?
G = C14.742- (1+4) order 448 = 26·7
29th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.742- (1+4), C14.492+ (1+4), C4⋊D4⋊24D7, C4⋊C4.93D14, C28⋊2D4⋊29C2, (C2×D4).96D14, D14⋊Q8⋊16C2, Dic7.Q8⋊14C2, (C2×C28).47C23, C22⋊C4.12D14, Dic7⋊D4⋊19C2, D14.D4⋊23C2, C28.48D4⋊45C2, C28.17D4⋊20C2, (C2×C14).165C24, D14⋊C4.18C22, (C22×C4).231D14, C4⋊Dic7.46C22, C2.51(D4⋊6D14), C22⋊Dic14⋊21C2, Dic7.D4⋊23C2, (D4×C14).129C22, Dic7⋊C4.79C22, (C2×Dic7).82C23, (C22×D7).72C23, C22.186(C23×D7), C23.115(C22×D7), C23.D7.29C22, C23.18D14⋊23C2, C23.23D14⋊13C2, (C22×C28).313C22, (C22×C14).193C23, C7⋊1(C22.56C24), (C4×Dic7).100C22, (C2×Dic14).34C22, C2.32(D4.10D14), (C22×Dic7).116C22, (C7×C4⋊D4)⋊27C2, (C2×C4×D7).90C22, (C2×C4).43(C22×D7), (C7×C4⋊C4).151C22, (C2×C7⋊D4).36C22, (C7×C22⋊C4).20C22, SmallGroup(448,1074)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1004 in 220 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22, C22 [×12], C7, C2×C4 [×4], C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×3], C23, D7, C14 [×3], C14 [×3], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×9], C22×C4, C22×C4 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×2], Dic7 [×7], C28 [×4], D14 [×3], C2×C14, C2×C14 [×9], C4⋊D4, C4⋊D4 [×3], C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C42.C2, Dic14 [×2], C4×D7, C2×Dic7 [×7], C2×Dic7 [×2], C7⋊D4 [×3], C2×C28 [×4], C2×C28, C7×D4 [×3], C22×D7, C22×C14 [×3], C22.56C24, C4×Dic7, Dic7⋊C4 [×7], C4⋊Dic7 [×2], D14⋊C4 [×3], C23.D7 [×7], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7, C22×Dic7 [×2], C2×C7⋊D4 [×3], C22×C28, D4×C14 [×3], C22⋊Dic14 [×2], D14.D4, Dic7.D4, Dic7.Q8, D14⋊Q8, C28.48D4, C23.23D14, C23.18D14 [×2], C28.17D4, C28⋊2D4, Dic7⋊D4 [×2], C7×C4⋊D4, C14.742- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4) [×2], 2- (1+4), C22×D7 [×7], C22.56C24, C23×D7, D4⋊6D14 [×2], D4.10D14, C14.742- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, bd=db, ebe-1=a7b, 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 85 114 63)(2 98 115 62)(3 97 116 61)(4 96 117 60)(5 95 118 59)(6 94 119 58)(7 93 120 57)(8 92 121 70)(9 91 122 69)(10 90 123 68)(11 89 124 67)(12 88 125 66)(13 87 126 65)(14 86 113 64)(15 137 143 195)(16 136 144 194)(17 135 145 193)(18 134 146 192)(19 133 147 191)(20 132 148 190)(21 131 149 189)(22 130 150 188)(23 129 151 187)(24 128 152 186)(25 127 153 185)(26 140 154 184)(27 139 141 183)(28 138 142 196)(29 178 78 206)(30 177 79 205)(31 176 80 204)(32 175 81 203)(33 174 82 202)(34 173 83 201)(35 172 84 200)(36 171 71 199)(37 170 72 198)(38 169 73 197)(39 182 74 210)(40 181 75 209)(41 180 76 208)(42 179 77 207)(43 102 220 164)(44 101 221 163)(45 100 222 162)(46 99 223 161)(47 112 224 160)(48 111 211 159)(49 110 212 158)(50 109 213 157)(51 108 214 156)(52 107 215 155)(53 106 216 168)(54 105 217 167)(55 104 218 166)(56 103 219 165)
