?
G = C14.422- (1+4) order 448 = 26·7
42nd non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.422- (1+4), (Q8×C14)⋊12C4, (C2×Q8)⋊9Dic7, (Q8×Dic7)⋊25C2, Q8.8(C2×Dic7), (C22×Q8).9D7, C14.47(C23×C4), C28.96(C22×C4), (C2×Q8).207D14, C2.9(C23×Dic7), (C2×C28).551C23, (C2×C14).304C24, (C22×C4).276D14, C4.19(C22×Dic7), C22.47(C23×D7), C4⋊Dic7.389C22, (Q8×C14).233C22, C23.237(C22×D7), (C22×C14).422C23, (C22×C28).284C22, C2.4(Q8.10D14), C7⋊3(C23.32C23), (C4×Dic7).169C22, (C2×Dic7).288C23, C23.D7.145C22, C22.10(C22×Dic7), C23.21D14.25C2, (Q8×C2×C14).9C2, (C7×Q8).26(C2×C4), (C2×C28).134(C2×C4), (C2×C4).30(C2×Dic7), (C2×C4).632(C22×D7), (C2×C14).210(C22×C4), SmallGroup(448,1265)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 692 in 266 conjugacy classes, 191 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×12], C4 [×8], C22, C22 [×2], C22 [×2], C7, C2×C4 [×18], C2×C4 [×8], Q8 [×16], C23, C14, C14 [×2], C14 [×2], C42 [×12], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×12], Dic7 [×8], C28 [×12], C2×C14, C2×C14 [×2], C2×C14 [×2], C42⋊C2 [×6], C4×Q8 [×8], C22×Q8, C2×Dic7 [×8], C2×C28 [×18], C7×Q8 [×16], C22×C14, C23.32C23, C4×Dic7 [×12], C4⋊Dic7 [×12], C23.D7 [×4], C22×C28 [×3], Q8×C14 [×12], C23.21D14 [×6], Q8×Dic7 [×8], Q8×C2×C14, C14.422- (1+4)
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C24, Dic7 [×8], D14 [×7], C23×C4, 2- (1+4) [×2], C2×Dic7 [×28], C22×D7 [×7], C23.32C23, C22×Dic7 [×14], C23×D7, Q8.10D14 [×2], C23×Dic7, C14.422- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=a7b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, 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)
(1 171 193 34)(2 172 194 35)(3 173 195 36)(4 174 196 37)(5 175 183 38)(6 176 184 39)(7 177 185 40)(8 178 186 41)(9 179 187 42)(10 180 188 29)(11 181 189 30)(12 182 190 31)(13 169 191 32)(14 170 192 33)(15 67 123 54)(16 68 124 55)(17 69 125 56)(18 70 126 43)(19 57 113 44)(20 58 114 45)(21 59 115 46)(22 60 116 47)(23 61 117 48)(24 62 118 49)(25 63 119 50)(26 64 120 51)(27 65 121 52)(28 66 122 53)(71 209 109 162)(72 210 110 163)(73 197 111 164)(74 198 112 165)(75 199 99 166)(76 200 100 167)(77 201 101 168)(78 202 102 155)(79 203 103 156)(80 204 104 157)(81 205 105 158)(82 206 106 159)(83 207 107 160)(84 208 108 161)(85 128 146 212)(86 129 147 213)(87 130 148 214)(88 131 149 215)(89 132 150 216)(90 133 151 217)(91 134 152 218)(92 135 153 219)(93 136 154 220)(94 137 141 221)(95 138 142 222)(96 139 143 223)(97 140 144 224)(98 127 145 211)
(1 52 8 45)(2 51 9 44)(3 50 10 43)(4 49 11 56)(5 48 12 55)(6 47 13 54)(7 46 14 53)(15 39 22 32)(16 38 23 31)(17 37 24 30)(18 36 25 29)(19 35 26 42)(20 34 27 41)(21 33 28 40)(57 194 64 187)(58 193 65 186)(59 192 66 185)(60 191 67 184)(61 190 68 183)(62 189 69 196)(63 188 70 195)(71 137 78 130)(72 136 79 129)(73 135 80 128)(74 134 81 127)(75 133 82 140)(76 132 83 139)(77 131 84 138)(85 197 92 204)(86 210 93 203)(87 209 94 202)(88 208 95 201)(89 207 96 200)(90 206 97 199)(91 205 98 198)(99 217 106 224)(100 216 107 223)(101 215 108 222)(102 214 109 221)(103 213 110 220)(104 212 111 219)(105 211 112 218)(113 172 120 179)(114 171 121 178)(115 170 122 177)(116 169 123 176)(117 182 124 175)(118 181 125 174)(119 180 126 173)(141 155 148 162)(142 168 149 161)(143 167 150 160)(144 166 151 159)(145 165 152 158)(146 164 153 157)(147 163 154 156)
