?
G = C14.712- (1+4) order 448 = 26·7
26th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.712- (1+4), C14.362+ (1+4), C28⋊Q8⋊19C2, C4⋊D4.9D7, C4⋊C4.179D14, (D4×Dic7)⋊19C2, (C2×D4).91D14, C22⋊C4.7D14, (C2×C28).37C23, Dic7⋊3Q8⋊22C2, C28.202(C4○D4), C28.48D4⋊32C2, C4.68(D4⋊2D7), C28.17D4⋊16C2, (C2×C14).147C24, (C22×C4).222D14, C2.38(D4⋊6D14), C23.13(C22×D7), C22⋊Dic14⋊18C2, (D4×C14).121C22, C23.D14⋊17C2, C23.18D14⋊8C2, Dic7⋊C4.17C22, C4⋊Dic7.310C22, (C4×Dic7).94C22, (C2×Dic7).68C23, C22.168(C23×D7), C23.D7.24C22, C23.21D14⋊25C2, (C22×C14).185C23, (C22×C28).239C22, C7⋊4(C22.36C24), C2.29(D4.10D14), (C2×Dic14).153C22, (C22×Dic7).108C22, C14.83(C2×C4○D4), (C7×C4⋊D4).9C2, C2.35(C2×D4⋊2D7), (C2×C4).36(C22×D7), (C7×C4⋊C4).143C22, (C7×C22⋊C4).12C22, SmallGroup(448,1056)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 844 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×12], D4 [×4], Q8 [×4], C23, C23 [×2], C14 [×3], C14 [×3], C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8 [×3], Dic7 [×8], C28 [×2], C28 [×3], C2×C14, C2×C14 [×9], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, Dic14 [×4], C2×Dic7 [×4], C2×Dic7 [×4], C2×Dic7 [×2], C2×C28 [×2], C2×C28 [×2], C2×C28 [×2], C7×D4 [×4], C22×C14, C22×C14 [×2], C22.36C24, C4×Dic7 [×2], C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7 [×3], C23.D7 [×2], C23.D7 [×8], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14, C2×Dic14 [×2], C22×Dic7 [×2], C22×C28, D4×C14, D4×C14 [×2], C22⋊Dic14 [×2], C23.D14 [×2], Dic7⋊3Q8, C28⋊Q8, C28.48D4, C23.21D14, D4×Dic7, C23.18D14 [×2], C28.17D4, C28.17D4 [×2], C7×C4⋊D4, C14.712- (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], C22.36C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, D4⋊6D14, D4.10D14, C14.712- (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=a7b-1, dbd-1=a7b, be=eb, dcd-1=a7c, 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 65 201 123)(2 64 202 122)(3 63 203 121)(4 62 204 120)(5 61 205 119)(6 60 206 118)(7 59 207 117)(8 58 208 116)(9 57 209 115)(10 70 210 114)(11 69 197 113)(12 68 198 126)(13 67 199 125)(14 66 200 124)(15 71 105 140)(16 84 106 139)(17 83 107 138)(18 82 108 137)(19 81 109 136)(20 80 110 135)(21 79 111 134)(22 78 112 133)(23 77 99 132)(24 76 100 131)(25 75 101 130)(26 74 102 129)(27 73 103 128)(28 72 104 127)(29 189 98 145)(30 188 85 144)(31 187 86 143)(32 186 87 142)(33 185 88 141)(34 184 89 154)(35 183 90 153)(36 196 91 152)(37 195 92 151)(38 194 93 150)(39 193 94 149)(40 192 95 148)(41 191 96 147)(42 190 97 146)(43 165 216 182)(44 164 217 181)(45 163 218 180)(46 162 219 179)(47 161 220 178)(48 160 221 177)(49 159 222 176)(50 158 223 175)(51 157 224 174)(52 156 211 173)(53 155 212 172)(54 168 213 171)(55 167 214 170)(56 166 215 169)
(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(57 122)(58 123)(59 124)(60 125)(61 126)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 121)(71 140)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(141 185)(142 186)(143 187)(144 188)(145 189)(146 190)(147 191)(148 192)(149 193)(150 194)(151 195)(152 196)(153 183)(154 184)(155 179)(156 180)(157 181)(158 182)(159 169)(160 170)(161 171)(162 172)(163 173)(164 174)(165 175)(166 176)(167 177)(168 178)
