?
G = C14.1072- (1+4) order 448 = 26·7
62nd non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.1072- (1+4), (C7×Q8)⋊18D4, Q8⋊9(C7⋊D4), C7⋊7(Q8⋊5D4), C28⋊2D4⋊42C2, (Q8×Dic7)⋊29C2, C28.267(C2×D4), D14⋊16(C4○D4), (C2×D4).236D14, (C2×Q8).210D14, C28.48D4⋊40C2, C28.17D4⋊31C2, (C2×C14).317C24, (C2×C28).561C23, (C22×C4).287D14, C14.167(C22×D4), D14⋊C4.164C22, (D4×C14).276C22, C4⋊Dic7.260C22, (Q8×C14).243C22, C22.326(C23×D7), C23.138(C22×D7), C23.D7.76C22, Dic7⋊C4.101C22, (C22×C28).296C22, (C22×C14).243C23, (C4×Dic7).176C22, (C2×Dic7).164C23, (C22×D7).244C23, C2.70(D4.10D14), (C2×Dic14).204C22, (C2×Q8×D7)⋊19C2, (C2×C4○D4)⋊9D7, (C14×C4○D4)⋊9C2, (C4×C7⋊D4)⋊30C2, C4.73(C2×C7⋊D4), C2.105(D7×C4○D4), C14.217(C2×C4○D4), (C2×C4×D7).169C22, C2.40(C22×C7⋊D4), (C2×C4).640(C22×D7), (C2×C7⋊D4).141C22, SmallGroup(448,1284)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1140 in 290 conjugacy classes, 113 normal (22 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×13], C7, C2×C4, C2×C4 [×3], C2×C4 [×19], D4 [×12], Q8 [×4], Q8 [×6], C23 [×3], C23, D7 [×2], C14 [×3], C14 [×3], C42 [×3], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic7 [×7], C28 [×6], C28, D14 [×2], D14 [×2], C2×C14, C2×C14 [×9], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, Dic14 [×6], C4×D7 [×6], C2×Dic7, C2×Dic7 [×6], C7⋊D4 [×6], C2×C28, C2×C28 [×3], C2×C28 [×6], C7×D4 [×6], C7×Q8 [×4], C22×D7, C22×C14 [×3], Q8⋊5D4, C4×Dic7 [×3], Dic7⋊C4 [×3], C4⋊Dic7 [×3], D14⋊C4, C23.D7 [×9], C2×Dic14 [×3], C2×C4×D7 [×3], Q8×D7 [×4], C2×C7⋊D4 [×3], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×4], C28.48D4 [×3], C4×C7⋊D4 [×3], C28.17D4 [×3], C28⋊2D4 [×3], Q8×Dic7, C2×Q8×D7, C14×C4○D4, C14.1072- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2- (1+4), C7⋊D4 [×4], C22×D7 [×7], Q8⋊5D4, C2×C7⋊D4 [×6], C23×D7, D7×C4○D4, D4.10D14, C22×C7⋊D4, C14.1072- (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=b-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)
(1 224 140 72)(2 223 127 71)(3 222 128 84)(4 221 129 83)(5 220 130 82)(6 219 131 81)(7 218 132 80)(8 217 133 79)(9 216 134 78)(10 215 135 77)(11 214 136 76)(12 213 137 75)(13 212 138 74)(14 211 139 73)(15 207 170 113)(16 206 171 126)(17 205 172 125)(18 204 173 124)(19 203 174 123)(20 202 175 122)(21 201 176 121)(22 200 177 120)(23 199 178 119)(24 198 179 118)(25 197 180 117)(26 210 181 116)(27 209 182 115)(28 208 169 114)(29 67 108 158)(30 66 109 157)(31 65 110 156)(32 64 111 155)(33 63 112 168)(34 62 99 167)(35 61 100 166)(36 60 101 165)(37 59 102 164)(38 58 103 163)(39 57 104 162)(40 70 105 161)(41 69 106 160)(42 68 107 159)(43 94 146 187)(44 93 147 186)(45 92 148 185)(46 91 149 184)(47 90 150 183)(48 89 151 196)(49 88 152 195)(50 87 153 194)(51 86 154 193)(52 85 141 192)(53 98 142 191)(54 97 143 190)(55 96 144 189)(56 95 145 188)
