?
G = C14.322+ (1+4) order 448 = 26·7
32nd non-split extension by C14 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.322+ (1+4), C14.682- (1+4), C4⋊D4⋊5D7, C4⋊C4.89D14, (C2×Dic7)⋊9D4, C28⋊7D4⋊43C2, C22.2(D4×D7), (C2×D4).89D14, C22⋊C4.4D14, D14⋊D4⋊16C2, D14⋊Q8⋊13C2, Dic7⋊D4⋊8C2, Dic7.45(C2×D4), C14.60(C22×D4), C23.8(C22×D7), (C2×C28).172C23, (C2×C14).141C24, D14⋊C4.69C22, (C2×D28).30C22, (C22×C4).217D14, C4⋊Dic7.44C22, C2.34(D4⋊6D14), C22⋊Dic14⋊15C2, (D4×C14).115C22, (C22×C14).12C23, (C22×D7).60C23, C22.162(C23×D7), C23.D7.19C22, Dic7⋊C4.158C22, (C22×C28).310C22, C7⋊2(C22.31C24), (C2×Dic7).224C23, C2.26(D4.10D14), (C2×Dic14).151C22, (C22×Dic7).102C22, C2.33(C2×D4×D7), (C7×C4⋊D4)⋊6C2, (C2×C14).4(C2×D4), (C2×D4⋊2D7)⋊9C2, (C2×C4×D7).80C22, (C2×Dic7⋊C4)⋊28C2, (C2×C4).35(C22×D7), (C7×C4⋊C4).137C22, (C2×C7⋊D4).24C22, (C7×C22⋊C4).6C22, SmallGroup(448,1050)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1420 in 294 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×12], C22, C22 [×2], C22 [×14], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×20], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×2], D7 [×2], C14 [×3], C14 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×7], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic7 [×4], Dic7 [×4], C28 [×4], D14 [×6], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C4⋊C4, C4⋊D4, C4⋊D4 [×7], C22⋊Q8 [×4], C2×C4○D4 [×2], Dic14 [×4], C4×D7 [×4], D28, C2×Dic7 [×10], C2×Dic7 [×5], C7⋊D4 [×10], C2×C28 [×2], C2×C28 [×2], C2×C28, C7×D4 [×5], C22×D7 [×2], C22×C14, C22×C14 [×2], C22.31C24, Dic7⋊C4 [×6], C4⋊Dic7, D14⋊C4 [×4], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7 [×2], C2×D28, D4⋊2D7 [×8], C22×Dic7 [×2], C22×Dic7 [×2], C2×C7⋊D4 [×6], C22×C28, D4×C14, D4×C14 [×2], C22⋊Dic14 [×2], D14⋊D4 [×2], D14⋊Q8 [×2], C2×Dic7⋊C4, C28⋊7D4, Dic7⋊D4 [×4], C7×C4⋊D4, C2×D4⋊2D7 [×2], C14.322+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2+ (1+4), 2- (1+4), C22×D7 [×7], C22.31C24, D4×D7 [×2], C23×D7, C2×D4×D7, D4⋊6D14, D4.10D14, C14.322+ (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=a7b-1, dbd-1=a7b, be=eb, dcd-1=ece=a7c, ede=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)
(1 160 198 68)(2 161 199 69)(3 162 200 70)(4 163 201 57)(5 164 202 58)(6 165 203 59)(7 166 204 60)(8 167 205 61)(9 168 206 62)(10 155 207 63)(11 156 208 64)(12 157 209 65)(13 158 210 66)(14 159 197 67)(15 181 153 71)(16 182 154 72)(17 169 141 73)(18 170 142 74)(19 171 143 75)(20 172 144 76)(21 173 145 77)(22 174 146 78)(23 175 147 79)(24 176 148 80)(25 177 149 81)(26 178 150 82)(27 179 151 83)(28 180 152 84)(29 139 211 54)(30 140 212 55)(31 127 213 56)(32 128 214 43)(33 129 215 44)(34 130 216 45)(35 131 217 46)(36 132 218 47)(37 133 219 48)(38 134 220 49)(39 135 221 50)(40 136 222 51)(41 137 223 52)(42 138 224 53)(85 114 99 193)(86 115 100 194)(87 116 101 195)(88 117 102 196)(89 118 103 183)(90 119 104 184)(91 120 105 185)(92 121 106 186)(93 122 107 187)(94 123 108 188)(95 124 109 189)(96 125 110 190)(97 126 111 191)(98 113 112 192)
