?
G = C14.252- (1+4) order 448 = 26·7
25th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.252- (1+4), C14.582+ (1+4), C28⋊Q8⋊29C2, C22⋊Q8⋊23D7, C4⋊C4.101D14, (C2×Q8).79D14, D14⋊3Q8⋊25C2, D14⋊Q8⋊24C2, Dic7.Q8⋊22C2, D14⋊C4.8C22, C22⋊C4.23D14, C4.Dic14⋊27C2, Dic7⋊Q8⋊18C2, C28.48D4⋊47C2, (C2×C28).176C23, (C2×C14).190C24, D14.D4.3C2, (C22×C4).252D14, C2.60(D4⋊6D14), C23.D14⋊26C2, Dic7⋊C4.81C22, C4⋊Dic7.222C22, Dic7.D4.3C2, (Q8×C14).119C22, (C2×Dic7).96C23, (C22×D7).81C23, C22.211(C23×D7), C23.126(C22×D7), C23.D7.36C22, (C22×C28).318C22, (C22×C14).218C23, C7⋊1(C22.57C24), (C4×Dic7).117C22, (C2×Dic14).35C22, C23.23D14.2C2, C2.39(D4.10D14), C2.26(Q8.10D14), C4⋊C4⋊D7⋊25C2, (C7×C22⋊Q8)⋊26C2, (C2×C4×D7).106C22, (C7×C4⋊C4).170C22, (C2×C4).187(C22×D7), (C2×C7⋊D4).42C22, (C7×C22⋊C4).45C22, SmallGroup(448,1099)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 828 in 196 conjugacy classes, 91 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4 [×6], C2×C4 [×9], D4, Q8 [×3], C23, C23, D7, C14 [×3], C14, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8 [×2], Dic7 [×7], C28 [×6], D14 [×3], C2×C14, C2×C14 [×3], C22⋊Q8, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], Dic14 [×2], C4×D7, C2×Dic7 [×7], C7⋊D4, C2×C28 [×6], C2×C28, C7×Q8, C22×D7, C22×C14, C22.57C24, C4×Dic7 [×3], Dic7⋊C4 [×9], C4⋊Dic7 [×4], D14⋊C4 [×5], C23.D7 [×3], C7×C22⋊C4 [×2], C7×C4⋊C4 [×3], C2×Dic14 [×2], C2×C4×D7, C2×C7⋊D4, C22×C28, Q8×C14, C23.D14 [×2], D14.D4, Dic7.D4, C28⋊Q8, Dic7.Q8, C4.Dic14, D14⋊Q8, C4⋊C4⋊D7 [×2], C28.48D4, C23.23D14, Dic7⋊Q8, D14⋊3Q8, C7×C22⋊Q8, C14.252- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C24, D14 [×7], 2+ (1+4), 2- (1+4) [×2], C22×D7 [×7], C22.57C24, C23×D7, D4⋊6D14, Q8.10D14, D4.10D14, C14.252- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=a7b-1, bd=db, ebe-1=a7b, dcd-1=a7c, ce=ec, ede-1=b2d >
►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)
(1 188 158 66)(2 189 159 67)(3 190 160 68)(4 191 161 69)(5 192 162 70)(6 193 163 57)(7 194 164 58)(8 195 165 59)(9 196 166 60)(10 183 167 61)(11 184 168 62)(12 185 155 63)(13 186 156 64)(14 187 157 65)(15 71 56 176)(16 72 43 177)(17 73 44 178)(18 74 45 179)(19 75 46 180)(20 76 47 181)(21 77 48 182)(22 78 49 169)(23 79 50 170)(24 80 51 171)(25 81 52 172)(26 82 53 173)(27 83 54 174)(28 84 55 175)(29 203 116 95)(30 204 117 96)(31 205 118 97)(32 206 119 98)(33 207 120 85)(34 208 121 86)(35 209 122 87)(36 210 123 88)(37 197 124 89)(38 198 125 90)(39 199 126 91)(40 200 113 92)(41 201 114 93)(42 202 115 94)(99 222 152 132)(100 223 153 133)(101 224 154 134)(102 211 141 135)(103 212 142 136)(104 213 143 137)(105 214 144 138)(106 215 145 139)(107 216 146 140)(108 217 147 127)(109 218 148 128)(110 219 149 129)(111 220 150 130)(112 221 151 131)
