?
G = C14.152- (1+4) order 448 = 26·7
15th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.152- (1+4), C4⋊C4.188D14, C22⋊Q8.9D7, (Q8×Dic7)⋊12C2, Dic7.Q8⋊17C2, (C2×C28).50C23, (C2×Q8).123D14, C22⋊C4.13D14, C4.Dic14⋊23C2, Dic7⋊Q8⋊13C2, Dic7⋊3Q8⋊24C2, C28.209(C4○D4), C4.72(D4⋊2D7), (C2×C14).169C24, (C22×C4).233D14, C28.48D4.16C2, Dic7⋊C4.24C22, C4⋊Dic7.311C22, (Q8×C14).104C22, (C2×Dic7).84C23, C23.D14.2C2, C22.190(C23×D7), C23.116(C22×D7), (C22×C14).197C23, (C22×C28).249C22, C7⋊3(C22.35C24), (C4×Dic7).103C22, C23.D7.114C22, C2.34(D4.10D14), C2.16(Q8.10D14), (C2×Dic14).159C22, C23.21D14.24C2, C14.89(C2×C4○D4), (C7×C22⋊Q8).9C2, C2.45(C2×D4⋊2D7), (C7×C4⋊C4).155C22, (C2×C4).182(C22×D7), (C7×C22⋊C4).24C22, SmallGroup(448,1078)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 668 in 192 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×13], C22, C22 [×3], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×4], C23, C14 [×3], C14, C42 [×6], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×17], C22×C4, C2×Q8, C2×Q8, Dic7 [×8], C28 [×2], C28 [×5], C2×C14, C2×C14 [×3], C42⋊C2, C4×Q8 [×2], C22⋊Q8, C22⋊Q8, C42.C2 [×5], C42⋊2C2 [×4], C4⋊Q8, Dic14 [×2], C2×Dic7 [×4], C2×Dic7 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×C14, C22.35C24, C4×Dic7 [×2], C4×Dic7 [×4], Dic7⋊C4 [×10], C4⋊Dic7 [×3], C4⋊Dic7 [×4], C23.D7 [×2], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14, C22×C28, Q8×C14, C23.D14 [×4], Dic7⋊3Q8, Dic7.Q8 [×2], C4.Dic14, C4.Dic14 [×2], C28.48D4, C23.21D14, Dic7⋊Q8, Q8×Dic7, C7×C22⋊Q8, C14.152- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2- (1+4) [×2], C22×D7 [×7], C22.35C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, Q8.10D14, D4.10D14, C14.152- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=b-1, dbd-1=a7b, be=eb, 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 78 18 65)(2 79 19 66)(3 80 20 67)(4 81 21 68)(5 82 22 69)(6 83 23 70)(7 84 24 57)(8 71 25 58)(9 72 26 59)(10 73 27 60)(11 74 28 61)(12 75 15 62)(13 76 16 63)(14 77 17 64)(29 109 53 87)(30 110 54 88)(31 111 55 89)(32 112 56 90)(33 99 43 91)(34 100 44 92)(35 101 45 93)(36 102 46 94)(37 103 47 95)(38 104 48 96)(39 105 49 97)(40 106 50 98)(41 107 51 85)(42 108 52 86)(113 177 130 196)(114 178 131 183)(115 179 132 184)(116 180 133 185)(117 181 134 186)(118 182 135 187)(119 169 136 188)(120 170 137 189)(121 171 138 190)(122 172 139 191)(123 173 140 192)(124 174 127 193)(125 175 128 194)(126 176 129 195)(141 197 156 222)(142 198 157 223)(143 199 158 224)(144 200 159 211)(145 201 160 212)(146 202 161 213)(147 203 162 214)(148 204 163 215)(149 205 164 216)(150 206 165 217)(151 207 166 218)(152 208 167 219)(153 209 168 220)(154 210 155 221)