(1 221)(2 222)(3 223)(4 224)(5 211)(6 212)(7 213)(8 214)(9 215)(10 216)(11 217)(12 218)(13 219)(14 220)(15 181)(16 182)(17 169)(18 170)(19 171)(20 172)(21 173)(22 174)(23 175)(24 176)(25 177)(26 178)(27 179)(28 180)(29 140)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 137)(41 138)(42 139)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 157)(58 158)(59 159)(60 160)(61 161)(62 162)(63 163)(64 164)(65 165)(66 166)(67 167)(68 168)(69 155)(70 156)(71 191)(72 192)(73 193)(74 194)(75 195)(76 196)(77 183)(78 184)(79 185)(80 186)(81 187)(82 188)(83 189)(84 190)(85 101)(86 102)(87 103)(88 104)(89 105)(90 106)(91 107)(92 108)(93 109)(94 110)(95 111)(96 112)(97 99)(98 100)(141 207)(142 208)(143 209)(144 210)(145 197)(146 198)(147 199)(148 200)(149 201)(150 202)(151 203)(152 204)(153 205)(154 206)
(1 85 114 63)(2 98 115 62)(3 97 116 61)(4 96 117 60)(5 95 118 59)(6 94 119 58)(7 93 120 57)(8 92 121 70)(9 91 122 69)(10 90 123 68)(11 89 124 67)(12 88 125 66)(13 87 126 65)(14 86 113 64)(15 130 143 188)(16 129 144 187)(17 128 145 186)(18 127 146 185)(19 140 147 184)(20 139 148 183)(21 138 149 196)(22 137 150 195)(23 136 151 194)(24 135 152 193)(25 134 153 192)(26 133 154 191)(27 132 141 190)(28 131 142 189)(29 206 78 178)(30 205 79 177)(31 204 80 176)(32 203 81 175)(33 202 82 174)(34 201 83 173)(35 200 84 172)(36 199 71 171)(37 198 72 170)(38 197 73 169)(39 210 74 182)(40 209 75 181)(41 208 76 180)(42 207 77 179)(43 157 220 109)(44 156 221 108)(45 155 222 107)(46 168 223 106)(47 167 224 105)(48 166 211 104)(49 165 212 103)(50 164 213 102)(51 163 214 101)(52 162 215 100)(53 161 216 99)(54 160 217 112)(55 159 218 111)(56 158 219 110)
(1 210 114 182)(2 197 115 169)(3 198 116 170)(4 199 117 171)(5 200 118 172)(6 201 119 173)(7 202 120 174)(8 203 121 175)(9 204 122 176)(10 205 123 177)(11 206 124 178)(12 207 125 179)(13 208 126 180)(14 209 113 181)(15 220 143 43)(16 221 144 44)(17 222 145 45)(18 223 146 46)(19 224 147 47)(20 211 148 48)(21 212 149 49)(22 213 150 50)(23 214 151 51)(24 215 152 52)(25 216 153 53)(26 217 154 54)(27 218 141 55)(28 219 142 56)(29 60 78 96)(30 61 79 97)(31 62 80 98)(32 63 81 85)(33 64 82 86)(34 65 83 87)(35 66 84 88)(36 67 71 89)(37 68 72 90)(38 69 73 91)(39 70 74 92)(40 57 75 93)(41 58 76 94)(42 59 77 95)(99 127 161 185)(100 128 162 186)(101 129 163 187)(102 130 164 188)(103 131 165 189)(104 132 166 190)(105 133 167 191)(106 134 168 192)(107 135 155 193)(108 136 156 194)(109 137 157 195)(110 138 158 196)(111 139 159 183)(112 140 160 184)
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), (1,85,114,63)(2,98,115,62)(3,97,116,61)(4,96,117,60)(5,95,118,59)(6,94,119,58)(7,93,120,57)(8,92,121,70)(9,91,122,69)(10,90,123,68)(11,89,124,67)(12,88,125,66)(13,87,126,65)(14,86,113,64)(15,137,143,195)(16,136,144,194)(17,135,145,193)(18,134,146,192)(19,133,147,191)(20,132,148,190)(21,131,149,189)(22,130,150,188)(23,129,151,187)(24,128,152,186)(25,127,153,185)(26,140,154,184)(27,139,141,183)(28,138,142,196)(29,178,78,206)(30,177,79,205)(31,176,80,204)(32,175,81,203)(33,174,82,202)(34,173,83,201)(35,172,84,200)(36,171,71,199)(37,170,72,198)(38,169,73,197)(39,182,74,210)(40,181,75,209)(41,180,76,208)(42,179,77,207)(43,102,220,164)(44,101,221,163)(45,100,222,162