?
G = C14.782- (1+4) order 448 = 26·7
33rd non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.782- (1+4), C14.572+ (1+4), C22⋊Q8⋊22D7, C4⋊C4.100D14, (C2×Q8).78D14, D14⋊3Q8⋊24C2, C28⋊7D4.19C2, (C2×C28).65C23, C22⋊C4.65D14, C4.Dic14⋊26C2, Dic7⋊4D4⋊16C2, D14.5D4⋊23C2, (C2×C14).189C24, D14⋊C4.72C22, (C2×D28).31C22, (C22×C4).251D14, C2.59(D4⋊6D14), C22.D28⋊17C2, C4⋊Dic7.221C22, (Q8×C14).118C22, C22.5(Q8⋊2D7), (C2×Dic7).95C23, (C22×D7).80C23, C22.210(C23×D7), C23.197(C22×D7), Dic7⋊C4.119C22, (C22×C28).317C22, (C22×C14).217C23, C7⋊5(C22.33C24), (C4×Dic7).116C22, C2.38(D4.10D14), (C22×Dic7).125C22, C4⋊C4⋊D7⋊24C2, (C7×C22⋊Q8)⋊25C2, (C2×Dic7⋊C4)⋊30C2, C14.117(C2×C4○D4), C2.21(C2×Q8⋊2D7), (C2×C4×D7).105C22, (C2×C14).29(C4○D4), (C7×C4⋊C4).169C22, (C2×C4).186(C22×D7), (C2×C7⋊D4).41C22, (C7×C22⋊C4).44C22, SmallGroup(448,1098)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 988 in 218 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C4 [×12], C22, C22 [×2], C22 [×8], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×2], D7 [×2], C14 [×3], C14 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×11], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, Dic7 [×6], C28 [×6], D14 [×6], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], C4×D7 [×2], D28, C2×Dic7 [×6], C2×Dic7 [×3], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28, C7×Q8, C22×D7 [×2], C22×C14, C22.33C24, C4×Dic7 [×2], Dic7⋊C4 [×6], C4⋊Dic7, C4⋊Dic7 [×4], D14⋊C4 [×8], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×C4×D7 [×2], C2×D28, C22×Dic7 [×2], C2×C7⋊D4 [×2], C22×C28, Q8×C14, Dic7⋊4D4 [×2], C22.D28 [×2], C4.Dic14 [×2], D14.5D4 [×2], C4⋊C4⋊D7 [×2], C2×Dic7⋊C4, C28⋊7D4, D14⋊3Q8 [×2], C7×C22⋊Q8, C14.782- (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, Q8⋊2D7 [×2], C23×D7, D4⋊6D14, C2×Q8⋊2D7, D4.10D14, C14.782- (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=a7b-1, dbd-1=ebe-1=a7b, cd=dc, 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 52 124 40)(2 53 125 41)(3 54 126 42)(4 55 113 29)(5 56 114 30)(6 43 115 31)(7 44 116 32)(8 45 117 33)(9 46 118 34)(10 47 119 35)(11 48 120 36)(12 49 121 37)(13 50 122 38)(14 51 123 39)(15 217 92 81)(16 218 93 82)(17 219 94 83)(18 220 95 84)(19 221 96 71)(20 222 97 72)(21 223 98 73)(22 224 85 74)(23 211 86 75)(24 212 87 76)(25 213 88 77)(26 214 89 78)(27 215 90 79)(28 216 91 80)(57 205 130 142)(58 206 131 143)(59 207 132 144)(60 208 133 145)(61 209 134 146)(62 210 135 147)(63 197 136 148)(64 198 137 149)(65 199 138 150)(66 200 139 151)(67 201 140 152)(68 202 127 153)(69 203 128 154)(70 204 129 141)(99 177 159 184)(100 178 160 185)(101 179 161 186)(102 180 162 187)(103 181 163 188)(104 182 164 189)(105 169 165 190)(106 170 166 191)(107 171 167 192)(108 172 168 193)(109 173 155 194)(110 174 156 195)(111 175 157 196)(112 176 158 183)
(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 48)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 56)(38 43)(39 44)(40 45)(41 46)(42 47)(99 166)(100 167)(101 168)(102 155)(103 156)(104 157)(105 158)(106 159)(107 160)(108 161)(109 162)(110 163)(111 164)(112 165)(141 197)(142 198)(143 199)(144 200)(145 201)(146 202)(147 203)(148 204)(149 205)(150 206)(151 207)(152 208)(153 209)(154 210)