(1 35 8 42)(2 34 9 41)(3 33 10 40)(4 32 11 39)(5 31 12 38)(6 30 13 37)(7 29 14 36)(15 219 22 212)(16 218 23 211)(17 217 24 224)(18 216 25 223)(19 215 26 222)(20 214 27 221)(21 213 28 220)(43 128 50 135)(44 127 51 134)(45 140 52 133)(46 139 53 132)(47 138 54 131)(48 137 55 130)(49 136 56 129)(57 197 64 204)(58 210 65 203)(59 209 66 202)(60 208 67 201)(61 207 68 200)(62 206 69 199)(63 205 70 198)(71 103 78 110)(72 102 79 109)(73 101 80 108)(74 100 81 107)(75 99 82 106)(76 112 83 105)(77 111 84 104)(85 190 92 183)(86 189 93 196)(87 188 94 195)(88 187 95 194)(89 186 96 193)(90 185 97 192)(91 184 98 191)(113 160 120 167)(114 159 121 166)(115 158 122 165)(116 157 123 164)(117 156 124 163)(118 155 125 162)(119 168 126 161)(141 170 148 177)(142 169 149 176)(143 182 150 175)(144 181 151 174)(145 180 152 173)(146 179 153 172)(147 178 154 171)
(1 188 158 66)(2 189 159 67)(3 190 160 68)(4 191 161 69)(5 192 162 70)(6 193 163 57)(7 194 164 58)(8 195 165 59)(9 196 166 60)(10 183 167 61)(11 184 168 62)(12 185 155 63)(13 186 156 64)(14 187 157 65)(15 78 56 169)(16 79 43 170)(17 80 44 171)(18 81 45 172)(19 82 46 173)(20 83 47 174)(21 84 48 175)(22 71 49 176)(23 72 50 177)(24 73 51 178)(25 74 52 179)(26 75 53 180)(27 76 54 181)(28 77 55 182)(29 95 116 203)(30 96 117 204)(31 97 118 205)(32 98 119 206)(33 85 120 207)(34 86 121 208)(35 87 122 209)(36 88 123 210)(37 89 124 197)(38 90 125 198)(39 91 126 199)(40 92 113 200)(41 93 114 201)(42 94 115 202)(99 139 152 215)(100 140 153 216)(101 127 154 217)(102 128 141 218)(103 129 142 219)(104 130 143 220)(105 131 144 221)(106 132 145 222)(107 133 146 223)(108 134 147 224)(109 135 148 211)(110 136 149 212)(111 137 150 213)(112 138 151 214)
(1 214 158 138)(2 213 159 137)(3 212 160 136)(4 211 161 135)(5 224 162 134)(6 223 163 133)(7 222 164 132)(8 221 165 131)(9 220 166 130)(10 219 167 129)(11 218 168 128)(12 217 155 127)(13 216 156 140)(14 215 157 139)(15 120 56 33)(16 119 43 32)(17 118 44 31)(18 117 45 30)(19 116 46 29)(20 115 47 42)(21 114 48 41)(22 113 49 40)(23 126 50 39)(24 125 51 38)(25 124 52 37)(26 123 53 36)(27 122 54 35)(28 121 55 34)(57 107 193 146)(58 106 194 145)(59 105 195 144)(60 104 196 143)(61 103 183 142)(62 102 184 141)(63 101 185 154)(64 100 186 153)(65 99 187 152)(66 112 188 151)(67 111 189 150)(68 110 190 149)(69 109 191 148)(70 108 192 147)(71 92 176 200)(72 91 177 199)(73 90 178 198)(74 89 179 197)(75 88 180 210)(76 87 181 209)(77 86 182 208)(78 85 169 207)(79 98 170 206)(80 97 171 205)(81 96 172 204)(82 95 173 203)(83 94 174 202)(84 93 175 201)
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), (1,188,158,66)(2,189,159,67)(3,190,160,68)(4,191,161,69)(5,192,162,70)(6,193,163,57)(7,194,164,58)(8,195,165,59)(9,196,166,60)(10,183,167,61)(11,184,168,62)(12,185,155,63)(13,186,156,64)(14,187,157,65)(15,71,56,176)(16,72,43,177)(17,73,44,178)(18,74,45,179)(19,75,46,180)(20,76,47,181)(21,77,48,182)(22,78,49,169)(23,79,50,170)(24,80,51,171)(25,81,52,172)(26,82,53,173)(27,83,54,174)(28,84,55,175)(29,203,116,95)(30,204,117,96)(31,205,118,97)(32,206,119,98)(33,207,120,85)(34,208,121,86)(35,209,122,87)(36,210,123,88)(37,197,124,89)(38,198,125,90)(39,199,126,91)(40,200,113,92)(41,201,114,93)(42,202,115,94)(99,222,152,132)(100,223,153,133)(101,224,154,134)(102,211,141,135)(103,212,142,136)(104,213,143,137)(105,214,144,138)(106,215,145,139)(107,216