(1 90 186 144)(2 89 187 143)(3 88 188 142)(4 87 189 141)(5 86 190 154)(6 85 191 153)(7 98 192 152)(8 97 193 151)(9 96 194 150)(10 95 195 149)(11 94 196 148)(12 93 183 147)(13 92 184 146)(14 91 185 145)(15 104 116 73)(16 103 117 72)(17 102 118 71)(18 101 119 84)(19 100 120 83)(20 99 121 82)(21 112 122 81)(22 111 123 80)(23 110 124 79)(24 109 125 78)(25 108 126 77)(26 107 113 76)(27 106 114 75)(28 105 115 74)(29 222 173 131)(30 221 174 130)(31 220 175 129)(32 219 176 128)(33 218 177 127)(34 217 178 140)(35 216 179 139)(36 215 180 138)(37 214 181 137)(38 213 182 136)(39 212 169 135)(40 211 170 134)(41 224 171 133)(42 223 172 132)(43 201 63 161)(44 200 64 160)(45 199 65 159)(46 198 66 158)(47 197 67 157)(48 210 68 156)(49 209 69 155)(50 208 70 168)(51 207 57 167)(52 206 58 166)(53 205 59 165)(54 204 60 164)(55 203 61 163)(56 202 62 162)
(1 171 193 34)(2 172 194 35)(3 173 195 36)(4 174 196 37)(5 175 183 38)(6 176 184 39)(7 177 185 40)(8 178 186 41)(9 179 187 42)(10 180 188 29)(11 181 189 30)(12 182 190 31)(13 169 191 32)(14 170 192 33)(15 54 123 67)(16 55 124 68)(17 56 125 69)(18 43 126 70)(19 44 113 57)(20 45 114 58)(21 46 115 59)(22 47 116 60)(23 48 117 61)(24 49 118 62)(25 50 119 63)(26 51 120 64)(27 52 121 65)(28 53 122 66)(71 209 109 162)(72 210 110 163)(73 197 111 164)(74 198 112 165)(75 199 99 166)(76 200 100 167)(77 201 101 168)(78 202 102 155)(79 203 103 156)(80 204 104 157)(81 205 105 158)(82 206 106 159)(83 207 107 160)(84 208 108 161)(85 212 146 128)(86 213 147 129)(87 214 148 130)(88 215 149 131)(89 216 150 132)(90 217 151 133)(91 218 152 134)(92 219 153 135)(93 220 154 136)(94 221 141 137)(95 222 142 138)(96 223 143 139)(97 224 144 140)(98 211 145 127)
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), (1,171,193,34)(2,172,194,35)(3,173,195,36)(4,174,196,37)(5,175,183,38)(6,176,184,39)(7,177,185,40)(8,178,186,41)(9,179,187,42)(10,180,188,29)(11,181,189,30)(12,182,190,31)(13,169,191,32)(14,170,192,33)(15,67,123,54)(16,68,124,55)(17,69,125,56)(18,70,126,43)(19,57,113,44)(20,58,114,45)(21,59,115,46)(22,60,116,47)(23,61,117,48)(24,62,118,49)(25,63,119,50)(26,64,120,51)(27,65,121,52)(28,66,122,53)(71,209,109,162)(72,210,110,163)(73,197,111,164)(74,198,112,165)(75,199,99,166)(76,200,100,167)(77,201,101,168)(78,202,102,155)(79,203,103,156)(80,204,104,157)(81,205,105,158)(82,206,106,159)(83,207,107,160)(84,208,108,161)(85,128,146,212)(86,129,147,213)(87,130,148,214)(88,131,149,215)(89,132,150,216)(90,133,151,217)(91,134,152,218)(92,135,153,219)(93,136,154,220)(94,137,141