(1 138 8 131)(2 137 9 130)(3 136 10 129)(4 135 11 128)(5 134 12 127)(6 133 13 140)(7 132 14 139)(15 101 22 108)(16 100 23 107)(17 99 24 106)(18 112 25 105)(19 111 26 104)(20 110 27 103)(21 109 28 102)(29 159 36 166)(30 158 37 165)(31 157 38 164)(32 156 39 163)(33 155 40 162)(34 168 41 161)(35 167 42 160)(43 201 50 208)(44 200 51 207)(45 199 52 206)(46 198 53 205)(47 197 54 204)(48 210 55 203)(49 209 56 202)(57 214 64 221)(58 213 65 220)(59 212 66 219)(60 211 67 218)(61 224 68 217)(62 223 69 216)(63 222 70 215)(71 188 78 195)(72 187 79 194)(73 186 80 193)(74 185 81 192)(75 184 82 191)(76 183 83 190)(77 196 84 189)(85 148 92 141)(86 147 93 154)(87 146 94 153)(88 145 95 152)(89 144 96 151)(90 143 97 150)(91 142 98 149)(113 170 120 177)(114 169 121 176)(115 182 122 175)(116 181 123 174)(117 180 124 173)(118 179 125 172)(119 178 126 171)
(1 150 198 26)(2 151 199 27)(3 152 200 28)(4 153 201 15)(5 154 202 16)(6 141 203 17)(7 142 204 18)(8 143 205 19)(9 144 206 20)(10 145 207 21)(11 146 208 22)(12 147 209 23)(13 148 210 24)(14 149 197 25)(29 185 211 120)(30 186 212 121)(31 187 213 122)(32 188 214 123)(33 189 215 124)(34 190 216 125)(35 191 217 126)(36 192 218 113)(37 193 219 114)(38 194 220 115)(39 195 221 116)(40 196 222 117)(41 183 223 118)(42 184 224 119)(43 101 128 87)(44 102 129 88)(45 103 130 89)(46 104 131 90)(47 105 132 91)(48 106 133 92)(49 107 134 93)(50 108 135 94)(51 109 136 95)(52 110 137 96)(53 111 138 97)(54 112 139 98)(55 99 140 85)(56 100 127 86)(57 174 163 78)(58 175 164 79)(59 176 165 80)(60 177 166 81)(61 178 167 82)(62 179 168 83)(63 180 155 84)(64 181 156 71)(65 182 157 72)(66 169 158 73)(67 170 159 74)(68 171 160 75)(69 172 161 76)(70 173 162 77)
(1 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 98)(8 97)(9 96)(10 95)(11 94)(12 93)(13 92)(14 91)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 177)(30 176)(31 175)(32 174)(33 173)(34 172)(35 171)(36 170)(37 169)(38 182)(39 181)(40 180)(41 179)(42 178)(57 195)(58 194)(59 193)(60 192)(61 191)(62 190)(63 189)(64 188)(65 187)(66 186)(67 185)(68 184)(69 183)(70 196)(71 221)(72 220)(73 219)(74 218)(75 217)(76 216)(77 215)(78 214)(79 213)(80 212)(81 211)(82 224)(83 223)(84 222)(99 203)(100 202)(101 201)(102 200)(103 199)(104 198)(105 197)(106 210)(107 209)(108 208)(109 207)(110 206)(111 205)(112 204)(113 166)(114 165)(115 164)(116 163)(117 162)(118 161)(119 160)(120 159)(121 158)(122 157)(123 156)(124 155)(125 168)(126 167)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 141)(134 154)(135 153)(136 152)(137 151)(138 150)(139 149)(140 148)
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), (1,160,198,68)(2,161,199,69)(3,162,200,70)(4,163,201,57)(5,164,202,58)(6,165,203,59)(7,166,204,60)(8,167,205,61)(9,168,206,62)(10,155,207,63)(11,156,208,64)(12,157,209,65)(13,158,210,66)(14,159,197,67)(15,181,153,71)(16,182,154,72)(17,169,141,73)(18,170,142,74)(19,171,143,75)(20,172,144,76)(21,173,145,77)(22,174,146,78)(23,175,147,79)(24,176,148,80)(25,177,149,81)(26,178,150,82)(27,179,151,83)(28,180,152