(1 224)(2 211)(3 212)(4 213)(5 214)(6 215)(7 216)(8 217)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 205)(16 206)(17 207)(18 208)(19 209)(20 210)(21 197)(22 198)(23 199)(24 200)(25 201)(26 202)(27 203)(28 204)(29 160)(30 161)(31 162)(32 163)(33 164)(34 165)(35 166)(36 167)(37 168)(38 155)(39 156)(40 157)(41 158)(42 159)(43 195)(44 196)(45 183)(46 184)(47 185)(48 186)(49 187)(50 188)(51 189)(52 190)(53 191)(54 192)(55 193)(56 194)(57 110)(58 111)(59 112)(60 99)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 139)(72 140)(73 127)(74 128)(75 129)(76 130)(77 131)(78 132)(79 133)(80 134)(81 135)(82 136)(83 137)(84 138)(85 143)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 141)(98 142)(113 172)(114 173)(115 174)(116 175)(117 176)(118 177)(119 178)(120 179)(121 180)(122 181)(123 182)(124 169)(125 170)(126 171)
(1 61 133 159)(2 60 134 158)(3 59 135 157)(4 58 136 156)(5 57 137 155)(6 70 138 168)(7 69 139 167)(8 68 140 166)(9 67 127 165)(10 66 128 164)(11 65 129 163)(12 64 130 162)(13 63 131 161)(14 62 132 160)(15 92 177 192)(16 91 178 191)(17 90 179 190)(18 89 180 189)(19 88 181 188)(20 87 182 187)(21 86 169 186)(22 85 170 185)(23 98 171 184)(24 97 172 183)(25 96 173 196)(26 95 174 195)(27 94 175 194)(28 93 176 193)(29 216 101 71)(30 215 102 84)(31 214 103 83)(32 213 104 82)(33 212 105 81)(34 211 106 80)(35 224 107 79)(36 223 108 78)(37 222 109 77)(38 221 110 76)(39 220 111 75)(40 219 112 74)(41 218 99 73)(42 217 100 72)(43 209 153 122)(44 208 154 121)(45 207 141 120)(46 206 142 119)(47 205 143 118)(48 204 144 117)(49 203 145 116)(50 202 146 115)(51 201 147 114)(52 200 148 113)(53 199 149 126)(54 198 150 125)(55 197 151 124)(56 210 152 123)
(1 98 133 184)(2 85 134 185)(3 86 135 186)(4 87 136 187)(5 88 137 188)(6 89 138 189)(7 90 139 190)(8 91 140 191)(9 92 127 192)(10 93 128 193)(11 94 129 194)(12 95 130 195)(13 96 131 196)(14 97 132 183)(15 165 177 67)(16 166 178 68)(17 167 179 69)(18 168 180 70)(19 155 181 57)(20 156 182 58)(21 157 169 59)(22 158 170 60)(23 159 171 61)(24 160 172 62)(25 161 173 63)(26 162 174 64)(27 163 175 65)(28 164 176 66)(29 113 101 200)(30 114 102 201)(31 115 103 202)(32 116 104 203)(33 117 105 204)(34 118 106 205)(35 119 107 206)(36 120 108 207)(37 121 109 208)(38 122 110 209)(39 123 111 210)(40 124 112 197)(41 125 99 198)(42 126 100 199)(43 221 153 76)(44 222 154 77)(45 223 141 78)(46 224 142 79)(47 211 143 80)(48 212 144 81)(49 213 145 82)(50 214 146 83)(51 215 147 84)(52 216 148 71)(53 217 149 72)(54 218 150 73)(55 219 151 74)(56 220 152 75)
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), (1,224,140,72)(2,223,127,71)(3,222,128,84)(4,221,129,83)(5,220,130,82)(6,219,131,81)(7,218,132,80)(8,217,133,79)(9,216,134,78)(10,215,135,77)(11,214,136,76)(12,213,137,75)(13,212,138,74)(14,211,139,73)(15,207,170,113)(16,206,171,126)(17,205,172,125)(18,204,173,124)(19,203,174,123)(20,202,175,122)(21,201,176,121)(22,200,177,120)(23,199,178,119)(24,198,179,118)(25,197,180,117)(26,210,181,116)(27