(1 32 208 94)(2 31 209 93)(3 30 210 92)(4 29 197 91)(5 42 198 90)(6 41 199 89)(7 40 200 88)(8 39 201 87)(9 38 202 86)(10 37 203 85)(11 36 204 98)(12 35 205 97)(13 34 206 96)(14 33 207 95)(15 216 112 50)(16 215 99 49)(17 214 100 48)(18 213 101 47)(19 212 102 46)(20 211 103 45)(21 224 104 44)(22 223 105 43)(23 222 106 56)(24 221 107 55)(25 220 108 54)(26 219 109 53)(27 218 110 52)(28 217 111 51)(57 187 122 150)(58 186 123 149)(59 185 124 148)(60 184 125 147)(61 183 126 146)(62 196 113 145)(63 195 114 144)(64 194 115 143)(65 193 116 142)(66 192 117 141)(67 191 118 154)(68 190 119 153)(69 189 120 152)(70 188 121 151)(71 175 133 165)(72 174 134 164)(73 173 135 163)(74 172 136 162)(75 171 137 161)(76 170 138 160)(77 169 139 159)(78 182 140 158)(79 181 127 157)(80 180 128 156)(81 179 129 155)(82 178 130 168)(83 177 131 167)(84 176 132 166)
(1 215 208 49)(2 216 209 50)(3 217 210 51)(4 218 197 52)(5 219 198 53)(6 220 199 54)(7 221 200 55)(8 222 201 56)(9 223 202 43)(10 224 203 44)(11 211 204 45)(12 212 205 46)(13 213 206 47)(14 214 207 48)(15 93 112 31)(16 94 99 32)(17 95 100 33)(18 96 101 34)(19 97 102 35)(20 98 103 36)(21 85 104 37)(22 86 105 38)(23 87 106 39)(24 88 107 40)(25 89 108 41)(26 90 109 42)(27 91 110 29)(28 92 111 30)(57 175 122 165)(58 176 123 166)(59 177 124 167)(60 178 125 168)(61 179 126 155)(62 180 113 156)(63 181 114 157)(64 182 115 158)(65 169 116 159)(66 170 117 160)(67 171 118 161)(68 172 119 162)(69 173 120 163)(70 174 121 164)(71 150 133 187)(72 151 134 188)(73 152 135 189)(74 153 136 190)(75 154 137 191)(76 141 138 192)(77 142 139 193)(78 143 140 194)(79 144 127 195)(80 145 128 196)(81 146 129 183)(82 147 130 184)(83 148 131 185)(84 149 132 186)
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), (1,65,201,123)(2,64,202,122)(3,63,203,121)(4,62,204,120)(5,61,205,119)(6,60,206,118)(7,59,207,117)(8,58,208,116)(9,57,209,115)(10,70,210,114)(11,69,197,113)(12,68,198,126)(13,67,199,125)(14,66,200,124)(15,71,105,140)(16,84,106,139)(17,83,107,138)(18,82,108,137)(19,81,109,136)(20,80,110,135)(21,79,111,134)(22,78,112,133)(23,77,99,132)(24,76,100,131)(25,75,101,130)(26,74,102,129)(27,73,103,128)(28,72,104,127)(29,189,98,145)(30,188,85,144)(31,187,86,143)(32,186,87,142)(33,185,88,141)(34,184,89,154)(35,183,90,153)(36,196,91,152)(37,195,92,151)(38,194,93,150)(39,193,94,149)(40,192,95,148)(41,191,96,147)(42,190,97,146)(43,165,216,182)(44,164,217,181)(45,163,218,180)(46,162,219,179)(47,161,220,178)(48,160,221,177)(49,159,222,176)(50,158,223,175)(51,157,224,174)(52,156,211,173)(53,155,212,172)(54,168,213,171)(55,167,214,170)(56,166,215,169), (15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,122)(58,123)(59,124)(60,125)(61,126)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,121)(71,140)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(141,185)(142,186)(143,187)(144,188)(145,189)(146,190)(147,191)(148,192)(149,193)(150,194)(151,195)(152,196)(153,183)(154,184)(155,179)(156,180)(157,181)(158,182)(159,169)(160,170)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178), (1,32,208,94)(2,31,209,93)(3,30,210,92)(4,29,197,91)(5,42,198,90)(6,41,199,89)(7,40,200,88)(8,39,201,87)(9,38,202,86)(10,37,203,85)(11,36,204,98)(12,35,205,97)(13,34,206,96)(14,33,207,95)(15,216,112,50)(16,215,99,49)(17,214,100,48)(18,213,101,47)(19,212,102,46)(20,211,103,45)(21,224,104,44)(22,223,105,43)(23,222,106,56)(24,221,107,55)(25,220,108,54)(26,219,109,53)(27,218