148,199,91)(64,147,200,90)(65,146,201,89)(66,145,202,88)(67,144,203,87)(68,143,204,86)(69,142,205,85)(70,141,206,98)(71,166,222,189)(72,165,223,188)(73,164,224,187)(74,163,211,186)(75,162,212,185)(76,161,213,184)(77,160,214,183)(78,159,215,196)(79,158,216,195)(80,157,217,194)(81,156,218,193)(82,155,219,192)(83,168,220,191)(84,167,221,190), (1,223,49,79)(2,224,50,80)(3,211,51,81)(4,212,52,82)(5,213,53,83)(6,214,54,84)(7,215,55,71)(8,216,56,72)(9,217,43,73)(10,218,44,74)(11,219,45,75)(12,220,46,76)(13,221,47,77)(14,222,48,78)(15,160,169,190)(16,161,170,191)(17,162,171,192)(18,163,172,193)(19,164,173,194)(20,165,174,195)(21,166,175,196)(22,167,176,183)(23,168,177,184)(24,155,178,185)(25,156,179,186)(26,157,180,187)(27,158,181,188)(28,159,182,189)(29,203,115,60)(30,204,116,61)(31,205,117,62)(32,206,118,63)(33,207,119,64)(34,208,120,65)(35,209,121,66)(36,210,122,67)(37,197,123,68)(38,198,124,69)(39,199,125,70)(40,200,126,57)(41,201,113,58)(42,202,114,59)(85,103,149,128)(86,104,150,129)(87,105,151,130)(88,106,152,131)(89,107,153,132)(90,108,154,133)(91,109,141,134)(92,110,142,135)(93,111,143,136)(94,112,144,137)(95,99,145,138)(96,100,146,139)(97,101,147,140)(98,102,148,127)>;
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), (1,121,49,35)(2,122,50,36)(3,123,51,37)(4,124,52,38)(5,125,53,39)(6,126,54,40)(7,113,55,41)(8,114,56,42)(9,115,43,29)(10,116,44,30)(11,117,45,31)(12,118,46,32)(13,119,47,33)(14,120,48,34)(15,133,169,108)(16,134,170,109)(17,135,171,110)(18,136,172,111)(19,137,173,112)(20,138,174,99)(21,139,175,100)(22,140,176,101)(23,127,177,102)(24,128,178,103)(25,129,179,104)(26,130,180,105)(27,131,181,106)(28,132,182,107)(57,77,200,221)(58,78,201,222)(59,79,202,223)(60,80,203,224)(61,81,204,211)(62,82,205,212)(63,83,206,213)(64,84,207,214)(65,71,208,215)(66,72,209,216)(67,73,210,217)(68,74,197,218)(69,75,198,219)(70,76,199,220)(85,192,149,162)(86,193,150,163)(87,194,151,164)(88,195,152,165)(89,196,153,166)(90,183,154,167)(91,184,141,168)(92,185,142,155)(93,186,143,156)(94,187,144,157)(95,188,145,158)(96,189,146,159)(97,190,147,160)(98,191,148,161), (29,115)(30,116)(31,117)(32,118)(33,119)(34,120)(35,121)(36,122)(37,123)(38,124)(39,125)(40,126)(41,113)(42,114)(57,200)(58,201)(59,202)(60,203)(61,204)(62,205)(63,206)(64,207)(65,208)(66,209)(67,210)(68,197)(69,198)(70,199)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,141)(92,142)(93,143)(94,144)(95,145)(96,146)(97,147)(98,148)(99,138)(100,139)(101,140)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137), (1,27,56,174)(2,26,43,173)(3,25,44,172)(4,24,45,171)(5,23,46,170)(6,22,47,169)(7,21,48,182)(8,20,49,181)(9,19,50,180)(10,18,51,179)(11,17,52,178)(12,16,53,177)(13,15,54,176)(14,28,55,175)(29,105,122,137)(30,104,123,136)(31,103,124,135)(32,102,125,134)(33,101,126,133)(34,100,113,132)(35,99,114,131)(36,112,115,130)(37,111,116,129)(38,110,117,128)(39,109,118,127)(40,108,119,140)(41,107,120,139)(42,106,121,138)(57,154,207,97)(58,153,208,96)(59,152,209,95)(60,151,210,94)(61,150,197,93)(62,149,198,92)(63,148,199,91)(64,147,200,90)(65,146,201,89)(66,145,202,88)(67,144,203,87)(68,143,204,86)(69,142,205,85)(70,141,206,98)(71,166,222,189)(72,165,223,188)(73,164,224,187)(74,163,211,