,221)(95,138,142,222)(96,139,143,223)(97,140,144,224)(98,127,145,211), (1,52,8,45)(2,51,9,44)(3,50,10,43)(4,49,11,56)(5,48,12,55)(6,47,13,54)(7,46,14,53)(15,39,22,32)(16,38,23,31)(17,37,24,30)(18,36,25,29)(19,35,26,42)(20,34,27,41)(21,33,28,40)(57,194,64,187)(58,193,65,186)(59,192,66,185)(60,191,67,184)(61,190,68,183)(62,189,69,196)(63,188,70,195)(71,137,78,130)(72,136,79,129)(73,135,80,128)(74,134,81,127)(75,133,82,140)(76,132,83,139)(77,131,84,138)(85,197,92,204)(86,210,93,203)(87,209,94,202)(88,208,95,201)(89,207,96,200)(90,206,97,199)(91,205,98,198)(99,217,106,224)(100,216,107,223)(101,215,108,222)(102,214,109,221)(103,213,110,220)(104,212,111,219)(105,211,112,218)(113,172,120,179)(114,171,121,178)(115,170,122,177)(116,169,123,176)(117,182,124,175)(118,181,125,174)(119,180,126,173)(141,155,148,162)(142,168,149,161)(143,167,150,160)(144,166,151,159)(145,165,152,158)(146,164,153,157)(147,163,154,156), (1,90,186,144)(2,89,187,143)(3,88,188,142)(4,87,189,141)(5,86,190,154)(6,85,191,153)(7,98,192,152)(8,97,193,151)(9,96,194,150)(10,95,195,149)(11,94,196,148)(12,93,183,147)(13,92,184,146)(14,91,185,145)(15,104,116,73)(16,103,117,72)(17,102,118,71)(18,101,119,84)(19,100,120,83)(20,99,121,82)(21,112,122,81)(22,111,123,80)(23,110,124,79)(24,109,125,78)(25,108,126,77)(26,107,113,76)(27,106,114,75)(28,105,115,74)(29,222,173,131)(30,221,174,130)(31,220,175,129)(32,219,176,128)(33,218,177,127)(34,217,178,140)(35,216,179,139)(36,215,180,138)(37,214,181,137)(38,213,182,136)(39,212,169,135)(40,211,170,134)(41,224,171,133)(42,223,172,132)(43,201,63,161)(44,200,64,160)(45,199,65,159)(46,198,66,158)(47,197,67,157)(48,210,68,156)(49,209,69,155)(50,208,70,168)(51,207,57,167)(52,206,58,166)(53,205,59,165)(54,204,60,164)(55,203,61,163)(56,202,62,162), (1,171,193,34)(2,172,194,35)(3,173,195,36)(4,174,196,37)(5,175,183,38)(6,176,184,39)(7,177,185,40)(8,178,186,41)(9,179,187,42)(10,180,188,29)(11,181,189,30)(12,182,190,31)(13,169,191,32)(14,170,192,33)(15,54,123,67)(16,55,124,68)(17,56,125,69)(18,43,126,70)(19,44,113,57)(20,45,114,58)(21,46,115,59)(22,47,116,60)(23,48,117,61)(24,49,118,62)(25,50,119,63)(26,51,120,64)(27,52,121,65)(28,53,122,66)(71,209,109,162)(72,210,110,163)(73,197,111,164)(74,198,112,165)(75,199,99,166)(76,200,100,167)(77,201,101,168)(78,202,102,155)(79,203,103,156)(80,204,104,157)(81,205,105,158)(82,206,106,159)(83,207,107,160)(84,208,108,161)(85,212,146,128)(86,213,147,129)(87,214,148,130)(88,215,149,131)(89,216,150,132)(90,217,151,133)(91,218,152,134)(92,219,153,135)(93,220,154,136)(94,221,141,137)(95,222,142,138)(96,223,143,139)(97,224,144,140)(98,211,145,127)>;