(1 190 8 183)(2 189 9 196)(3 188 10 195)(4 187 11 194)(5 186 12 193)(6 185 13 192)(7 184 14 191)(15 174 22 181)(16 173 23 180)(17 172 24 179)(18 171 25 178)(19 170 26 177)(20 169 27 176)(21 182 28 175)(29 223 36 216)(30 222 37 215)(31 221 38 214)(32 220 39 213)(33 219 40 212)(34 218 41 211)(35 217 42 224)(43 208 50 201)(44 207 51 200)(45 206 52 199)(46 205 53 198)(47 204 54 197)(48 203 55 210)(49 202 56 209)(57 115 64 122)(58 114 65 121)(59 113 66 120)(60 126 67 119)(61 125 68 118)(62 124 69 117)(63 123 70 116)(71 131 78 138)(72 130 79 137)(73 129 80 136)(74 128 81 135)(75 127 82 134)(76 140 83 133)(77 139 84 132)(85 144 92 151)(86 143 93 150)(87 142 94 149)(88 141 95 148)(89 154 96 147)(90 153 97 146)(91 152 98 145)(99 167 106 160)(100 166 107 159)(101 165 108 158)(102 164 109 157)(103 163 110 156)(104 162 111 155)(105 161 112 168)
(1 152 18 167)(2 151 19 166)(3 150 20 165)(4 149 21 164)(5 148 22 163)(6 147 23 162)(7 146 24 161)(8 145 25 160)(9 144 26 159)(10 143 27 158)(11 142 28 157)(12 141 15 156)(13 154 16 155)(14 153 17 168)(29 135 53 118)(30 134 54 117)(31 133 55 116)(32 132 56 115)(33 131 43 114)(34 130 44 113)(35 129 45 126)(36 128 46 125)(37 127 47 124)(38 140 48 123)(39 139 49 122)(40 138 50 121)(41 137 51 120)(42 136 52 119)(57 220 84 209)(58 219 71 208)(59 218 72 207)(60 217 73 206)(61 216 74 205)(62 215 75 204)(63 214 76 203)(64 213 77 202)(65 212 78 201)(66 211 79 200)(67 224 80 199)(68 223 81 198)(69 222 82 197)(70 221 83 210)(85 177 107 196)(86 176 108 195)(87 175 109 194)(88 174 110 193)(89 173 111 192)(90 172 112 191)(91 171 99 190)(92 170 100 189)(93 169 101 188)(94 182 102 187)(95 181 103 186)(96 180 104 185)(97 179 105 184)(98 178 106 183)
(1 71 18 58)(2 72 19 59)(3 73 20 60)(4 74 21 61)(5 75 22 62)(6 76 23 63)(7 77 24 64)(8 78 25 65)(9 79 26 66)(10 80 27 67)(11 81 28 68)(12 82 15 69)(13 83 16 70)(14 84 17 57)(29 109 53 87)(30 110 54 88)(31 111 55 89)(32 112 56 90)(33 99 43 91)(34 100 44 92)(35 101 45 93)(36 102 46 94)(37 103 47 95)(38 104 48 96)(39 105 49 97)(40 106 50 98)(41 107 51 85)(42 108 52 86)(113 189 130 170)(114 190 131 171)(115 191 132 172)(116 192 133 173)(117 193 134 174)(118 194 135 175)(119 195 136 176)(120 196 137 177)(121 183 138 178)(122 184 139 179)(123 185 140 180)(124 186 127 181)(125 187 128 182)(126 188 129 169)(141 222 156 197)(142 223 157 198)(143 224 158 199)(144 211 159 200)(145 212 160 201)(146 213 161 202)(147 214 162 203)(148 215 163 204)(149 216 164 205)(150 217 165 206)(151 218 166 207)(152 219 167 208)(153 220 168 209)(154 221 155 210)
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), (1,78,18,65)(2,79,19,66)(3,80,20,67)(4,81,21,68)(5,82,22,69)(6,83,23,70)(7,84,24,57)(8,71,25,58)(9,72,26,59)(10,73,27,60)(11,74,28,61)(12,75,15,62)(13,76,16,63)(14,77,17,64)(29,109,53,87)(30,110,54,88)(31,111,55,89)(32,112,56,90)(33,99,43,91)(34,100,44,92)(35,101,45,93)(36,102,46,94)(37,103,47,95)(38,104,48,96)(39,105,49,97)(40,106,50,98)(41,107,51,85)(42,108,52,86)(113,177,130,196)(114,178,131,183)(115,179,132,184)(116,180,133,185)(117,181,134,186)(118,182,135,187)(119,169,136,188)(120,170,137,189)(121,171,138,190)(122,172,139,191)(123,173,140,192)(124,174,127,193)(125,175,128,194)(126,176,129,195)(141,197,156,222)(142,198,157,223)(143,199,158,224)(144,200,159,211)(145,201,160,212)(146,202,161,213)(147,203,162,214)(148,204,163,215)(149,205,164,216)(150,206,165,217)(151,207,166,218)(152,208,167,219)(153,209,168,220)(154,210,155,221), (1,190,8,183)(2,189,9,196)(3,188,10,