,209,182,115)(28,208,169,114)(29,67,108,158)(30,66,109,157)(31,65,110,156)(32,64,111,155)(33,63,112,168)(34,62,99,167)(35,61,100,166)(36,60,101,165)(37,59,102,164)(38,58,103,163)(39,57,104,162)(40,70,105,161)(41,69,106,160)(42,68,107,159)(43,94,146,187)(44,93,147,186)(45,92,148,185)(46,91,149,184)(47,90,150,183)(48,89,151,196)(49,88,152,195)(50,87,153,194)(51,86,154,193)(52,85,141,192)(53,98,142,191)(54,97,143,190)(55,96,144,189)(56,95,145,188), (1,224)(2,211)(3,212)(4,213)(5,214)(6,215)(7,216)(8,217)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,197)(22,198)(23,199)(24,200)(25,201)(26,202)(27,203)(28,204)(29,160)(30,161)(31,162)(32,163)(33,164)(34,165)(35,166)(36,167)(37,168)(38,155)(39,156)(40,157)(41,158)(42,159)(43,195)(44,196)(45,183)(46,184)(47,185)(48,186)(49,187)(50,188)(51,189)(52,190)(53,191)(54,192)(55,193)(56,194)(57,110)(58,111)(59,112)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,139)(72,140)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135)(82,136)(83,137)(84,138)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,141)(98,142)(113,172)(114,173)(115,174)(116,175)(117,176)(118,177)(119,178)(120,179)(121,180)(122,181)(123,182)(124,169)(125,170)(126,171), (1,61,133,159)(2,60,134,158)(3,59,135,157)(4,58,136,156)(5,57,137,155)(6,70,138,168)(7,69,139,167)(8,68,140,166)(9,67,127,165)(10,66,128,164)(11,65,129,163)(12,64,130,162)(13,63,131,161)(14,62,132,160)(15,92,177,192)(16,91,178,191)(17,90,179,190)(18,89,180,189)(19,88,181,188)(20,87,182,187)(21,86,169,186)(22,85,170,185)(23,98,171,184)(24,97,172,183)(25,96,173,196)(26,95,174,195)(27,94,175,194)(28,93,176,193)(29,216,101,71)(30,215,102,84)(31,214,103,83)(32,213,104,82)(33,212,105,81)(34,211,106,80)(35,224,107,79)(36,223,108,78)(37,222,109,77)(38,221,110,76)(39,220,111,75)(40,219,112,74)(41,218,99,73)(42,217,100,72)(43,209,153,122)(44,208,154,121)(45,207,141,120)(46,206,142,119)(47,205,143,118)(48,204,144,117)(49,203,145,116)(50,202,146,115)(51,201,147,114)(52,200,148,113)(53,199,149,126)(54,198,150,125)(55,197,151,124)(56,210,152,123), (1,98,133,184)(2,85,134,185)(3,86,135,186)(4,87,136,187)(5,88,137,188)(6,89,138,189)(7,90,139,190)(8,91,140,191)(9,92,127,192)(10,93,128,193)(11,94,129,194)(12,95,130,195)(13,96,131,196)(14,97,132,183)(15,165,177,67)(16,166,178,68)(17,167,179,69)(18,168,180,70)(19,155,181,57)(20,156,182,58)(21,157,169,59)(22,158,170,60)(23,159,171,61)(24,160,172,62)(25,161,173,63)(26,162,174,64)(27,163,175,65)(28,164,176,66)(29,113,101,200)(30,114,102,201)(31,115,103,202)(32,116,104,203)(33,117,105,204)(34,118,106,205)(35,119,107,206)(36,120,108,207)(37,121,109,208)(38,122,110,209)(39,123,111,210)(40,124,112,197)(41,125,99,198)(42,126,100,199)(43,221,153,76)(44,222,154,77)(45,223,141,78)(46,224,142,79)(47,211,143,80)(48,212,144,81)(49,213,145,82)(50,214,146,83)(51,215,147,84)(52,216,148,71)(53,217,149,72)(54,218,150,73)(55,219,151,74)(56,220,152,75)>;