)(46,99,223,161)(47,112,224,160)(48,111,211,159)(49,110,212,158)(50,109,213,157)(51,108,214,156)(52,107,215,155)(53,106,216,168)(54,105,217,167)(55,104,218,166)(56,103,219,165), (1,221)(2,222)(3,223)(4,224)(5,211)(6,212)(7,213)(8,214)(9,215)(10,216)(11,217)(12,218)(13,219)(14,220)(15,181)(16,182)(17,169)(18,170)(19,171)(20,172)(21,173)(22,174)(23,175)(24,176)(25,177)(26,178)(27,179)(28,180)(29,140)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,157)(58,158)(59,159)(60,160)(61,161)(62,162)(63,163)(64,164)(65,165)(66,166)(67,167)(68,168)(69,155)(70,156)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,183)(78,184)(79,185)(80,186)(81,187)(82,188)(83,189)(84,190)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,99)(98,100)(141,207)(142,208)(143,209)(144,210)(145,197)(146,198)(147,199)(148,200)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206), (1,85,114,63)(2,98,115,62)(3,97,116,61)(4,96,117,60)(5,95,118,59)(6,94,119,58)(7,93,120,57)(8,92,121,70)(9,91,122,69)(10,90,123,68)(11,89,124,67)(12,88,125,66)(13,87,126,65)(14,86,113,64)(15,130,143,188)(16,129,144,187)(17,128,145,186)(18,127,146,185)(19,140,147,184)(20,139,148,183)(21,138,149,196)(22,137,150,195)(23,136,151,194)(24,135,152,193)(25,134,153,192)(26,133,154,191)(27,132,141,190)(28,131,142,189)(29,206,78,178)(30,205,79,177)(31,204,80,176)(32,203,81,175)(33,202,82,174)(34,201,83,173)(35,200,84,172)(36,199,71,171)(37,198,72,170)(38,197,73,169)(39,210,74,182)(40,209,75,181)(41,208,76,180)(42,207,77,179)(43,157,220,109)(44,156,221,108)(45,155,222,107)(46,168,223,106)(47,167,224,105)(48,166,211,104)(49,165,212,103)(50,164,213,102)(51,163,214,101)(52,162,215,100)(53,161,216,99)(54,160,217,112)(55,159,218,111)(56,158,219,110), (1,210,114,182)(2,197,115,169)(3,198,116,170)(4,199,117,171)(5,200,118,172)(6,201,119,173)(7,202,120,174)(8,203,121,175)(9,204,122,176)(10,205,123,177)(11,206,124,178)(12,207,125,179)(13,208,126,180)(14,209,113,181)(15,220,143,43)(16,221,144,44)(17,222,145,45)(18,223,146,46)(19,224,147,47)(20,211,148,48)(21,212,149,49)(22,213,150,50)(23,214,151,51)(24,215,152,52)(25,216,153,53)(26,217,154,54)(27,218,141,55)(28,219,142,56)(29,60,78,96)(30,61,79,97)(31,62,80,98)(32,63,81,85)(33,64,82,86)(34,65,83,87)(35,66,84,88)(36,67,71,89)(37,68,72,90)(38,69,73,91)(39,70,74,92)(40,57,75,93)(41,58,76,94)(42,59,77,95)(99,127,161,185)(100,128,162,186)(101,129,163,187)(102,130,164,188)(103,131,165,189)(104,132,166,190)(105,133,167,191)(106,134,168,192)(107,135,155,193)(108,136,156,194)(109,137,157,195)(110,138,158,196)(111,139,159,183)(112,140,160,184)>;
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), (1,85,114,63)(2,98,115,62)(3,97,116,61)(4,96,117,60)(5,95,118,59)(6,94,119,58)(7,93,120,57)(8,92,121,70)(9,91,122,69)(10,90,123,68)(11,89,124,67)(12,88,125,66)(13,87,126,65)(14,86,113,64)(15,137,143,195)(16,136,144,194)(17,135,145,193)(18,134,146,192)(19,133,147,191)(20,132,148,190)(21,131,149,189)(22,130,150,188)(23,129,151,187)(24,128,152,186)(25,127,153,185)(26,140,154,184)(27,139,141,183)(28,138,142,196