,146,140)(108,217,147,127)(109,218,148,128)(110,219,149,129)(111,220,150,130)(112,221,151,131), (1,35,8,42)(2,34,9,41)(3,33,10,40)(4,32,11,39)(5,31,12,38)(6,30,13,37)(7,29,14,36)(15,219,22,212)(16,218,23,211)(17,217,24,224)(18,216,25,223)(19,215,26,222)(20,214,27,221)(21,213,28,220)(43,128,50,135)(44,127,51,134)(45,140,52,133)(46,139,53,132)(47,138,54,131)(48,137,55,130)(49,136,56,129)(57,197,64,204)(58,210,65,203)(59,209,66,202)(60,208,67,201)(61,207,68,200)(62,206,69,199)(63,205,70,198)(71,103,78,110)(72,102,79,109)(73,101,80,108)(74,100,81,107)(75,99,82,106)(76,112,83,105)(77,111,84,104)(85,190,92,183)(86,189,93,196)(87,188,94,195)(88,187,95,194)(89,186,96,193)(90,185,97,192)(91,184,98,191)(113,160,120,167)(114,159,121,166)(115,158,122,165)(116,157,123,164)(117,156,124,163)(118,155,125,162)(119,168,126,161)(141,170,148,177)(142,169,149,176)(143,182,150,175)(144,181,151,174)(145,180,152,173)(146,179,153,172)(147,178,154,171), (1,188,158,66)(2,189,159,67)(3,190,160,68)(4,191,161,69)(5,192,162,70)(6,193,163,57)(7,194,164,58)(8,195,165,59)(9,196,166,60)(10,183,167,61)(11,184,168,62)(12,185,155,63)(13,186,156,64)(14,187,157,65)(15,78,56,169)(16,79,43,170)(17,80,44,171)(18,81,45,172)(19,82,46,173)(20,83,47,174)(21,84,48,175)(22,71,49,176)(23,72,50,177)(24,73,51,178)(25,74,52,179)(26,75,53,180)(27,76,54,181)(28,77,55,182)(29,95,116,203)(30,96,117,204)(31,97,118,205)(32,98,119,206)(33,85,120,207)(34,86,121,208)(35,87,122,209)(36,88,123,210)(37,89,124,197)(38,90,125,198)(39,91,126,199)(40,92,113,200)(41,93,114,201)(42,94,115,202)(99,139,152,215)(100,140,153,216)(101,127,154,217)(102,128,141,218)(103,129,142,219)(104,130,143,220)(105,131,144,221)(106,132,145,222)(107,133,146,223)(108,134,147,224)(109,135,148,211)(110,136,149,212)(111,137,150,213)(112,138,151,214), (1,214,158,138)(2,213,159,137)(3,212,160,136)(4,211,161,135)(5,224,162,134)(6,223,163,133)(7,222,164,132)(8,221,165,131)(9,220,166,130)(10,219,167,129)(11,218,168,128)(12,217,155,127)(13,216,156,140)(14,215,157,139)(15,120,56,33)(16,119,43,32)(17,118,44,31)(18,117,45,30)(19,116,46,29)(20,115,47,42)(21,114,48,41)(22,113,49,40)(23,126,50,39)(24,125,51,38)(25,124,52,37)(26,123,53,36)(27,122,54,35)(28,121,55,34)(57,107,193,146)(58,106,194,145)(59,105,195,144)(60,104,196,143)(61,103,183,142)(62,102,184,141)(63,101,185,154)(64,100,186,153)(65,99,187,152)(66,112,188,151)(67,111,189,150)(68,110,190,149)(69,109,191,148)(70,108,192,147)(71,92,176,200)(72,91,177,199)(73,90,178,198)(74,89,179,197)(75,88,180,210)(76,87,181,209)(77,86,182,208)(78,85,169,207)(79,98,170,206)(80,97,171,205)(81,96,172,204)(82,95,173,203)(83,94,174,202)(84,93,175,201)>;