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), (1,224,140,72)(2,223,127,71)(3,222,128,84)(4,221,129,83)(5,220,130,82)(6,219,131,81)(7,218,132,80)(8,217,133,79)(9,216,134,78)(10,215,135,77)(11,214,136,76)(12,213,137,75)(13,212,138,74)(14,211,139,73)(15,207,170,113)(16,206,171,126)(17,205,172,125)(18,204,173,124)(19,203,174,123)(20,202,175,122)(21,201,176,121)(22,200,177,120)(23,199,178,119)(24,198,179,118)(25,197,180,117)(26,210,181,116)(27,209,182,115)(28,208,169,114)(29,67,108,158)(30,66,109,157)(31,65,110,156)(32,64,111,155)(33,63,112,168)(34,62,99,167)(35,61,100,166)(36,60,101,165)(37,59,102,164)(38,58,103,163)(39,57,104,162)(40,70,105,161)(41,69,106,160)(42,68,107,159)(43,94,146,187)(44,93,147,186)(45,92,148,185)(46,91,149,184)(47,90,150,183)(48,89,151,196)(49,88,152,195)(50,87,153,194)(51,86,154,193)(52,85,141,192)(53,98,142,191)(54,97,143,190)(55,96,144,189)(56,95,145,188), (1,224)(2,211)(3,212)(4,213)(5,214)(6,215)(7,216)(8,217)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,197)(22,198)(23,199)(24,200)(25,201)(26,202)(27,203)(28,204)(29,160)(30,161)(31,162)(32,163)(33,164)(34,165)(35,166)(36,167)(37,168)(38,155)(39,156)(40,157)(41,158)(42,159)(43,195)(44,196)(45,183)(46,184)(47,185)(48,186)(49,187)(50,188)(51,189)(52,190)(53,191)(54,192)(55,193)(56,194)(57,110)(58,111)(59,112)(60,99)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,139)(72,140)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135)(82,136)(83,137)(84,138)(85,143)(86,144)(87,145)(88,146)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,141)(98,142)(113,172)(114,173)(115,174)(116,175)(117,176)(118,177)(119,178)(120,179)(121,180)(122,181)(123,182)(124,169)(125,170)(126,171), (1,61,133,159)(2,60,134,158)(3,59,135,157)(4,58,136,156)(5,57,137,155)(6,70,138,168)(7,69,139,167)(8,68,140,166)(9,67,127,165)(10,66,128,164)(11,65,129,163)(12,64,130,162)(13,63,131,161)(14,62,132,160)(15,92,177,192)(16,91,178,191)(17,90,179,190)(18,89,180,189)(19,88,181,188)(20,87,182,187)(21,86,169,186)(22,85,170,185)(23,98,171,184)(24,97,172,183)(25,96,173,196)(26,95,174,195)(27,94,175,194)(28,93,176,193)(29,216,101,71)(30,215,102,84)(31,214,103,83)(32,213,104,82)(33,212,105,81)(34,211,106,80)(35,224,107,79)(36,223,108,78)(37,222,109,77)(38,221,110,76)(39,220,111,75)(40,219,112,74)(41,218,99,73)(42,217,100,72)(43,209,153,122)(44,208,154,121)(45,207,141,120)(46,206,142,119)(47,205,143,118)(48,204,144,117)(49,203,145,116)(50,202,146,115)(51,201,147,114)(52,200,148,113)(53,199,149,126)(54,198,150,125)(55,197,151,124)(56,210,152,123), (1,98,133,184)(2,85,134,185)(3,86,135,186)(4,87,136,187)(5,88,137,188)(6,89,138,189)(7,90,139,190)(8,91,140,191)(9,92,127,192)(10,93,128,193)(11,94,129,194)(12,95,130,195)(13,96,131,196)(14,97,132,183)(15,165,177,67)(16,166,178,68)(17,167,179,69)(18,168,180,70)(19,155,181,57)(20,156,182,58)(21,157,169,59)(22,158,170,60)(23,159,171,61