186)(75,162,212,185)(76,161,213,184)(77,160,214,183)(78,159,215,196)(79,158,216,195)(80,157,217,194)(81,156,218,193)(82,155,219,192)(83,168,220,191)(84,167,221,190), (1,223,49,79)(2,224,50,80)(3,211,51,81)(4,212,52,82)(5,213,53,83)(6,214,54,84)(7,215,55,71)(8,216,56,72)(9,217,43,73)(10,218,44,74)(11,219,45,75)(12,220,46,76)(13,221,47,77)(14,222,48,78)(15,160,169,190)(16,161,170,191)(17,162,171,192)(18,163,172,193)(19,164,173,194)(20,165,174,195)(21,166,175,196)(22,167,176,183)(23,168,177,184)(24,155,178,185)(25,156,179,186)(26,157,180,187)(27,158,181,188)(28,159,182,189)(29,203,115,60)(30,204,116,61)(31,205,117,62)(32,206,118,63)(33,207,119,64)(34,208,120,65)(35,209,121,66)(36,210,122,67)(37,197,123,68)(38,198,124,69)(39,199,125,70)(40,200,126,57)(41,201,113,58)(42,202,114,59)(85,103,149,128)(86,104,150,129)(87,105,151,130)(88,106,152,131)(89,107,153,132)(90,108,154,133)(91,109,141,134)(92,110,142,135)(93,111,143,136)(94,112,144,137)(95,99,145,138)(96,100,146,139)(97,101,147,140)(98,102,148,127) );
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)], [(1,121,49,35),(2,122,50,36),(3,123,51,37),(4,124,52,38),(5,125,53,39),(6,126,54,40),(7,113,55,41),(8,114,56,42),(9,115,43,29),(10,116,44,30),(11,117,45,31),(12,118,46,32),(13,119,47,33),(14,120,48,34),(15,133,169,108),(16,134,170,109),(17,135,171,110),(18,136,172,111),(19,137,173,112),(20,138,174,99),(21,139,175,100),(22,140,176,101),(23,127,177,102),(24,128,178,103),(25,129,179,104),(26,130,180,105),(27,131,181,106),(28,132,182,107),(57,77,200,221),(58,78,201,222),(59,79,202,223),(60,80,203,224),(61,81,204,211),(62,82,205,212),(63,83,206,213),(64,84,207,214),(65,71,208,215),(66,72,209,216),(67,73,210,217),(68,74,197,218),(69,75,198,219),(70,76,199,220),(85,192,149,162),(86,193,150,163),(87,194,151,164),(88,195,152,165),(89,196,153,166),(90,183,154,167),(91,184,141,168),(92,185,142,155),(93,186,143,156),(94,187,144,157),(95,188,145,158),(96,189,146,159),(97,190,147,160),(98,191,148,161)], [(29,115),(30,116),(31,117),(32,118),(33,119),(34,120),(35,121),(36,122),(37,123),(38,124),(39,125),(40,126),(41,113),(42,114),(57,200),(58,201),(59,202),(60,203),(61,204),(62,205),(63,206),(64,207),(65,208),(66,209),(67,210),(68,197),(69,198),(70,199),(85,149),(86,150),(87,151),(88,152),(89,153),(90,154),(91,141),(92,142),(93,143),(94,144),(95,145),(96,146),(97,147),(98,148),(99,138),(100,139),(101,140),(102,127),(103,128),(104,129),(105,130),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,137)], [(1,27,56,174),(2,26,43,173),(3,25,44,172),(4,24,45,171),(5,23,46,170),(6,22,47,169),(7,21,48,182),(8,20,49,181),(9,19,50,180),(10,18,51,179),(11,17,52,178),(12,16,53,177),(13,15,54,176),(14,28,55,175),(29,105,122,137),(30,104,123,136),(31,103,124,135),(32,102,125,134),(33,101,126,133),(34,100,113,132),(35,99,114,131),(36,112,115,130),(37,111,116,129),(38,110,117,128),(39,109,118,127),(40,108,119,140),(41,107,120,139),(42,106,121,138),(57,154,207,97),(58,153,208,96),(59,152,209,95),(60,151,210,94),(61,150,197,93),(62,149,198,92),(63,148,199,91),(64,147,200,90),(65,146,201,89),(66,145,202,88),(67,144,203,87),(68,143,204,86),(69,142,205,85),(70,141,206,98),(71,166,222,189),(72,165,223,188),(73,164,224,187