)(29,178,78,206)(30,177,79,205)(31,176,80,204)(32,175,81,203)(33,174,82,202)(34,173,83,201)(35,172,84,200)(36,171,71,199)(37,170,72,198)(38,169,73,197)(39,182,74,210)(40,181,75,209)(41,180,76,208)(42,179,77,207)(43,102,220,164)(44,101,221,163)(45,100,222,162)(46,99,223,161)(47,112,224,160)(48,111,211,159)(49,110,212,158)(50,109,213,157)(51,108,214,156)(52,107,215,155)(53,106,216,168)(54,105,217,167)(55,104,218,166)(56,103,219,165), (1,221)(2,222)(3,223)(4,224)(5,211)(6,212)(7,213)(8,214)(9,215)(10,216)(11,217)(12,218)(13,219)(14,220)(15,181)(16,182)(17,169)(18,170)(19,171)(20,172)(21,173)(22,174)(23,175)(24,176)(25,177)(26,178)(27,179)(28,180)(29,140)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,157)(58,158)(59,159)(60,160)(61,161)(62,162)(63,163)(64,164)(65,165)(66,166)(67,167)(68,168)(69,155)(70,156)(71,191)(72,192)(73,193)(74,194)(75,195)(76,196)(77,183)(78,184)(79,185)(80,186)(81,187)(82,188)(83,189)(84,190)(85,101)(86,102)(87,103)(88,104)(89,105)(90,106)(91,107)(92,108)(93,109)(94,110)(95,111)(96,112)(97,99)(98,100)(141,207)(142,208)(143,209)(144,210)(145,197)(146,198)(147,199)(148,200)(149,201)(150,202)(151,203)(152,204)(153,205)(154,206), (1,85,114,63)(2,98,115,62)(3,97,116,61)(4,96,117,60)(5,95,118,59)(6,94,119,58)(7,93,120,57)(8,92,121,70)(9,91,122,69)(10,90,123,68)(11,89,124,67)(12,88,125,66)(13,87,126,65)(14,86,113,64)(15,130,143,188)(16,129,144,187)(17,128,145,186)(18,127,146,185)(19,140,147,184)(20,139,148,183)(21,138,149,196)(22,137,150,195)(23,136,151,194)(24,135,152,193)(25,134,153,192)(26,133,154,191)(27,132,141,190)(28,131,142,189)(29,206,78,178)(30,205,79,177)(31,204,80,176)(32,203,81,175)(33,202,82,174)(34,201,83,173)(35,200,84,172)(36,199,71,171)(37,198,72,170)(38,197,73,169)(39,210,74,182)(40,209,75,181)(41,208,76,180)(42,207,77,179)(43,157,220,109)(44,156,221,108)(45,155,222,107)(46,168,223,106)(47,167,224,105)(48,166,211,104)(49,165,212,103)(50,164,213,102)(51,163,214,101)(52,162,215,100)(53,161,216,99)(54,160,217,112)(55,159,218,111)(56,158,219,110), (1,210,114,182)(2,197,115,169)(3,198,116,170)(4,199,117,171)(5,200,118,172)(6,201,119,173)(7,202,120,174)(8,203,121,175)(9,204,122,176)(10,205,123,177)(11,206,124,178)(12,207,125,179)(13,208,126,180)(14,209,113,181)(15,220,143,43)(16,221,144,44)(17,222,145,45)(18,223,146,46)(19,224,147,47)(20,211,148,48)(21,212,149,49)(22,213,150,50)(23,214,151,51)(24,215,152,52)(25,216,153,53)(26,217,154,54)(27,218,141,55)(28,219,142,56)(29,60,78,96)(30,61,79,97)(31,62,80,98)(32,63,81,85)(33,64,82,86)(34,65,83,87)(35,66,84,88)(36,67,71,89)(37,68,72,90)(38,69,73,91)(39,70,74,92)(40,57,75,93)(41,58,76,94)(42,59,77,95)(99,127,161,185)(100,128,162,186)(101,129,163,187)(102,130,164,188)(103,131,165,189)(104,132,166,190)(105,133,167,191)(106,134,168,192)(107,135,155,193)(108,136,156,194)(109,137,157,195)(110,138,158,196)(111,139,159,183)(112,140,160,184) );