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), (1,171,193,34)(2,172,194,35)(3,173,195,36)(4,174,196,37)(5,175,183,38)(6,176,184,39)(7,177,185,40)(8,178,186,41)(9,179,187,42)(10,180,188,29)(11,181,189,30)(12,182,190,31)(13,169,191,32)(14,170,192,33)(15,67,123,54)(16,68,124,55)(17,69,125,56)(18,70,126,43)(19,57,113,44)(20,58,114,45)(21,59,115,46)(22,60,116,47)(23,61,117,48)(24,62,118,49)(25,63,119,50)(26,64,120,51)(27,65,121,52)(28,66,122,53)(71,209,109,162)(72,210,110,163)(73,197,111,164)(74,198,112,165)(75,199,99,166)(76,200,100,167)(77,201,101,168)(78,202,102,155)(79,203,103,156)(80,204,104,157)(81,205,105,158)(82,206,106,159)(83,207,107,160)(84,208,108,161)(85,128,146,212)(86,129,147,213)(87,130,148,214)(88,131,149,215)(89,132,150,216)(90,133,151,217)(91,134,152,218)(92,135,153,219)(93,136,154,220)(94,137,141,221)(95,138,142,222)(96,139,143,223)(97,140,144,224)(98,127,145,211), (1,52,8,45)(2,51,9,44)(3,50,10,43)(4,49,11,56)(5,48,12,55)(6,47,13,54)(7,46,14,53)(15,39,22,32)(16,38,23,31)(17,37,24,30)(18,36,25,29)(19,35,26,42)(20,34,27,41)(21,33,28,40)(57,194,64,187)(58,193,65,186)(59,192,66,185)(60,191,67,184)(61,190,68,183)(62,189,69,196)(63,188,70,195)(71,137,78,130)(72,136,79,129)(73,135,80,128)(74,134,81,127)(75,133,82,140)(76,132,83,139)(77,131,84,138)(85,197,92,204)(86,210,93,203)(87,209,94,202)(88,208,95,201)(89,207,96,200)(90,206,97,199)(91,205,98,198)(99,217,106,224)(100,216,107,223)(101,215,108,222)(102,214,109,221)(103,213,110,220)(104,212,111,219)(105,211,112,218)(113,172,120,179)(114,171,121,178)(115,170,122,177)(116,169,123,176)(117,182,124,175)(118,181,125,174)(119,180,126,173)(141,155,148,162)(142,168,149,161)(143,167,150,160)(144,166,151,159)(145,165,152,158)(146,164,153,157)(147,163,154,156), (1,90,186,144)(2,89,187,143)(3,88,188,142)(4,87,189,141)(5,86,190,154)(6,85,191,153)(7,98,192,152)(8,97,193,151)(9,96,194,150)(10,95,195,149)(11,94,196,148)(12,93,183,147)(13,92,184,146)(14,91,185,145)(15,104,116,73)(16,103,117,72)(17,102,118,71)(18,101,119,84)(19,100,120,83)(20,99,121,82)(21,112,122,81)(22,111,123,80)(23,110,124,79)(24,109,125,78)(25,108,126,77)(26,107,113,76)(27,106,114,75)(28,105,115,74)(29,222,173,131)(30,221,174,130)(31,220,175,129)(32,219,176,128)(33,218,177,127)(34,217,178,140)(35,216,179,139)(36,215,180,138)(37,214,181,137)(38,213,182,136)(39,212,169,135)(40,211,170,134)(41,224,171,133)(42,223,172,132)(43,201,63,161)(44,200,64,160)(45,199,65,159)(46,198,66,158)(47,197,67,157)(48,210,68,156)(49,209,69,155)(50,208,70,168)(51,207,57,167)(52,206,58,166)(53,205,59,165)(54,204,60,164)(55,203,61,163)(56,202,62,162), (1,171,193,34)(2,172,194,35)(3,173,195,36)(4,174,196,37)(5,175,183,38)(6,176,184,39)(7,177,185,40)(8,178,186,41)(9,179,187,42)(10,180,188,29)(11,181,189,30)(12,182,190,31)(13,169,191,32)(14,170,192,33)(15,54,123,67)(16,55,124,68)(17,56,125,69)(18,43,126,70)(19,44,113,57)(20,45,114,58)(21,46,115,59)(22,47,116,60)(23,48,117,61)(24,49,118,62)(25,50,119,63)(26,51,120,64)(27,52,121,65)(28,53,122,66)(71,209,109,162)(72,210,110,163)(73,197,111,164)(74,198,112,165)(75,199,99,166)(76,200,100,167)(77,201,101,168)(78,202,102,155)(79,203,103,156)(80,204,104,157)(81,205,105,158)(82,206,106,159)(83,207,107,160)(84,208,108,161)(85,212,146,128)(86,213,147,129)(87,214,148,130)(88,215,149,131)(89,216,150,132)(90,217,151,133)(91,218,152,134)(92,219,153,135)(93,220,154,136)(94,221,141,137)(95,222,142,138)(96,223,143,139)(97,224,144,140)(98,211,145,127) );