,84)(29,139,211,54)(30,140,212,55)(31,127,213,56)(32,128,214,43)(33,129,215,44)(34,130,216,45)(35,131,217,46)(36,132,218,47)(37,133,219,48)(38,134,220,49)(39,135,221,50)(40,136,222,51)(41,137,223,52)(42,138,224,53)(85,114,99,193)(86,115,100,194)(87,116,101,195)(88,117,102,196)(89,118,103,183)(90,119,104,184)(91,120,105,185)(92,121,106,186)(93,122,107,187)(94,123,108,188)(95,124,109,189)(96,125,110,190)(97,126,111,191)(98,113,112,192), (1,138,8,131)(2,137,9,130)(3,136,10,129)(4,135,11,128)(5,134,12,127)(6,133,13,140)(7,132,14,139)(15,101,22,108)(16,100,23,107)(17,99,24,106)(18,112,25,105)(19,111,26,104)(20,110,27,103)(21,109,28,102)(29,159,36,166)(30,158,37,165)(31,157,38,164)(32,156,39,163)(33,155,40,162)(34,168,41,161)(35,167,42,160)(43,201,50,208)(44,200,51,207)(45,199,52,206)(46,198,53,205)(47,197,54,204)(48,210,55,203)(49,209,56,202)(57,214,64,221)(58,213,65,220)(59,212,66,219)(60,211,67,218)(61,224,68,217)(62,223,69,216)(63,222,70,215)(71,188,78,195)(72,187,79,194)(73,186,80,193)(74,185,81,192)(75,184,82,191)(76,183,83,190)(77,196,84,189)(85,148,92,141)(86,147,93,154)(87,146,94,153)(88,145,95,152)(89,144,96,151)(90,143,97,150)(91,142,98,149)(113,170,120,177)(114,169,121,176)(115,182,122,175)(116,181,123,174)(117,180,124,173)(118,179,125,172)(119,178,126,171), (1,150,198,26)(2,151,199,27)(3,152,200,28)(4,153,201,15)(5,154,202,16)(6,141,203,17)(7,142,204,18)(8,143,205,19)(9,144,206,20)(10,145,207,21)(11,146,208,22)(12,147,209,23)(13,148,210,24)(14,149,197,25)(29,185,211,120)(30,186,212,121)(31,187,213,122)(32,188,214,123)(33,189,215,124)(34,190,216,125)(35,191,217,126)(36,192,218,113)(37,193,219,114)(38,194,220,115)(39,195,221,116)(40,196,222,117)(41,183,223,118)(42,184,224,119)(43,101,128,87)(44,102,129,88)(45,103,130,89)(46,104,131,90)(47,105,132,91)(48,106,133,92)(49,107,134,93)(50,108,135,94)(51,109,136,95)(52,110,137,96)(53,111,138,97)(54,112,139,98)(55,99,140,85)(56,100,127,86)(57,174,163,78)(58,175,164,79)(59,176,165,80)(60,177,166,81)(61,178,167,82)(62,179,168,83)(63,180,155,84)(64,181,156,71)(65,182,157,72)(66,169,158,73)(67,170,159,74)(68,171,160,75)(69,172,161,76)(70,173,162,77), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,98)(8,97)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,177)(30,176)(31,175)(32,174)(33,173)(34,172)(35,171)(36,170)(37,169)(38,182)(39,181)(40,180)(41,179)(42,178)(57,195)(58,194)(59,193)(60,192)(61,191)(62,190)(63,189)(64,188)(65,187)(66,186)(67,185)(68,184)(69,183)(70,196)(71,221)(72,220)(73,219)(74,218)(75,217)(76,216)(77,215)(78,214)(79,213)(80,212)(81,211)(82,224)(83,223)(84,222)(99,203)(100,202)(101,201)(102,200)(103,199)(104,198)(105,197)(106,210)(107,209)(108,208)(109,207)(110,206)(111,205)(112,204)(113,166)(114,165)(115,164)(116,163)(117,162)(118,161)(119,160)(120,159)(121,158)(122,157)(123,156)(124,155)(125,168)(126,167)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)>;