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)], [(1,85,114,63),(2,98,115,62),(3,97,116,61),(4,96,117,60),(5,95,118,59),(6,94,119,58),(7,93,120,57),(8,92,121,70),(9,91,122,69),(10,90,123,68),(11,89,124,67),(12,88,125,66),(13,87,126,65),(14,86,113,64),(15,137,143,195),(16,136,144,194),(17,135,145,193),(18,134,146,192),(19,133,147,191),(20,132,148,190),(21,131,149,189),(22,130,150,188),(23,129,151,187),(24,128,152,186),(25,127,153,185),(26,140,154,184),(27,139,141,183),(28,138,142,196),(29,178,78,206),(30,177,79,205),(31,176,80,204),(32,175,81,203),(33,174,82,202),(34,173,83,201),(35,172,84,200),(36,171,71,199),(37,170,72,198),(38,169,73,197),(39,182,74,210),(40,181,75,209),(41,180,76,208),(42,179,77,207),(43,102,220,164),(44,101,221,163),(45,100,222,162),(46,99,223,161),(47,112,224,160),(48,111,211,159),(49,110,212,158),(50,109,213,157),(51,108,214,156),(52,107,215,155),(53,106,216,168),(54,105,217,167),(55,104,218,166),(56,103,219,165)], [(1,221),(2,222),(3,223),(4,224),(5,211),(6,212),(7,213),(8,214),(9,215),(10,216),(11,217),(12,218),(13,219),(14,220),(15,181),(16,182),(17,169),(18,170),(19,171),(20,172),(21,173),(22,174),(23,175),(24,176),(25,177),(26,178),(27,179),(28,180),(29,140),(30,127),(31,128),(32,129),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,137),(41,138),(42,139),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,157),(58,158),(59,159),(60,160),(61,161),(62,162),(63,163),(64,164),(65,165),(66,166),(67,167),(68,168),(69,155),(70,156),(71,191),(72,192),(73,193),(74,194),(75,195),(76,196),(77,183),(78,184),(79,185),(80,186),(81,187),(82,188),(83,189),(84,190),(85,101),(86,102),(87,103),(88,104),(89,105),(90,106),(91,107),(92,108),(93,109),(94,110),(95,111),(96,112),(97,99),(98,100),(141,207),(142,208),(143,209),(144,210),(145,197),(146,198),(147,199),(148,200),(149,201),(150,202),(151,203),(152,204),(153,205),(154,206)], [(1,85,114,63),(2,98,115,62),(3,97,116,61),(4,96,117,60),(5,95,118,59),(6,94,119,58),(7,93,120,57),(8,92,121,70),(9,91,122,69),(10,90,123,68),(11,89,124,67),(12,88,125,66),(13,87,126,65),(14,86,113,64),(15,130,143,188),(16,129,144,187),(17,128,145,186),(18,127,146,185),(19,140,147,184),(20,139,148,183),(21,138,149,196),(22,137,150,195),(23,136,151,194),(24,135,152,193),(25,134,153,192),(26,133,154,191),(27,132,141,190),(28,131,142,189),(29,206,78,178),(30,205,79,177),(31,204,80,176),(32,203,81,175),(33,202,82,174),(34,201,83,173),(35,200,84,172),(36,199,71,171),(37,198,72,170),(38,197,73,169),(39,210,74,182),(40,209,75,181),(41,208,76,180),(42,207,77,179),(43,157,220,109),(44,156,221,108),(45,155,222,107),(46,168,223,106),(47,167,224,105),(48,166,211,104),(49,165,212,103),(50,164,213,102),(51,163,214,101),(52,162,215,100),(53,161,216,99),(54,160,217,112),(55,159,218,111),(56,158,219,110)], [(1,210,114,182),(2,197,115,169),(3,198,116,170),(4,199,117,171),(5,200,118,172),(6,201,119,173),(7,202,120,174),(8,203,121,175),(9,204,122,176),(10,205,123,177),(11,206,124,178),(12,207,125,179),(13,208,126,180),(14,209,113,181),(15,220,143,43),(16,221,144,44),(17,222,145,45),(18,223,146,46),(19,224,147,47),(20,211,148,48),(21,212,149,49),(22,213,150,50),(23,214,151,51),(24,215,152,52),(25,216,153,53),