),(74,163,211,186),(75,162,212,185),(76,161,213,184),(77,160,214,183),(78,159,215,196),(79,158,216,195),(80,157,217,194),(81,156,218,193),(82,155,219,192),(83,168,220,191),(84,167,221,190)], [(1,223,49,79),(2,224,50,80),(3,211,51,81),(4,212,52,82),(5,213,53,83),(6,214,54,84),(7,215,55,71),(8,216,56,72),(9,217,43,73),(10,218,44,74),(11,219,45,75),(12,220,46,76),(13,221,47,77),(14,222,48,78),(15,160,169,190),(16,161,170,191),(17,162,171,192),(18,163,172,193),(19,164,173,194),(20,165,174,195),(21,166,175,196),(22,167,176,183),(23,168,177,184),(24,155,178,185),(25,156,179,186),(26,157,180,187),(27,158,181,188),(28,159,182,189),(29,203,115,60),(30,204,116,61),(31,205,117,62),(32,206,118,63),(33,207,119,64),(34,208,120,65),(35,209,121,66),(36,210,122,67),(37,197,123,68),(38,198,124,69),(39,199,125,70),(40,200,126,57),(41,201,113,58),(42,202,114,59),(85,103,149,128),(86,104,150,129),(87,105,151,130),(88,106,152,131),(89,107,153,132),(90,108,154,133),(91,109,141,134),(92,110,142,135),(93,111,143,136),(94,112,144,137),(95,99,145,138),(96,100,146,139),(97,101,147,140),(98,102,148,127)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 8 | 0 | 0 |
| 0 | 0 | 21 | 8 | 0 | 0 |
| 0 | 0 | 2 | 16 | 1 | 8 |
| 0 | 0 | 25 | 2 | 12 | 10 |
| 24 | 16 | 0 | 0 | 0 | 0 |
| 13 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 13 | 22 | 21 |
| 0 | 0 | 11 | 2 | 26 | 15 |
| 0 | 0 | 9 | 28 | 28 | 0 |
| 0 | 0 | 3 | 17 | 15 | 0 |
| 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 | 9 | 28 | 28 | 0 |
| 0 | 0 | 10 | 15 | 0 | 28 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 8 | 0 | 0 |
| 0 | 0 | 3 | 8 | 0 | 0 |
| 0 | 0 | 4 | 0 | 7 | 26 |
| 0 | 0 | 18 | 1 | 16 | 22 |
| 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 | 19 | 21 | 8 | 18 |
| 0 | 0 | 20 | 12 | 27 | 21 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,3,21,2,25,0,0,8,8,16,2,0,0,0,0,1,12,0,0,0,0,8,10],[24,13,0,0,0,0,16,5,0,0,0,0,0,0,28,11,9,3,0,0,13,2,28,17,0,0,22,26,28,15,0,0,21,15,0,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,9,10,0,0,0,1,28,15,0,0,0,0,28,0,0,0,0,0,0,28],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,21,3,4,18,0,0,8,8,0,1,0,0,0,0,7,16,0,0,0,0,26,22],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,18,19,20,0,0,11,27,21,12,0,0,0,0,8,27,0,0,0,0,18,21] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | 14Q | 14R | 28A | ··· | 28L | 28M | ··· | 28U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 28 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊2D7 | D4⋊8D14 | D4.10D14 |
| kernel | C14.852- (1+4) | C22⋊Dic14 | C23.D14 | Dic7⋊4D4 | D14.D4 | C22.D28 | Dic7.Q8 | C4.Dic14 | D14⋊2Q8 | C4⋊C4⋊D7 | C2×C4⋊Dic7 | C23.23D14 | D4×Dic7 | Dic7⋊D4 | C7×C22.D4 | C22.D4 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 9 | 6 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{85}2_-^{(1+4)} % in TeX
G:=Group("C14.85ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1118);
// by ID
G=gap.SmallGroup(448,1118);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,100,675,409,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=a^7*b^2,e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^7*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