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)], [(1,171,193,34),(2,172,194,35),(3,173,195,36),(4,174,196,37),(5,175,183,38),(6,176,184,39),(7,177,185,40),(8,178,186,41),(9,179,187,42),(10,180,188,29),(11,181,189,30),(12,182,190,31),(13,169,191,32),(14,170,192,33),(15,67,123,54),(16,68,124,55),(17,69,125,56),(18,70,126,43),(19,57,113,44),(20,58,114,45),(21,59,115,46),(22,60,116,47),(23,61,117,48),(24,62,118,49),(25,63,119,50),(26,64,120,51),(27,65,121,52),(28,66,122,53),(71,209,109,162),(72,210,110,163),(73,197,111,164),(74,198,112,165),(75,199,99,166),(76,200,100,167),(77,201,101,168),(78,202,102,155),(79,203,103,156),(80,204,104,157),(81,205,105,158),(82,206,106,159),(83,207,107,160),(84,208,108,161),(85,128,146,212),(86,129,147,213),(87,130,148,214),(88,131,149,215),(89,132,150,216),(90,133,151,217),(91,134,152,218),(92,135,153,219),(93,136,154,220),(94,137,141,221),(95,138,142,222),(96,139,143,223),(97,140,144,224),(98,127,145,211)], [(1,52,8,45),(2,51,9,44),(3,50,10,43),(4,49,11,56),(5,48,12,55),(6,47,13,54),(7,46,14,53),(15,39,22,32),(16,38,23,31),(17,37,24,30),(18,36,25,29),(19,35,26,42),(20,34,27,41),(21,33,28,40),(57,194,64,187),(58,193,65,186),(59,192,66,185),(60,191,67,184),(61,190,68,183),(62,189,69,196),(63,188,70,195),(71,137,78,130),(72,136,79,129),(73,135,80,128),(74,134,81,127),(75,133,82,140),(76,132,83,139),(77,131,84,138),(85,197,92,204),(86,210,93,203),(87,209,94,202),(88,208,95,201),(89,207,96,200),(90,206,97,199),(91,205,98,198),(99,217,106,224),(100,216,107,223),(101,215,108,222),(102,214,109,221),(103,213,110,220),(104,212,111,219),(105,211,112,218),(113,172,120,179),(114,171,121,178),(115,170,122,177),(116,169,123,176),(117,182,124,175),(118,181,125,174),(119,180,126,173),(141,155,148,162),(142,168,149,161),(143,167,150,160),(144,166,151,159),(145,165,152,158),(146,164,153,157),(147,163,154,156)], [(1,90,186,144),(2,89,187,143),(3,88,188,142),(4,87,189,141),(5,86,190,154),(6,85,191,153),(7,98,192,152),(8,97,193,151),(9,96,194,150),(10,95,195,149),(11,94,196,148),(12,93,183,147),(13,92,184,146),(14,91,185,145),(15,104,116,73),(16,103,117,72),(17,102,118,71),(18,101,119,84),(19,100,120,83),(20,99,121,82),(21,112,122,81),(22,111,123,80),(23,110,124,79),(24,109,125,78),(25,108,126,77),(26,107,113,76),(27,106,114,75),(28,105,115,74),(29,222,173,131),(30,221,174,130),(31,220,175,129),(32,219,176,128),(33,218,177,127),(34,217,178,140),(35,216,179,139),(36,215,180,138),(37,214,181,137),(38,213,182,136),(39,212,169,135),(40,211,170,134),(41,224,171,133),(42,223,172,132),(43,201,63,161),(44,200,64,160),(45,199,65,159),(46,198,66,158),(47,197,67,157),(48,210,68,156),(49,209,69,155),(50,208,70,168),(51,207,57,167),(52,206,58,166),(53,205,59,165),(54,204,