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), (1,160,198,68)(2,161,199,69)(3,162,200,70)(4,163,201,57)(5,164,202,58)(6,165,203,59)(7,166,204,60)(8,167,205,61)(9,168,206,62)(10,155,207,63)(11,156,208,64)(12,157,209,65)(13,158,210,66)(14,159,197,67)(15,181,153,71)(16,182,154,72)(17,169,141,73)(18,170,142,74)(19,171,143,75)(20,172,144,76)(21,173,145,77)(22,174,146,78)(23,175,147,79)(24,176,148,80)(25,177,149,81)(26,178,150,82)(27,179,151,83)(28,180,152,84)(29,139,211,54)(30,140,212,55)(31,127,213,56)(32,128,214,43)(33,129,215,44)(34,130,216,45)(35,131,217,46)(36,132,218,47)(37,133,219,48)(38,134,220,49)(39,135,221,50)(40,136,222,51)(41,137,223,52)(42,138,224,53)(85,114,99,193)(86,115,100,194)(87,116,101,195)(88,117,102,196)(89,118,103,183)(90,119,104,184)(91,120,105,185)(92,121,106,186)(93,122,107,187)(94,123,108,188)(95,124,109,189)(96,125,110,190)(97,126,111,191)(98,113,112,192), (1,138,8,131)(2,137,9,130)(3,136,10,129)(4,135,11,128)(5,134,12,127)(6,133,13,140)(7,132,14,139)(15,101,22,108)(16,100,23,107)(17,99,24,106)(18,112,25,105)(19,111,26,104)(20,110,27,103)(21,109,28,102)(29,159,36,166)(30,158,37,165)(31,157,38,164)(32,156,39,163)(33,155,40,162)(34,168,41,161)(35,167,42,160)(43,201,50,208)(44,200,51,207)(45,199,52,206)(46,198,53,205)(47,197,54,204)(48,210,55,203)(49,209,56,202)(57,214,64,221)(58,213,65,220)(59,212,66,219)(60,211,67,218)(61,224,68,217)(62,223,69,216)(63,222,70,215)(71,188,78,195)(72,187,79,194)(73,186,80,193)(74,185,81,192)(75,184,82,191)(76,183,83,190)(77,196,84,189)(85,148,92,141)(86,147,93,154)(87,146,94,153)(88,145,95,152)(89,144,96,151)(90,143,97,150)(91,142,98,149)(113,170,120,177)(114,169,121,176)(115,182,122,175)(116,181,123,174)(117,180,124,173)(118,179,125,172)(119,178,126,171), (1,150,198,26)(2,151,199,27)(3,152,200,28)(4,153,201,15)(5,154,202,16)(6,141,203,17)(7,142,204,18)(8,143,205,19)(9,144,206,20)(10,145,207,21)(11,146,208,22)(12,147,209,23)(13,148,210,24)(14,149,197,25)(29,185,211,120)(30,186,212,121)(31,187,213,122)(32,188,214,123)(33,189,215,124)(34,190,216,125)(35,191,217,126)(36,192,218,113)(37,193,219,114)(38,194,220,115)(39,195,221,116)(40,196,222,117)(41,183,223,118)(42,184,224,119)(43,101,128,87)(44,102,129,88)(45,103,130,89)(46,104,131,90)(47,105,132,91)(48,106,133,92)(49,107,134,93)(50,108,135,94)(51,109,136,95)(52,110,137,96)(53,111,138,97)(54,112,139,98)(55,99,140,85)(56,100,127,86)(57,174,163,78)(58,175,164,79)(59,176,165,80)(60,177,166,81)(61,178,167,82)(62,179,168,83)(63,180,155,84)(64,181,156,71)(65,182,157,72)(66,169,158,73)(67,170,159,74)(68,171,160,75)(69,172,161,76)(70,173,162,77), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,98)(8,97)(9,96)(10,95)(11,94)(12,93)(13,92)(14,91)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,177)(30,176)(31,175)(32,174)(33,173)(34,172)(35,171)(36,170)(37,169)(38,182)(39,181)(40,180)(41,179)(42,178)(57,195)(58,194)(59,193)(60,192)(61,191)(62,190)(63,189)(64,188)(65,187)(66,186)(67,185)(68,184)(69,183)(70,196)(71,221)(72,220)(73,219)(74,218)(75,217)(76,216)(77,215)(78,214)(79,213)(80,212)(81,211)(82,224)(83,223)(84,222)(99,203)(100,202)(101,201)(102,200)(103,199)(104,198)(105,197)(106,210)(107,209)(108,208)(109,207)(110,206)(111,205)(112,204)(113,166)(114,165)(115,164)(116,163)(117,162)(118,161)(119,160)(120,159)(121,158)(122,157)(123,156)(124,155)(125,168)(126,167)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148) );