(1 172 117 186)(2 171 118 185)(3 170 119 184)(4 169 120 183)(5 182 121 196)(6 181 122 195)(7 180 123 194)(8 179 124 193)(9 178 125 192)(10 177 126 191)(11 176 113 190)(12 175 114 189)(13 174 115 188)(14 173 116 187)(15 143 85 199)(16 142 86 198)(17 141 87 197)(18 154 88 210)(19 153 89 209)(20 152 90 208)(21 151 91 207)(22 150 92 206)(23 149 93 205)(24 148 94 204)(25 147 95 203)(26 146 96 202)(27 145 97 201)(28 144 98 200)(29 112 48 165)(30 111 49 164)(31 110 50 163)(32 109 51 162)(33 108 52 161)(34 107 53 160)(35 106 54 159)(36 105 55 158)(37 104 56 157)(38 103 43 156)(39 102 44 155)(40 101 45 168)(41 100 46 167)(42 99 47 166)(57 82 137 211)(58 81 138 224)(59 80 139 223)(60 79 140 222)(61 78 127 221)(62 77 128 220)(63 76 129 219)(64 75 130 218)(65 74 131 217)(66 73 132 216)(67 72 133 215)(68 71 134 214)(69 84 135 213)(70 83 136 212)
(1 67 124 140)(2 68 125 127)(3 69 126 128)(4 70 113 129)(5 57 114 130)(6 58 115 131)(7 59 116 132)(8 60 117 133)(9 61 118 134)(10 62 119 135)(11 63 120 136)(12 64 121 137)(13 65 122 138)(14 66 123 139)(15 163 92 103)(16 164 93 104)(17 165 94 105)(18 166 95 106)(19 167 96 107)(20 168 97 108)(21 155 98 109)(22 156 85 110)(23 157 86 111)(24 158 87 112)(25 159 88 99)(26 160 89 100)(27 161 90 101)(28 162 91 102)(29 148 55 197)(30 149 56 198)(31 150 43 199)(32 151 44 200)(33 152 45 201)(34 153 46 202)(35 154 47 203)(36 141 48 204)(37 142 49 205)(38 143 50 206)(39 144 51 207)(40 145 52 208)(41 146 53 209)(42 147 54 210)(71 178 221 185)(72 179 222 186)(73 180 223 187)(74 181 224 188)(75 182 211 189)(76 169 212 190)(77 170 213 191)(78 171 214 192)(79 172 215 193)(80 173 216 194)(81 174 217 195)(82 175 218 196)(83 176 219 183)(84 177 220 184)
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), (1,52,124,40)(2,53,125,41)(3,54,126,42)(4,55,113,29)(5,56,114,30)(6,43,115,31)(7,44,116,32)(8,45,117,33)(9,46,118,34)(10,47,119,35)(11,48,120,36)(12,49,121,37)(13,50,122,38)(14,51,123,39)(15,217,92,81)(16,218,93,82)(17,219,94,83)(18,220,95,84)(19,221,96,71)(20,222,97,72)(21,223,98,73)(22,224,85,74)(23,211,86,75)(24,212,87,76)(25,213,88,77)(26,214,89,78)(27,215,90,79)(28,216,91,80)(57,205,130,142)(58,206,131,143)(59,207,132,144)(60,208,133,145)(61,209,134,146)(62,210,135,147)(63,197,136,148)(64,198,137,149)(65,199,138,150)(66,200,139,151)(67,201,140,152)(68,202,127,153)(69,203,128,154)(70,204,129,141)(99,177,159,184)(100,178,160,185)(101,179,161,186)(102,180,162,187)(103,181,163,188)(104,182,164,189)(105,169,165,190)(106,170,166,191)(107,171,167,192)(108,172,168,193)(109,173,155,194)(110,174,156,195)(111,175,157,196)(112,176,158,183), (15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,43)(39,44)(40,45)(41,46)(42,47)(99,166)(100,167)(101,168)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165)(141,197)(142,198)(143,199)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210), (1,172,117,186)(2,171,118,185)(3,170,119,184)(4,169,120,183)(5,182,121,196)(6,181,122,195)(7,180,123,194)(8,179,124,193)(9,178,125,192)(10,177,126,191)(11,176,113,190)(12,175,114,189)(13,174,115,188)(14,173,116,187)(15,143,85,199)(16,142,86,198)(17,141,87,197)(18,154,88,210)(19,153,89,209)(20,152,90,208)(21,151,91,207)(