60,164),(55,203,61,163),(56,202,62,162)], [(1,171,193,34),(2,172,194,35),(3,173,195,36),(4,174,196,37),(5,175,183,38),(6,176,184,39),(7,177,185,40),(8,178,186,41),(9,179,187,42),(10,180,188,29),(11,181,189,30),(12,182,190,31),(13,169,191,32),(14,170,192,33),(15,54,123,67),(16,55,124,68),(17,56,125,69),(18,43,126,70),(19,44,113,57),(20,45,114,58),(21,46,115,59),(22,47,116,60),(23,48,117,61),(24,49,118,62),(25,50,119,63),(26,51,120,64),(27,52,121,65),(28,53,122,66),(71,209,109,162),(72,210,110,163),(73,197,111,164),(74,198,112,165),(75,199,99,166),(76,200,100,167),(77,201,101,168),(78,202,102,155),(79,203,103,156),(80,204,104,157),(81,205,105,158),(82,206,106,159),(83,207,107,160),(84,208,108,161),(85,212,146,128),(86,213,147,129),(87,214,148,130),(88,215,149,131),(89,216,150,132),(90,217,151,133),(91,218,152,134),(92,219,153,135),(93,220,154,136),(94,221,141,137),(95,222,142,138),(96,223,143,139),(97,224,144,140),(98,211,145,127)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 0 | 19 | 0 | 0 | 0 | 0 |
| 3 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 20 | 4 | 0 |
| 0 | 0 | 28 | 1 | 0 | 4 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 3 | 26 | 28 | 16 |
| 0 | 0 | 0 | 25 | 18 | 1 |
| 27 | 10 | 0 | 0 | 0 | 0 |
| 14 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 4 | 11 | 27 |
| 0 | 0 | 0 | 9 | 18 | 0 |
| 0 | 0 | 0 | 18 | 20 | 0 |
| 0 | 0 | 27 | 4 | 2 | 13 |
| 27 | 10 | 0 | 0 | 0 | 0 |
| 14 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 16 | 0 |
| 0 | 0 | 11 | 10 | 13 | 24 |
| 0 | 0 | 0 | 24 | 8 | 17 |
| 0 | 0 | 22 | 7 | 24 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 27 | 3 | 1 | 13 |
| 0 | 0 | 16 | 20 | 11 | 28 |
G:=sub<GL(6,GF(29))| [0,3,0,0,0,0,19,26,0,0,0,0,0,0,22,0,0,28,0,0,0,22,20,1,0,0,0,0,4,0,0,0,0,0,0,4],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,3,0,0,0,1,0,26,25,0,0,0,0,28,18,0,0,0,0,16,1],[27,14,0,0,0,0,10,2,0,0,0,0,0,0,16,0,0,27,0,0,4,9,18,4,0,0,11,18,20,2,0,0,27,0,0,13],[27,14,0,0,0,0,10,2,0,0,0,0,0,0,0,11,0,22,0,0,8,10,24,7,0,0,16,13,8,24,0,0,0,24,17,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,27,16,0,0,1,0,3,20,0,0,0,0,1,11,0,0,0,0,13,28] >;
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4L | 4M | ··· | 4AB | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D7 | D14 | Dic7 | D14 | 2- (1+4) | Q8.10D14 |
| kernel | C14.422- (1+4) | C23.21D14 | Q8×Dic7 | Q8×C2×C14 | Q8×C14 | C22×Q8 | C22×C4 | C2×Q8 | C2×Q8 | C14 | C2 |
| # reps | 1 | 6 | 8 | 1 | 16 | 3 | 9 | 24 | 12 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_{14}._{42}2_-^{(1+4)} % in TeX
G:=Group("C14.42ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1265);
// by ID
G=gap.SmallGroup(448,1265);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,387,184,1123,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=a^7*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations
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