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)], [(1,160,198,68),(2,161,199,69),(3,162,200,70),(4,163,201,57),(5,164,202,58),(6,165,203,59),(7,166,204,60),(8,167,205,61),(9,168,206,62),(10,155,207,63),(11,156,208,64),(12,157,209,65),(13,158,210,66),(14,159,197,67),(15,181,153,71),(16,182,154,72),(17,169,141,73),(18,170,142,74),(19,171,143,75),(20,172,144,76),(21,173,145,77),(22,174,146,78),(23,175,147,79),(24,176,148,80),(25,177,149,81),(26,178,150,82),(27,179,151,83),(28,180,152,84),(29,139,211,54),(30,140,212,55),(31,127,213,56),(32,128,214,43),(33,129,215,44),(34,130,216,45),(35,131,217,46),(36,132,218,47),(37,133,219,48),(38,134,220,49),(39,135,221,50),(40,136,222,51),(41,137,223,52),(42,138,224,53),(85,114,99,193),(86,115,100,194),(87,116,101,195),(88,117,102,196),(89,118,103,183),(90,119,104,184),(91,120,105,185),(92,121,106,186),(93,122,107,187),(94,123,108,188),(95,124,109,189),(96,125,110,190),(97,126,111,191),(98,113,112,192)], [(1,138,8,131),(2,137,9,130),(3,136,10,129),(4,135,11,128),(5,134,12,127),(6,133,13,140),(7,132,14,139),(15,101,22,108),(16,100,23,107),(17,99,24,106),(18,112,25,105),(19,111,26,104),(20,110,27,103),(21,109,28,102),(29,159,36,166),(30,158,37,165),(31,157,38,164),(32,156,39,163),(33,155,40,162),(34,168,41,161),(35,167,42,160),(43,201,50,208),(44,200,51,207),(45,199,52,206),(46,198,53,205),(47,197,54,204),(48,210,55,203),(49,209,56,202),(57,214,64,221),(58,213,65,220),(59,212,66,219),(60,211,67,218),(61,224,68,217),(62,223,69,216),(63,222,70,215),(71,188,78,195),(72,187,79,194),(73,186,80,193),(74,185,81,192),(75,184,82,191),(76,183,83,190),(77,196,84,189),(85,148,92,141),(86,147,93,154),(87,146,94,153),(88,145,95,152),(89,144,96,151),(90,143,97,150),(91,142,98,149),(113,170,120,177),(114,169,121,176),(115,182,122,175),(116,181,123,174),(117,180,124,173),(118,179,125,172),(119,178,126,171)], [(1,150,198,26),(2,151,199,27),(3,152,200,28),(4,153,201,15),(5,154,202,16),(6,141,203,17),(7,142,204,18),(8,143,205,19),(9,144,206,20),(10,145,207,21),(11,146,208,22),(12,147,209,23),(13,148,210,24),(14,149,197,25),(29,185,211,120),(30,186,212,121),(31,187,213,122),(32,188,214,123),(33,189,215,124),(34,190,216,125),(35,191,217,126),(36,192,218,113),(37,193,219,114),(38,194,220,115),(39,195,221,116),(40,196,222,117),(41,183,223,118),(42,184,224,119),(43,101,128,87),(44,102,129,88),(45,103,130,89),(46,104,131,90),(47,105,132,91),(48,106,133,92),(49,107,134,93),(50,108,135,94),(51,109,136,95),(52,110,137,96),(53,111,138,97),(54,112,139,98),(55,99,140,85),(56,100,127,86),(57,174,163,78),(58,175,164,79),(59,176,165,80),(60,177,166,81),(61,178,167,82),(62,179,168,83),(63,180,155,84),(64,181,156,71),(65,182,157,72),(66,169,158,73),(67,170,159,74),(68,171,160,75),(69,172,161,76),(70,173,162,77)], [(1,90),(2,89),(3,88),(4,87),