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), (1,188,158,66)(2,189,159,67)(3,190,160,68)(4,191,161,69)(5,192,162,70)(6,193,163,57)(7,194,164,58)(8,195,165,59)(9,196,166,60)(10,183,167,61)(11,184,168,62)(12,185,155,63)(13,186,156,64)(14,187,157,65)(15,71,56,176)(16,72,43,177)(17,73,44,178)(18,74,45,179)(19,75,46,180)(20,76,47,181)(21,77,48,182)(22,78,49,169)(23,79,50,170)(24,80,51,171)(25,81,52,172)(26,82,53,173)(27,83,54,174)(28,84,55,175)(29,203,116,95)(30,204,117,96)(31,205,118,97)(32,206,119,98)(33,207,120,85)(34,208,121,86)(35,209,122,87)(36,210,123,88)(37,197,124,89)(38,198,125,90)(39,199,126,91)(40,200,113,92)(41,201,114,93)(42,202,115,94)(99,222,152,132)(100,223,153,133)(101,224,154,134)(102,211,141,135)(103,212,142,136)(104,213,143,137)(105,214,144,138)(106,215,145,139)(107,216,146,140)(108,217,147,127)(109,218,148,128)(110,219,149,129)(111,220,150,130)(112,221,151,131), (1,35,8,42)(2,34,9,41)(3,33,10,40)(4,32,11,39)(5,31,12,38)(6,30,13,37)(7,29,14,36)(15,219,22,212)(16,218,23,211)(17,217,24,224)(18,216,25,223)(19,215,26,222)(20,214,27,221)(21,213,28,220)(43,128,50,135)(44,127,51,134)(45,140,52,133)(46,139,53,132)(47,138,54,131)(48,137,55,130)(49,136,56,129)(57,197,64,204)(58,210,65,203)(59,209,66,202)(60,208,67,201)(61,207,68,200)(62,206,69,199)(63,205,70,198)(71,103,78,110)(72,102,79,109)(73,101,80,108)(74,100,81,107)(75,99,82,106)(76,112,83,105)(77,111,84,104)(85,190,92,183)(86,189,93,196)(87,188,94,195)(88,187,95,194)(89,186,96,193)(90,185,97,192)(91,184,98,191)(113,160,120,167)(114,159,121,166)(115,158,122,165)(116,157,123,164)(117,156,124,163)(118,155,125,162)(119,168,126,161)(141,170,148,177)(142,169,149,176)(143,182,150,175)(144,181,151,174)(145,180,152,173)(146,179,153,172)(147,178,154,171), (1,188,158,66)(2,189,159,67)(3,190,160,68)(4,191,161,69)(5,192,162,70)(6,193,163,57)(7,194,164,58)(8,195,165,59)(9,196,166,60)(10,183,167,61)(11,184,168,62)(12,185,155,63)(13,186,156,64)(14,187,157,65)(15,78,56,169)(16,79,43,170)(17,80,44,171)(18,81,45,172)(19,82,46,173)(20,83,47,174)(21,84,48,175)(22,71,49,176)(23,72,50,177)(24,73,51,178)(25,74,52,179)(26,75,53,180)(27,76,54,181)(28,77,55,182)(29,95,116,203)(30,96,117,204)(31,97,118,205)(32,98,119,206)(33,85,120,207)(34,86,121,208)(35,87,122,209)(36,88,123,210)(37,89,124,197)(38,90,125,198)(39,91,126,199)(40,92,113,200)(41,93,114,201)(42,94,115,202)(99,139,152,215)(100,140,153,216)(101,127,154,217)(102,128,141,218)(103,129,142,219)(104,130,143,220)(105,131,144,221)(106,132,145,222)(107,133,146,223)(108,134,147,224)(109,135,148,211)(110,136,149,212)(111,137,150,213)(112,138,151,214), (1,214,158,138)(2,213,159,137)(3,212,160,136)(4,211,161,135)(5,224,162,134)(6,223,163,133)(7,222,164,132)(8,221,165,131)(9,220,166,130)(10,219,167,129)(11,218,168,128)(12,217,155,127)(13,216,156,140)(14,215,157,139)(15,120,56,33)(16,119,43,32)(17,118,44,31)(18,117,45,30)(19,116,46,29)(20,115,47,42)(21,114,48,41)(22,113,49,40)(23,126,50,39)(24,125,51,38)(25,124,52,37)(26,123,53,36)(27,122,54,35)(28,121,55,34)(57,107,193,146)(58,106,194,145)(59,105,195,144)(60,104,196,143)(61,103,183,142)(62,102,184,141)(63,101,185,154)(64,100,186,153)(65,99,187,152)(66,112,188,151)(67,111,189,150)(68,110,190,149)(69,109,191,148)(70,108,192,147)(71,92,176,200)(72,91,177,199)(73,90,178,198)(74,89,179,197)(75,88,180,210)(76,87,181,209)(77,86,182,208)(78,85,169,207)(79,98,170,206)(80,97,171,205)(81,96,172,204)(82,95,173,203)(83,94,174,202)(84,93,175,201) );