195)(4,187,11,194)(5,186,12,193)(6,185,13,192)(7,184,14,191)(15,174,22,181)(16,173,23,180)(17,172,24,179)(18,171,25,178)(19,170,26,177)(20,169,27,176)(21,182,28,175)(29,223,36,216)(30,222,37,215)(31,221,38,214)(32,220,39,213)(33,219,40,212)(34,218,41,211)(35,217,42,224)(43,208,50,201)(44,207,51,200)(45,206,52,199)(46,205,53,198)(47,204,54,197)(48,203,55,210)(49,202,56,209)(57,115,64,122)(58,114,65,121)(59,113,66,120)(60,126,67,119)(61,125,68,118)(62,124,69,117)(63,123,70,116)(71,131,78,138)(72,130,79,137)(73,129,80,136)(74,128,81,135)(75,127,82,134)(76,140,83,133)(77,139,84,132)(85,144,92,151)(86,143,93,150)(87,142,94,149)(88,141,95,148)(89,154,96,147)(90,153,97,146)(91,152,98,145)(99,167,106,160)(100,166,107,159)(101,165,108,158)(102,164,109,157)(103,163,110,156)(104,162,111,155)(105,161,112,168), (1,152,18,167)(2,151,19,166)(3,150,20,165)(4,149,21,164)(5,148,22,163)(6,147,23,162)(7,146,24,161)(8,145,25,160)(9,144,26,159)(10,143,27,158)(11,142,28,157)(12,141,15,156)(13,154,16,155)(14,153,17,168)(29,135,53,118)(30,134,54,117)(31,133,55,116)(32,132,56,115)(33,131,43,114)(34,130,44,113)(35,129,45,126)(36,128,46,125)(37,127,47,124)(38,140,48,123)(39,139,49,122)(40,138,50,121)(41,137,51,120)(42,136,52,119)(57,220,84,209)(58,219,71,208)(59,218,72,207)(60,217,73,206)(61,216,74,205)(62,215,75,204)(63,214,76,203)(64,213,77,202)(65,212,78,201)(66,211,79,200)(67,224,80,199)(68,223,81,198)(69,222,82,197)(70,221,83,210)(85,177,107,196)(86,176,108,195)(87,175,109,194)(88,174,110,193)(89,173,111,192)(90,172,112,191)(91,171,99,190)(92,170,100,189)(93,169,101,188)(94,182,102,187)(95,181,103,186)(96,180,104,185)(97,179,105,184)(98,178,106,183), (1,71,18,58)(2,72,19,59)(3,73,20,60)(4,74,21,61)(5,75,22,62)(6,76,23,63)(7,77,24,64)(8,78,25,65)(9,79,26,66)(10,80,27,67)(11,81,28,68)(12,82,15,69)(13,83,16,70)(14,84,17,57)(29,109,53,87)(30,110,54,88)(31,111,55,89)(32,112,56,90)(33,99,43,91)(34,100,44,92)(35,101,45,93)(36,102,46,94)(37,103,47,95)(38,104,48,96)(39,105,49,97)(40,106,50,98)(41,107,51,85)(42,108,52,86)(113,189,130,170)(114,190,131,171)(115,191,132,172)(116,192,133,173)(117,193,134,174)(118,194,135,175)(119,195,136,176)(120,196,137,177)(121,183,138,178)(122,184,139,179)(123,185,140,180)(124,186,127,181)(125,187,128,182)(126,188,129,169)(141,222,156,197)(142,223,157,198)(143,224,158,199)(144,211,159,200)(145,212,160,201)(146,213,161,202)(147,214,162,203)(148,215,163,204)(149,216,164,205)(150,217,165,206)(151,218,166,207)(152,219,167,208)(153,220,168,209)(154,221,155,210)>;
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), (1,78,18,65)(2,79,19,66)(3,80,20,67)(4,81,21,68)(5,82,22,69)(6,83,23,70)(7,84,24,57)(8,71,25,58)(9,72,26,59)(10,73,27,60)(11,74,28,61)(12,75,15,62)(13,76,16,63)(14,77,17,64)(29,109,53,87)(30,110,54,88)(31,111,55,89)(32,112,56,90)(33,99,43,91)(34,100,44,92)(35,101,45,93)(36,102,46,94)(37,103,47,95)(38,104,48,96)(39,105,49,97)(40,106,50,98)(41,107,51,85)(42,108,52,86)(113,177,130,196)(114,178,131,183)(115,179,132,184)(116,180,133,185)(117,181,134,186)(118,182,135,187)(119,169,136,188)(120,170,137,189)(121,171,138,190)