22,150,92,206)(23,149,93,205)(24,148,94,204)(25,147,95,203)(26,146,96,202)(27,145,97,201)(28,144,98,200)(29,112,48,165)(30,111,49,164)(31,110,50,163)(32,109,51,162)(33,108,52,161)(34,107,53,160)(35,106,54,159)(36,105,55,158)(37,104,56,157)(38,103,43,156)(39,102,44,155)(40,101,45,168)(41,100,46,167)(42,99,47,166)(57,82,137,211)(58,81,138,224)(59,80,139,223)(60,79,140,222)(61,78,127,221)(62,77,128,220)(63,76,129,219)(64,75,130,218)(65,74,131,217)(66,73,132,216)(67,72,133,215)(68,71,134,214)(69,84,135,213)(70,83,136,212), (1,67,124,140)(2,68,125,127)(3,69,126,128)(4,70,113,129)(5,57,114,130)(6,58,115,131)(7,59,116,132)(8,60,117,133)(9,61,118,134)(10,62,119,135)(11,63,120,136)(12,64,121,137)(13,65,122,138)(14,66,123,139)(15,163,92,103)(16,164,93,104)(17,165,94,105)(18,166,95,106)(19,167,96,107)(20,168,97,108)(21,155,98,109)(22,156,85,110)(23,157,86,111)(24,158,87,112)(25,159,88,99)(26,160,89,100)(27,161,90,101)(28,162,91,102)(29,148,55,197)(30,149,56,198)(31,150,43,199)(32,151,44,200)(33,152,45,201)(34,153,46,202)(35,154,47,203)(36,141,48,204)(37,142,49,205)(38,143,50,206)(39,144,51,207)(40,145,52,208)(41,146,53,209)(42,147,54,210)(71,178,221,185)(72,179,222,186)(73,180,223,187)(74,181,224,188)(75,182,211,189)(76,169,212,190)(77,170,213,191)(78,171,214,192)(79,172,215,193)(80,173,216,194)(81,174,217,195)(82,175,218,196)(83,176,219,183)(84,177,220,184)>;
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), (1,52,124,40)(2,53,125,41)(3,54,126,42)(4,55,113,29)(5,56,114,30)(6,43,115,31)(7,44,116,32)(8,45,117,33)(9,46,118,34)(10,47,119,35)(11,48,120,36)(12,49,121,37)(13,50,122,38)(14,51,123,39)(15,217,92,81)(16,218,93,82)(17,219,94,83)(18,220,95,84)(19,221,96,71)(20,222,97,72)(21,223,98,73)(22,224,85,74)(23,211,86,75)(24,212,87,76)(25,213,88,77)(26,214,89,78)(27,215,90,79)(28,216,91,80)(57,205,130,142)(58,206,131,143)(59,207,132,144)(60,208,133,145)(61,209,134,146)(62,210,135,147)(63,197,136,148)(64,198,137,149)(65,199,138,150)(66,200,139,151)(67,201,140,152)(68,202,127,153)(69,203,128,154)(70,204,129,141)(99,177,159,184)(100,178,160,185)(101,179,161,186)(102,180,162,187)(103,181,163,188)(104,182,164,189)(105,169,165,190)(106,170,166,191)(107,171,167,192)(108,172,168,193)(109,173,155,194)(110,174,156,195)(111,175,157,196)(112,176,158,183), (15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,48)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,56)(38,43)(39,44)(40,45)(41,46)(42,47)(99,166)(100,167)(101,168)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165)(141,197)(142,198)(143,199)(144,200)(145,201)(146,202)(147,203)(148,204)(149,205)(150,206)(151,207)(152,208)(153,209)(154,210), (1,172,117,186)(2,171,118,185)(3,170,119,184)(4,169,120,183)(5,182,121,196)(6,181,122,195)(7,180,123,194)(8,179,124,193)(9,178,125,192)(10,177,126,191)(11,176,113,190)(12,175,114,189)(13,174,115,188)(14,173,116,187)(15,143,85,199)(16,142,86,198)(17,141,87,197)(18,154,88,210)(19,153,89,209)(20,152,90,208)(21,151,91,207)(22,150,92,206)(23,149,93,205)(24,148,94,204)(25,147,95,203)(26,146,96,202)(27,145,97,201)(28,144,98,200)(29,112,48,165)(30,111,49,164)(31,110,50,163)(32,109,51,162)(33,108,52,161)(34,107,53,160)(35,106,54,159)(36,105,55,158)(37,104,56,