,110,52)(28,217,111,51)(57,187,122,150)(58,186,123,149)(59,185,124,148)(60,184,125,147)(61,183,126,146)(62,196,113,145)(63,195,114,144)(64,194,115,143)(65,193,116,142)(66,192,117,141)(67,191,118,154)(68,190,119,153)(69,189,120,152)(70,188,121,151)(71,175,133,165)(72,174,134,164)(73,173,135,163)(74,172,136,162)(75,171,137,161)(76,170,138,160)(77,169,139,159)(78,182,140,158)(79,181,127,157)(80,180,128,156)(81,179,129,155)(82,178,130,168)(83,177,131,167)(84,176,132,166), (1,215,208,49)(2,216,209,50)(3,217,210,51)(4,218,197,52)(5,219,198,53)(6,220,199,54)(7,221,200,55)(8,222,201,56)(9,223,202,43)(10,224,203,44)(11,211,204,45)(12,212,205,46)(13,213,206,47)(14,214,207,48)(15,93,112,31)(16,94,99,32)(17,95,100,33)(18,96,101,34)(19,97,102,35)(20,98,103,36)(21,85,104,37)(22,86,105,38)(23,87,106,39)(24,88,107,40)(25,89,108,41)(26,90,109,42)(27,91,110,29)(28,92,111,30)(57,175,122,165)(58,176,123,166)(59,177,124,167)(60,178,125,168)(61,179,126,155)(62,180,113,156)(63,181,114,157)(64,182,115,158)(65,169,116,159)(66,170,117,160)(67,171,118,161)(68,172,119,162)(69,173,120,163)(70,174,121,164)(71,150,133,187)(72,151,134,188)(73,152,135,189)(74,153,136,190)(75,154,137,191)(76,141,138,192)(77,142,139,193)(78,143,140,194)(79,144,127,195)(80,145,128,196)(81,146,129,183)(82,147,130,184)(83,148,131,185)(84,149,132,186)>;
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), (1,65,201,123)(2,64,202,122)(3,63,203,121)(4,62,204,120)(5,61,205,119)(6,60,206,118)(7,59,207,117)(8,58,208,116)(9,57,209,115)(10,70,210,114)(11,69,197,113)(12,68,198,126)(13,67,199,125)(14,66,200,124)(15,71,105,140)(16,84,106,139)(17,83,107,138)(18,82,108,137)(19,81,109,136)(20,80,110,135)(21,79,111,134)(22,78,112,133)(23,77,99,132)(24,76,100,131)(25,75,101,130)(26,74,102,129)(27,73,103,128)(28,72,104,127)(29,189,98,145)(30,188,85,144)(31,187,86,143)(32,186,87,142)(33,185,88,141)(34,184,89,154)(35,183,90,153)(36,196,91,152)(37,195,92,151)(38,194,93,150)(39,193,94,149)(40,192,95,148)(41,191,96,147)(42,190,97,146)(43,165,216,182)(44,164,217,181)(45,163,218,180)(46,162,219,179)(47,161,220,178)(48,160,221,177)(49,159,222,176)(50,158,223,175)(51,157,224,174)(52,156,211,173)(53,155,212,172)(54,168,213,171)(55,167,214,170)(56,166,215,169), (15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(57,122)(58,123)(59,124)(60,125)(61,126)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,121)(71,140)(72,127)(73,128)(74,129)(75,130)(76,131)(77,132)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(141,185)(142,186)(143,187)(144,188)(145,189)(146,190)(147,191)(148,192)(149,193)(150,194)(151,195)(152,196)(153,183)(154,184)(155,179)(156,180)(157,181)(158,182)(159,169)(160,170)(161,171)(162,172)(163,173)(164,174)(165,175)(166,176)(167,177)(168,178), (1,32,208,94)(2,31,209,93)(3,30,210,92)(4,29,197,91)(5,42,198,90)(6,41,199,89)(7,40,200,88)(8,39,201,87)(9,38,202,86)(10,37,203,85)(11,36,204,98)(12,35,205,97)(13,34,206,96)(14,33,207,95)(15,216,112,50)(16,215,99,49)(17,214,100,48)(18,213,101,47)(19,212,102,46)(20,211,103,45)(21,224,104,44)(22,223,105,43)(23,222,106,56)(24,221,107,55)(25,220,108,54)(26,219,109,53)(27,218,110,52)(28,217,111,51)(57,187,122,150)(58,186,123,149)(59,185,124,148)(60,184,125,147)(61,183,126,146)(62,196,113,145)(63,195,114,144)(64,194,115,143)(65,193,116,142)(66,192,117,141)(67,191,118,154)(