)(24,160,172,62)(25,161,173,63)(26,162,174,64)(27,163,175,65)(28,164,176,66)(29,113,101,200)(30,114,102,201)(31,115,103,202)(32,116,104,203)(33,117,105,204)(34,118,106,205)(35,119,107,206)(36,120,108,207)(37,121,109,208)(38,122,110,209)(39,123,111,210)(40,124,112,197)(41,125,99,198)(42,126,100,199)(43,221,153,76)(44,222,154,77)(45,223,141,78)(46,224,142,79)(47,211,143,80)(48,212,144,81)(49,213,145,82)(50,214,146,83)(51,215,147,84)(52,216,148,71)(53,217,149,72)(54,218,150,73)(55,219,151,74)(56,220,152,75) );
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)], [(1,224,140,72),(2,223,127,71),(3,222,128,84),(4,221,129,83),(5,220,130,82),(6,219,131,81),(7,218,132,80),(8,217,133,79),(9,216,134,78),(10,215,135,77),(11,214,136,76),(12,213,137,75),(13,212,138,74),(14,211,139,73),(15,207,170,113),(16,206,171,126),(17,205,172,125),(18,204,173,124),(19,203,174,123),(20,202,175,122),(21,201,176,121),(22,200,177,120),(23,199,178,119),(24,198,179,118),(25,197,180,117),(26,210,181,116),(27,209,182,115),(28,208,169,114),(29,67,108,158),(30,66,109,157),(31,65,110,156),(32,64,111,155),(33,63,112,168),(34,62,99,167),(35,61,100,166),(36,60,101,165),(37,59,102,164),(38,58,103,163),(39,57,104,162),(40,70,105,161),(41,69,106,160),(42,68,107,159),(43,94,146,187),(44,93,147,186),(45,92,148,185),(46,91,149,184),(47,90,150,183),(48,89,151,196),(49,88,152,195),(50,87,153,194),(51,86,154,193),(52,85,141,192),(53,98,142,191),(54,97,143,190),(55,96,144,189),(56,95,145,188)], [(1,224),(2,211),(3,212),(4,213),(5,214),(6,215),(7,216),(8,217),(9,218),(10,219),(11,220),(12,221),(13,222),(14,223),(15,205),(16,206),(17,207),(18,208),(19,209),(20,210),(21,197),(22,198),(23,199),(24,200),(25,201),(26,202),(27,203),(28,204),(29,160),(30,161),(31,162),(32,163),(33,164),(34,165),(35,166),(36,167),(37,168),(38,155),(39,156),(40,157),(41,158),(42,159),(43,195),(44,196),(45,183),(46,184),(47,185),(48,186),(49,187),(50,188),(51,189),(52,190),(53,191),(54,192),(55,193),(56,194),(57,110),(58,111),(59,112),(60,99),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,139),(72,140),(73,127),(74,128),(75,129),(76,130),(77,131),(78,132),(79,133),(80,134),(81,135),(82,136),(83,137),(84,138),(85,143),(86,144),(87,145),(88,146),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,153),(96,154),(97,141),(98,142),(113,172),(114,173),(115,174),(116,175),(117,176),(118,177),(119,178),(120,179),(121,180),(122,181),(123,182),(124,169),(125,170),(126,171)], [(1,61,133,159),(2,60,134,158),(3,59,135,157),(4,58,136,156),(5,57,137,155),(6,70,138,168),(7,69,139,167),(8,68,140,166),(9,67,127,165),(10,66,128,164),(11,65,129,163),(12,64,130,162),(13,63,131,161),(14,62,132,160),(15,92,177,192),(16,91,178,191),(17,90,179,190),(18,89,180,189),(19,88,181,188),(20,87,182,187),(21,86,169,186),(22,85,170,185),(23,98,171,184),(24,97,172,183),(25,96,173,196),(26,95,174,195),(27,94,175,194),(28,93,176,193),(29,216,