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)], [(1,188,158,66),(2,189,159,67),(3,190,160,68),(4,191,161,69),(5,192,162,70),(6,193,163,57),(7,194,164,58),(8,195,165,59),(9,196,166,60),(10,183,167,61),(11,184,168,62),(12,185,155,63),(13,186,156,64),(14,187,157,65),(15,71,56,176),(16,72,43,177),(17,73,44,178),(18,74,45,179),(19,75,46,180),(20,76,47,181),(21,77,48,182),(22,78,49,169),(23,79,50,170),(24,80,51,171),(25,81,52,172),(26,82,53,173),(27,83,54,174),(28,84,55,175),(29,203,116,95),(30,204,117,96),(31,205,118,97),(32,206,119,98),(33,207,120,85),(34,208,121,86),(35,209,122,87),(36,210,123,88),(37,197,124,89),(38,198,125,90),(39,199,126,91),(40,200,113,92),(41,201,114,93),(42,202,115,94),(99,222,152,132),(100,223,153,133),(101,224,154,134),(102,211,141,135),(103,212,142,136),(104,213,143,137),(105,214,144,138),(106,215,145,139),(107,216,146,140),(108,217,147,127),(109,218,148,128),(110,219,149,129),(111,220,150,130),(112,221,151,131)], [(1,35,8,42),(2,34,9,41),(3,33,10,40),(4,32,11,39),(5,31,12,38),(6,30,13,37),(7,29,14,36),(15,219,22,212),(16,218,23,211),(17,217,24,224),(18,216,25,223),(19,215,26,222),(20,214,27,221),(21,213,28,220),(43,128,50,135),(44,127,51,134),(45,140,52,133),(46,139,53,132),(47,138,54,131),(48,137,55,130),(49,136,56,129),(57,197,64,204),(58,210,65,203),(59,209,66,202),(60,208,67,201),(61,207,68,200),(62,206,69,199),(63,205,70,198),(71,103,78,110),(72,102,79,109),(73,101,80,108),(74,100,81,107),(75,99,82,106),(76,112,83,105),(77,111,84,104),(85,190,92,183),(86,189,93,196),(87,188,94,195),(88,187,95,194),(89,186,96,193),(90,185,97,192),(91,184,98,191),(113,160,120,167),(114,159,121,166),(115,158,122,165),(116,157,123,164),(117,156,124,163),(118,155,125,162),(119,168,126,161),(141,170,148,177),(142,169,149,176),(143,182,150,175),(144,181,151,174),(145,180,152,173),(146,179,153,172),(147,178,154,171)], [(1,188,158,66),(2,189,159,67),(3,190,160,68),(4,191,161,69),(5,192,162,70),(6,193,163,57),(7,194,164,58),(8,195,165,59),(9,196,166,60),(10,183,167,61),(11,184,168,62),(12,185,155,63),(13,186,156,64),(14,187,157,65),(15,78,56,169),(16,79,43,170),(17,80,44,171),(18,81,45,172),(19,82,46,173),(20,83,47,174),(21,84,48,175),(22,71,49,176),(23,72,50,177),(24,73,51,178),(25,74,52,179),(26,75,53,180),(27,76,54,181),(28,77,55,182),(29,95,116,203),(30,96,117,204),(31,97,118,205),(32,98,119,206),(33,85,120,207),(34,86,121,208),(35,87,122,209),(36,88,123,210),(37,89,124,197),(38,90,125,198),(39,91,126,199),(40,92,113,200),(41,93,114,201),(42,94,115,202),(99,139,152,215),(100,140,153,216),(101,127,154,217),(102,128,141,218),(103,129,142,219),(104,130,143,220),(105,131,144,221),(106,132,145,222),(107,133,146,223),(108,134,147,224),(109,135,148,211),(110,136,149,212),(111,137,150,213),(112,138,151,214)], [(1,214,158,138),(2,213,159,137),(3,212,160,136),(4,211,161,135),(5,224,162,134),(6,223,163,133),(7,222,164,132),(8,221,165,131),(9,220,166,130),(10,219,167,129),(11,218,168,128),(12,217,155,127),(13,216,156,140),(14,215,157,139),(15,120,56,33),(16,119,43,32),(17,118,44,31),(18,117,45,30),(