157)(38,103,43,156)(39,102,44,155)(40,101,45,168)(41,100,46,167)(42,99,47,166)(57,82,137,211)(58,81,138,224)(59,80,139,223)(60,79,140,222)(61,78,127,221)(62,77,128,220)(63,76,129,219)(64,75,130,218)(65,74,131,217)(66,73,132,216)(67,72,133,215)(68,71,134,214)(69,84,135,213)(70,83,136,212), (1,67,124,140)(2,68,125,127)(3,69,126,128)(4,70,113,129)(5,57,114,130)(6,58,115,131)(7,59,116,132)(8,60,117,133)(9,61,118,134)(10,62,119,135)(11,63,120,136)(12,64,121,137)(13,65,122,138)(14,66,123,139)(15,163,92,103)(16,164,93,104)(17,165,94,105)(18,166,95,106)(19,167,96,107)(20,168,97,108)(21,155,98,109)(22,156,85,110)(23,157,86,111)(24,158,87,112)(25,159,88,99)(26,160,89,100)(27,161,90,101)(28,162,91,102)(29,148,55,197)(30,149,56,198)(31,150,43,199)(32,151,44,200)(33,152,45,201)(34,153,46,202)(35,154,47,203)(36,141,48,204)(37,142,49,205)(38,143,50,206)(39,144,51,207)(40,145,52,208)(41,146,53,209)(42,147,54,210)(71,178,221,185)(72,179,222,186)(73,180,223,187)(74,181,224,188)(75,182,211,189)(76,169,212,190)(77,170,213,191)(78,171,214,192)(79,172,215,193)(80,173,216,194)(81,174,217,195)(82,175,218,196)(83,176,219,183)(84,177,220,184) );
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)], [(1,52,124,40),(2,53,125,41),(3,54,126,42),(4,55,113,29),(5,56,114,30),(6,43,115,31),(7,44,116,32),(8,45,117,33),(9,46,118,34),(10,47,119,35),(11,48,120,36),(12,49,121,37),(13,50,122,38),(14,51,123,39),(15,217,92,81),(16,218,93,82),(17,219,94,83),(18,220,95,84),(19,221,96,71),(20,222,97,72),(21,223,98,73),(22,224,85,74),(23,211,86,75),(24,212,87,76),(25,213,88,77),(26,214,89,78),(27,215,90,79),(28,216,91,80),(57,205,130,142),(58,206,131,143),(59,207,132,144),(60,208,133,145),(61,209,134,146),(62,210,135,147),(63,197,136,148),(64,198,137,149),(65,199,138,150),(66,200,139,151),(67,201,140,152),(68,202,127,153),(69,203,128,154),(70,204,129,141),(99,177,159,184),(100,178,160,185),(101,179,161,186),(102,180,162,187),(103,181,163,188),(104,182,164,189),(105,169,165,190),(106,170,166,191),(107,171,167,192),(108,172,168,193),(109,173,155,194),(110,174,156,195),(111,175,157,196),(112,176,158,183)], [(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,98),(29,48),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,56),(38,43),(39,44),(40,45),(41,46),(42,47),(99,166),(100,167),(101,168),(102,155),(103,156),(104,157),(105,158),(106,159),(107,160),(108,161),(109,162),(110,163),(111,164),(112,165),(141,197),(142,198),(143,199),(144,200),(145,201),(146,202),(147,203),(148,204),(149,205),(150,206),(151,207),(152,208),(153,209),(154,210)], [(1,172,117,186),(2,171,118,185),(3,170,119,184),(4,169,120,183),(5,182,121,196),(6,181,122,195),(7,180,123,194),(8,179,124,193),(9,178,125,192),(10,177,126,191),(11,176,113,190),(12,175,114,189),(13,174,115,188),(14,173,116,187),(15,143,85,199),(16,142,86,198),(17,141,87,197),(18,154,88,210),(19,153,89,209),(20,152,90,208),(21,151,91,207),(22,150,92,206),(23,149,93,205),(24,148,94,204),(25,147,95,203),(26,146,96,202),(27,145,97,201),(28,144,98,200),(29,112,48,165),(30,111,49,164),(31,110,50,163),(32,109,51,162),(33,108,52,161),(34,107,53,160),(35,106,54,159),(36,105,55,158),(37,104,56,157),(38,103,43,156),(39,102,44,155),(40,101,45,168),(