(26,217,154,54),(27,218,141,55),(28,219,142,56),(29,60,78,96),(30,61,79,97),(31,62,80,98),(32,63,81,85),(33,64,82,86),(34,65,83,87),(35,66,84,88),(36,67,71,89),(37,68,72,90),(38,69,73,91),(39,70,74,92),(40,57,75,93),(41,58,76,94),(42,59,77,95),(99,127,161,185),(100,128,162,186),(101,129,163,187),(102,130,164,188),(103,131,165,189),(104,132,166,190),(105,133,167,191),(106,134,168,192),(107,135,155,193),(108,136,156,194),(109,137,157,195),(110,138,158,196),(111,139,159,183),(112,140,160,184)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 25 |
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 1 | 4 |
| 0 | 0 | 0 | 0 | 17 | 0 | 8 | 27 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 23 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 10 |
| 1 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 21 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 18 |
| 19 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 19 | 19 | 10 | 10 | 0 | 0 | 0 | 0 |
| 7 | 10 | 22 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 3 | 17 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 26 |
| 0 | 0 | 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 19 |
| 20 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 1 | 27 | 8 |
| 0 | 0 | 0 | 0 | 28 | 12 | 14 | 0 |
| 0 | 0 | 0 | 0 | 25 | 27 | 21 | 13 |
| 0 | 0 | 0 | 0 | 1 | 21 | 1 | 6 |
G:=sub<GL(8,GF(29))| [19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,19,19,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,14,25],[19,7,0,0,0,0,0,0,19,10,0,0,0,0,0,0,0,0,19,7,0,0,0,0,0,0,19,10,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,1,8,19,12,0,0,0,0,4,27,23,10],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,27,0,28,0,0,0,0,0,0,0,0,20,15,0,0,0,0,0,0,14,9,0,0,0,0,0,0,0,0,11,15,0,0,0,0,0,0,21,18],[19,7,19,7,0,0,0,0,19,10,19,10,0,0,0,0,0,0,10,22,0,0,0,0,0,0,10,19,0,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,3,0,10,17,0,0,0,0,17,26,6,19],[20,14,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,20,14,0,0,0,0,0,0,15,9,0,0,0,0,0,0,0,0,19,28,25,1,0,0,0,0,1,12,27,21,0,0,0,0,27,14,21,1,0,0,0,0,8,0,13,6] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4K | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 28 | 4 | 4 | 4 | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊6D14 | D4.10D14 |
| kernel | C14.742- (1+4) | C22⋊Dic14 | D14.D4 | Dic7.D4 | Dic7.Q8 | D14⋊Q8 | C28.48D4 | C23.23D14 | C23.18D14 | C28.17D4 | C28⋊2D4 | Dic7⋊D4 | C7×C4⋊D4 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 3 | 6 | 3 | 3 | 9 | 2 | 1 | 12 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{74}2_-^{(1+4)} % in TeX
G:=Group("C14.74ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1074);
// by ID
G=gap.SmallGroup(448,1074);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,570,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=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,e*b*e^-1=a^7*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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