68,190,119,153)(69,189,120,152)(70,188,121,151)(71,175,133,165)(72,174,134,164)(73,173,135,163)(74,172,136,162)(75,171,137,161)(76,170,138,160)(77,169,139,159)(78,182,140,158)(79,181,127,157)(80,180,128,156)(81,179,129,155)(82,178,130,168)(83,177,131,167)(84,176,132,166), (1,215,208,49)(2,216,209,50)(3,217,210,51)(4,218,197,52)(5,219,198,53)(6,220,199,54)(7,221,200,55)(8,222,201,56)(9,223,202,43)(10,224,203,44)(11,211,204,45)(12,212,205,46)(13,213,206,47)(14,214,207,48)(15,93,112,31)(16,94,99,32)(17,95,100,33)(18,96,101,34)(19,97,102,35)(20,98,103,36)(21,85,104,37)(22,86,105,38)(23,87,106,39)(24,88,107,40)(25,89,108,41)(26,90,109,42)(27,91,110,29)(28,92,111,30)(57,175,122,165)(58,176,123,166)(59,177,124,167)(60,178,125,168)(61,179,126,155)(62,180,113,156)(63,181,114,157)(64,182,115,158)(65,169,116,159)(66,170,117,160)(67,171,118,161)(68,172,119,162)(69,173,120,163)(70,174,121,164)(71,150,133,187)(72,151,134,188)(73,152,135,189)(74,153,136,190)(75,154,137,191)(76,141,138,192)(77,142,139,193)(78,143,140,194)(79,144,127,195)(80,145,128,196)(81,146,129,183)(82,147,130,184)(83,148,131,185)(84,149,132,186) );
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)], [(1,65,201,123),(2,64,202,122),(3,63,203,121),(4,62,204,120),(5,61,205,119),(6,60,206,118),(7,59,207,117),(8,58,208,116),(9,57,209,115),(10,70,210,114),(11,69,197,113),(12,68,198,126),(13,67,199,125),(14,66,200,124),(15,71,105,140),(16,84,106,139),(17,83,107,138),(18,82,108,137),(19,81,109,136),(20,80,110,135),(21,79,111,134),(22,78,112,133),(23,77,99,132),(24,76,100,131),(25,75,101,130),(26,74,102,129),(27,73,103,128),(28,72,104,127),(29,189,98,145),(30,188,85,144),(31,187,86,143),(32,186,87,142),(33,185,88,141),(34,184,89,154),(35,183,90,153),(36,196,91,152),(37,195,92,151),(38,194,93,150),(39,193,94,149),(40,192,95,148),(41,191,96,147),(42,190,97,146),(43,165,216,182),(44,164,217,181),(45,163,218,180),(46,162,219,179),(47,161,220,178),(48,160,221,177),(49,159,222,176),(50,158,223,175),(51,157,224,174),(52,156,211,173),(53,155,212,172),(54,168,213,171),(55,167,214,170),(56,166,215,169)], [(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(57,122),(58,123),(59,124),(60,125),(61,126),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119),(69,120),(70,121),(71,140),(72,127),(73,128),(74,129),(75,130),(76,131),(77,132),(78,133),(79,134),(80,135),(81,136),(82,137),(83,138),(84,139),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(141,185),(142,186),(143,187),(144,188),(145,189),(146,190),(147,191),(148,192),(149,193),(150,194),(151,195),(152,196),(153,183),(154,184),(155,179),(156,180),(157,181),(158,182),(159,169),(160,170),(161,171),(162,172),(163,173),(164,174),(165,175),(166,176),(167,177),(168,178)], [(1,32,208,94),(2,31,209,93),(3,30,210,92),(4,29,197,91),(5,42,198,90),(6,41,199,89),(7,40,200,88),(8,39,201,87),(9,38,202,86),(10,37,203,85),(11,36,204,98),(12,35,205,97),(13,34,206,96),(14,33,207,95),(15,216,112,50),(16,215,99,49),(17,214,100,48),(18,213,101,47),(19,212,102,46),(20,211,103,45),(21,224,104,44),(22,223,105,43),(23,222,106,56),(24,221,107,55),(25,220,108,54),(26,219,109,53),(27,218,110,52),(28,217,111,51),(57,187,122,150),(58,186,123,149),(59,185,124,148),(60,184,125,147),(61,183,126,146),(62,196,113,145),(63,195,114,144),(64,194,115,143),(65,193,116,142),(66,192,117,141),