41,100,46,167),(42,99,47,166),(57,82,137,211),(58,81,138,224),(59,80,139,223),(60,79,140,222),(61,78,127,221),(62,77,128,220),(63,76,129,219),(64,75,130,218),(65,74,131,217),(66,73,132,216),(67,72,133,215),(68,71,134,214),(69,84,135,213),(70,83,136,212)], [(1,67,124,140),(2,68,125,127),(3,69,126,128),(4,70,113,129),(5,57,114,130),(6,58,115,131),(7,59,116,132),(8,60,117,133),(9,61,118,134),(10,62,119,135),(11,63,120,136),(12,64,121,137),(13,65,122,138),(14,66,123,139),(15,163,92,103),(16,164,93,104),(17,165,94,105),(18,166,95,106),(19,167,96,107),(20,168,97,108),(21,155,98,109),(22,156,85,110),(23,157,86,111),(24,158,87,112),(25,159,88,99),(26,160,89,100),(27,161,90,101),(28,162,91,102),(29,148,55,197),(30,149,56,198),(31,150,43,199),(32,151,44,200),(33,152,45,201),(34,153,46,202),(35,154,47,203),(36,141,48,204),(37,142,49,205),(38,143,50,206),(39,144,51,207),(40,145,52,208),(41,146,53,209),(42,147,54,210),(71,178,221,185),(72,179,222,186),(73,180,223,187),(74,181,224,188),(75,182,211,189),(76,169,212,190),(77,170,213,191),(78,171,214,192),(79,172,215,193),(80,173,216,194),(81,174,217,195),(82,175,218,196),(83,176,219,183),(84,177,220,184)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 7 | 22 | 0 | 0 |
| 0 | 0 | 25 | 0 | 4 | 4 |
| 0 | 0 | 11 | 4 | 25 | 18 |
| 22 | 22 | 0 | 0 | 0 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 27 | 13 | 13 |
| 0 | 0 | 12 | 9 | 0 | 28 |
| 0 | 0 | 27 | 0 | 11 | 2 |
| 0 | 0 | 18 | 27 | 11 | 2 |
| 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 | 24 | 18 | 28 | 0 |
| 0 | 0 | 24 | 18 | 0 | 28 |
| 12 | 23 | 0 | 0 | 0 | 0 |
| 19 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 26 | 0 | 0 |
| 0 | 0 | 24 | 19 | 0 | 0 |
| 0 | 0 | 11 | 0 | 26 | 18 |
| 0 | 0 | 7 | 26 | 22 | 3 |
| 1 | 14 | 0 | 0 | 0 | 0 |
| 4 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 27 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 9 | 27 | 11 | 2 |
| 0 | 0 | 2 | 0 | 27 | 18 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,7,25,11,0,0,4,22,0,4,0,0,0,0,4,25,0,0,0,0,4,18],[22,3,0,0,0,0,22,7,0,0,0,0,0,0,7,12,27,18,0,0,27,9,0,27,0,0,13,0,11,11,0,0,13,28,2,2],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,24,24,0,0,0,1,18,18,0,0,0,0,28,0,0,0,0,0,0,28],[12,19,0,0,0,0,23,17,0,0,0,0,0,0,10,24,11,7,0,0,26,19,0,26,0,0,0,0,26,22,0,0,0,0,18,3],[1,4,0,0,0,0,14,28,0,0,0,0,0,0,20,11,9,2,0,0,27,9,27,0,0,0,0,0,11,27,0,0,0,0,2,18] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
64 irreducible representations
| dim | 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 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | Q8⋊2D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.782- (1+4) | Dic7⋊4D4 | C22.D28 | C4.Dic14 | D14.5D4 | C4⋊C4⋊D7 | C2×Dic7⋊C4 | C28⋊7D4 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{78}2_-^{(1+4)} % in TeX
G:=Group("C14.78ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1098);
// by ID
G=gap.SmallGroup(448,1098);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,387,100,675,409,80,18822]);
// 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=a^7*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=a^7*b^2*d>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