19,116,46,29),(20,115,47,42),(21,114,48,41),(22,113,49,40),(23,126,50,39),(24,125,51,38),(25,124,52,37),(26,123,53,36),(27,122,54,35),(28,121,55,34),(57,107,193,146),(58,106,194,145),(59,105,195,144),(60,104,196,143),(61,103,183,142),(62,102,184,141),(63,101,185,154),(64,100,186,153),(65,99,187,152),(66,112,188,151),(67,111,189,150),(68,110,190,149),(69,109,191,148),(70,108,192,147),(71,92,176,200),(72,91,177,199),(73,90,178,198),(74,89,179,197),(75,88,180,210),(76,87,181,209),(77,86,182,208),(78,85,169,207),(79,98,170,206),(80,97,171,205),(81,96,172,204),(82,95,173,203),(83,94,174,202),(84,93,175,201)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 0 | 22 | 0 | 0 | 0 | 0 | 0 |
| 15 | 0 | 0 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 13 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 26 | 0 | 9 | 17 |
| 1 | 0 | 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 8 | 28 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 0 | 27 | 15 |
| 0 | 0 | 0 | 0 | 19 | 0 | 1 | 20 |
| 0 | 0 | 0 | 0 | 0 | 5 | 9 | 5 |
| 0 | 0 | 0 | 0 | 10 | 20 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 3 | 0 | 20 | 12 |
| 28 | 0 | 0 | 22 | 0 | 0 | 0 | 0 |
| 21 | 0 | 28 | 1 | 0 | 0 | 0 | 0 |
| 8 | 28 | 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 9 | 5 |
| 0 | 0 | 0 | 0 | 19 | 0 | 1 | 20 |
G:=sub<GL(8,GF(29))| [4,0,14,15,0,0,0,0,0,4,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,9,14,0,15,0,0,0,0,0,13,0,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,0,13],[1,0,21,8,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,17,26,21,26,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,9,0,0,0,0,0,0,0,17],[1,0,21,8,0,0,0,0,0,0,0,28,0,0,0,0,22,1,28,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,20,19,0,10,0,0,0,0,0,0,5,20,0,0,0,0,27,1,9,0,0,0,0,0,15,20,5,0],[1,8,21,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,26,0,3,0,0,0,0,0,12,0,0,0,0,0,0,0,0,17,20,0,0,0,0,0,0,0,12],[28,21,8,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,22,1,28,1,0,0,0,0,0,0,0,0,9,10,0,19,0,0,0,0,15,20,5,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,5,20] >;
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | ··· | 4M | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | 4 | ··· | 4 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | 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 | Q8.10D14 | D4.10D14 |
| kernel | C14.252- (1+4) | C23.D14 | D14.D4 | Dic7.D4 | C28⋊Q8 | Dic7.Q8 | C4.Dic14 | D14⋊Q8 | C4⋊C4⋊D7 | C28.48D4 | C23.23D14 | Dic7⋊Q8 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 6 | 9 | 3 | 3 | 1 | 2 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{25}2_-^{(1+4)} % in TeX
G:=Group("C14.25ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1099);
// by ID
G=gap.SmallGroup(448,1099);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,1571,570,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=a^7*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=b^2*d>;
// generators/relations
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