(5,86),(6,85),(7,98),(8,97),(9,96),(10,95),(11,94),(12,93),(13,92),(14,91),(15,50),(16,49),(17,48),(18,47),(19,46),(20,45),(21,44),(22,43),(23,56),(24,55),(25,54),(26,53),(27,52),(28,51),(29,177),(30,176),(31,175),(32,174),(33,173),(34,172),(35,171),(36,170),(37,169),(38,182),(39,181),(40,180),(41,179),(42,178),(57,195),(58,194),(59,193),(60,192),(61,191),(62,190),(63,189),(64,188),(65,187),(66,186),(67,185),(68,184),(69,183),(70,196),(71,221),(72,220),(73,219),(74,218),(75,217),(76,216),(77,215),(78,214),(79,213),(80,212),(81,211),(82,224),(83,223),(84,222),(99,203),(100,202),(101,201),(102,200),(103,199),(104,198),(105,197),(106,210),(107,209),(108,208),(109,207),(110,206),(111,205),(112,204),(113,166),(114,165),(115,164),(116,163),(117,162),(118,161),(119,160),(120,159),(121,158),(122,157),(123,156),(124,155),(125,168),(126,167),(127,147),(128,146),(129,145),(130,144),(131,143),(132,142),(133,141),(134,154),(135,153),(136,152),(137,151),(138,150),(139,149),(140,148)])
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 | 0 | 0 | 1 | 8 |
| 0 | 0 | 0 | 0 | 12 | 10 |
| 8 | 11 | 0 | 0 | 0 | 0 |
| 18 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 18 | 7 | 8 |
| 0 | 0 | 11 | 2 | 3 | 14 |
| 0 | 0 | 0 | 0 | 21 | 11 |
| 0 | 0 | 0 | 0 | 2 | 8 |
| 11 | 8 | 0 | 0 | 0 | 0 |
| 21 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 16 | 0 | 0 |
| 0 | 0 | 2 | 5 | 0 | 0 |
| 0 | 0 | 28 | 12 | 15 | 2 |
| 0 | 0 | 22 | 4 | 3 | 14 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 11 | 4 |
| 0 | 0 | 0 | 28 | 25 | 0 |
| 0 | 0 | 0 | 15 | 1 | 0 |
| 0 | 0 | 14 | 24 | 0 | 1 |
| 21 | 18 | 0 | 0 | 0 | 0 |
| 11 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 16 | 3 | 20 |
| 0 | 0 | 2 | 5 | 27 | 21 |
| 0 | 0 | 28 | 12 | 15 | 2 |
| 0 | 0 | 22 | 4 | 3 | 14 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,3,21,0,0,0,0,8,8,0,0,0,0,0,0,1,12,0,0,0,0,8,10],[8,18,0,0,0,0,11,21,0,0,0,0,0,0,27,11,0,0,0,0,18,2,0,0,0,0,7,3,21,2,0,0,8,14,11,8],[11,21,0,0,0,0,8,18,0,0,0,0,0,0,24,2,28,22,0,0,16,5,12,4,0,0,0,0,15,3,0,0,0,0,2,14],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,14,0,0,0,28,15,24,0,0,11,25,1,0,0,0,4,0,0,1],[21,11,0,0,0,0,18,8,0,0,0,0,0,0,24,2,28,22,0,0,16,5,12,4,0,0,3,27,15,3,0,0,20,21,2,14] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 4 | 4 | 28 | 28 | 4 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4×D7 | D4⋊6D14 | D4.10D14 |
| kernel | C14.322+ (1+4) | C22⋊Dic14 | D14⋊D4 | D14⋊Q8 | C2×Dic7⋊C4 | C28⋊7D4 | Dic7⋊D4 | C7×C4⋊D4 | C2×D4⋊2D7 | C2×Dic7 | C4⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C14 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 2 | 4 | 3 | 6 | 3 | 3 | 9 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{32}2_+^{(1+4)} % in TeX
G:=Group("C14.32ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1050);
// by ID
G=gap.SmallGroup(448,1050);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,387,1123,570,185,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations
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