101,71),(30,215,102,84),(31,214,103,83),(32,213,104,82),(33,212,105,81),(34,211,106,80),(35,224,107,79),(36,223,108,78),(37,222,109,77),(38,221,110,76),(39,220,111,75),(40,219,112,74),(41,218,99,73),(42,217,100,72),(43,209,153,122),(44,208,154,121),(45,207,141,120),(46,206,142,119),(47,205,143,118),(48,204,144,117),(49,203,145,116),(50,202,146,115),(51,201,147,114),(52,200,148,113),(53,199,149,126),(54,198,150,125),(55,197,151,124),(56,210,152,123)], [(1,98,133,184),(2,85,134,185),(3,86,135,186),(4,87,136,187),(5,88,137,188),(6,89,138,189),(7,90,139,190),(8,91,140,191),(9,92,127,192),(10,93,128,193),(11,94,129,194),(12,95,130,195),(13,96,131,196),(14,97,132,183),(15,165,177,67),(16,166,178,68),(17,167,179,69),(18,168,180,70),(19,155,181,57),(20,156,182,58),(21,157,169,59),(22,158,170,60),(23,159,171,61),(24,160,172,62),(25,161,173,63),(26,162,174,64),(27,163,175,65),(28,164,176,66),(29,113,101,200),(30,114,102,201),(31,115,103,202),(32,116,104,203),(33,117,105,204),(34,118,106,205),(35,119,107,206),(36,120,108,207),(37,121,109,208),(38,122,110,209),(39,123,111,210),(40,124,112,197),(41,125,99,198),(42,126,100,199),(43,221,153,76),(44,222,154,77),(45,223,141,78),(46,224,142,79),(47,211,143,80),(48,212,144,81),(49,213,145,82),(50,214,146,83),(51,215,147,84),(52,216,148,71),(53,217,149,72),(54,218,150,73),(55,219,151,74),(56,220,152,75)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 8 | 8 | 0 | 0 |
| 21 | 3 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 5 | 16 | 0 | 0 |
| 2 | 24 | 0 | 0 |
| 0 | 0 | 26 | 20 |
| 0 | 0 | 17 | 3 |
| 5 | 16 | 0 | 0 |
| 13 | 24 | 0 | 0 |
| 0 | 0 | 26 | 20 |
| 0 | 0 | 17 | 3 |
| 28 | 0 | 0 | 0 |
| 26 | 1 | 0 | 0 |
| 0 | 0 | 28 | 28 |
| 0 | 0 | 2 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 22 | 8 |
| 0 | 0 | 1 | 7 |
G:=sub<GL(4,GF(29))| [8,21,0,0,8,3,0,0,0,0,28,0,0,0,0,28],[5,2,0,0,16,24,0,0,0,0,26,17,0,0,20,3],[5,13,0,0,16,24,0,0,0,0,26,17,0,0,20,3],[28,26,0,0,0,1,0,0,0,0,28,2,0,0,28,1],[1,0,0,0,0,1,0,0,0,0,22,1,0,0,8,7] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4H | 4I | 4J | 4K | ··· | 4P | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 14 | 14 | 2 | ··· | 2 | 14 | 14 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | 2- (1+4) | D7×C4○D4 | D4.10D14 |
| kernel | C14.1072- (1+4) | C28.48D4 | C4×C7⋊D4 | C28.17D4 | C28⋊2D4 | Q8×Dic7 | C2×Q8×D7 | C14×C4○D4 | C7×Q8 | C2×C4○D4 | D14 | C22×C4 | C2×D4 | C2×Q8 | Q8 | C14 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 4 | 3 | 4 | 9 | 9 | 3 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{107}2_-^{(1+4)} % in TeX
G:=Group("C14.107ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1284);
// by ID
G=gap.SmallGroup(448,1284);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,387,184,675,136,18822]);
// 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=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
×
⋊
ℤ
𝔽
○
≀