(67,191,118,154),(68,190,119,153),(69,189,120,152),(70,188,121,151),(71,175,133,165),(72,174,134,164),(73,173,135,163),(74,172,136,162),(75,171,137,161),(76,170,138,160),(77,169,139,159),(78,182,140,158),(79,181,127,157),(80,180,128,156),(81,179,129,155),(82,178,130,168),(83,177,131,167),(84,176,132,166)], [(1,215,208,49),(2,216,209,50),(3,217,210,51),(4,218,197,52),(5,219,198,53),(6,220,199,54),(7,221,200,55),(8,222,201,56),(9,223,202,43),(10,224,203,44),(11,211,204,45),(12,212,205,46),(13,213,206,47),(14,214,207,48),(15,93,112,31),(16,94,99,32),(17,95,100,33),(18,96,101,34),(19,97,102,35),(20,98,103,36),(21,85,104,37),(22,86,105,38),(23,87,106,39),(24,88,107,40),(25,89,108,41),(26,90,109,42),(27,91,110,29),(28,92,111,30),(57,175,122,165),(58,176,123,166),(59,177,124,167),(60,178,125,168),(61,179,126,155),(62,180,113,156),(63,181,114,157),(64,182,115,158),(65,169,116,159),(66,170,117,160),(67,171,118,161),(68,172,119,162),(69,173,120,163),(70,174,121,164),(71,150,133,187),(72,151,134,188),(73,152,135,189),(74,153,136,190),(75,154,137,191),(76,141,138,192),(77,142,139,193),(78,143,140,194),(79,144,127,195),(80,145,128,196),(81,146,129,183),(82,147,130,184),(83,148,131,185),(84,149,132,186)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 0 | 0 | 22 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 24 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 |
| 0 | 0 | 0 | 0 | 20 | 14 |
| 0 | 0 | 14 | 20 | 0 | 0 |
| 0 | 0 | 9 | 15 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 27 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 17 | 12 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 |
| 0 | 0 | 0 | 0 | 14 | 20 |
| 0 | 0 | 20 | 14 | 0 | 0 |
| 0 | 0 | 15 | 9 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 9 | 0 | 0 |
| 0 | 0 | 20 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 |
| 0 | 0 | 0 | 0 | 20 | 14 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,22,0,0,0,0,0,0,22],[12,24,0,0,0,0,0,17,0,0,0,0,0,0,0,0,14,9,0,0,0,0,20,15,0,0,15,20,0,0,0,0,9,14,0,0],[28,27,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[17,5,0,0,0,0,12,12,0,0,0,0,0,0,0,0,20,15,0,0,0,0,14,9,0,0,9,14,0,0,0,0,15,20,0,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,15,20,0,0,0,0,9,14,0,0,0,0,0,0,15,20,0,0,0,0,9,14] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 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 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 14 | 14 | 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 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊2D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.712- (1+4) | C22⋊Dic14 | C23.D14 | Dic7⋊3Q8 | C28⋊Q8 | C28.48D4 | C23.21D14 | D4×Dic7 | C23.18D14 | C28.17D4 | C7×C4⋊D4 | C4⋊D4 | C28 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 3 | 4 | 6 | 3 | 3 | 9 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{71}2_-^{(1+4)} % in TeX
G:=Group("C14.71ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1056);
// by ID
G=gap.SmallGroup(448,1056);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,219,675,570,297,18822]);
// 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=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=a^7*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations
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