(122,172,139,191)(123,173,140,192)(124,174,127,193)(125,175,128,194)(126,176,129,195)(141,197,156,222)(142,198,157,223)(143,199,158,224)(144,200,159,211)(145,201,160,212)(146,202,161,213)(147,203,162,214)(148,204,163,215)(149,205,164,216)(150,206,165,217)(151,207,166,218)(152,208,167,219)(153,209,168,220)(154,210,155,221), (1,190,8,183)(2,189,9,196)(3,188,10,195)(4,187,11,194)(5,186,12,193)(6,185,13,192)(7,184,14,191)(15,174,22,181)(16,173,23,180)(17,172,24,179)(18,171,25,178)(19,170,26,177)(20,169,27,176)(21,182,28,175)(29,223,36,216)(30,222,37,215)(31,221,38,214)(32,220,39,213)(33,219,40,212)(34,218,41,211)(35,217,42,224)(43,208,50,201)(44,207,51,200)(45,206,52,199)(46,205,53,198)(47,204,54,197)(48,203,55,210)(49,202,56,209)(57,115,64,122)(58,114,65,121)(59,113,66,120)(60,126,67,119)(61,125,68,118)(62,124,69,117)(63,123,70,116)(71,131,78,138)(72,130,79,137)(73,129,80,136)(74,128,81,135)(75,127,82,134)(76,140,83,133)(77,139,84,132)(85,144,92,151)(86,143,93,150)(87,142,94,149)(88,141,95,148)(89,154,96,147)(90,153,97,146)(91,152,98,145)(99,167,106,160)(100,166,107,159)(101,165,108,158)(102,164,109,157)(103,163,110,156)(104,162,111,155)(105,161,112,168), (1,152,18,167)(2,151,19,166)(3,150,20,165)(4,149,21,164)(5,148,22,163)(6,147,23,162)(7,146,24,161)(8,145,25,160)(9,144,26,159)(10,143,27,158)(11,142,28,157)(12,141,15,156)(13,154,16,155)(14,153,17,168)(29,135,53,118)(30,134,54,117)(31,133,55,116)(32,132,56,115)(33,131,43,114)(34,130,44,113)(35,129,45,126)(36,128,46,125)(37,127,47,124)(38,140,48,123)(39,139,49,122)(40,138,50,121)(41,137,51,120)(42,136,52,119)(57,220,84,209)(58,219,71,208)(59,218,72,207)(60,217,73,206)(61,216,74,205)(62,215,75,204)(63,214,76,203)(64,213,77,202)(65,212,78,201)(66,211,79,200)(67,224,80,199)(68,223,81,198)(69,222,82,197)(70,221,83,210)(85,177,107,196)(86,176,108,195)(87,175,109,194)(88,174,110,193)(89,173,111,192)(90,172,112,191)(91,171,99,190)(92,170,100,189)(93,169,101,188)(94,182,102,187)(95,181,103,186)(96,180,104,185)(97,179,105,184)(98,178,106,183), (1,71,18,58)(2,72,19,59)(3,73,20,60)(4,74,21,61)(5,75,22,62)(6,76,23,63)(7,77,24,64)(8,78,25,65)(9,79,26,66)(10,80,27,67)(11,81,28,68)(12,82,15,69)(13,83,16,70)(14,84,17,57)(29,109,53,87)(30,110,54,88)(31,111,55,89)(32,112,56,90)(33,99,43,91)(34,100,44,92)(35,101,45,93)(36,102,46,94)(37,103,47,95)(38,104,48,96)(39,105,49,97)(40,106,50,98)(41,107,51,85)(42,108,52,86)(113,189,130,170)(114,190,131,171)(115,191,132,172)(116,192,133,173)(117,193,134,174)(118,194,135,175)(119,195,136,176)(120,196,137,177)(121,183,138,178)(122,184,139,179)(123,185,140,180)(124,186,127,181)(125,187,128,182)(126,188,129,169)(141,222,156,197)(142,223,157,198)(143,224,158,199)(144,211,159,200)(145,212,160,201)(146,213,161,202)(147,214,162,203)(148,215,163,204)(149,216,164,205)(150,217,165,206)(151,218,166,207)(152,219,167,208)(153,220,168,209)(154,221,155,210) );
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)], [(1,78,18,65),(2,79,19,66),(3,80,20,67),(4,81,21,68),(5,82,22,69),(6,83,23,70),(7,84,24,57),(8,71,25,58),(9,72,26,59),(10,73,27,60),(11,74,28,61),(12,75,15,62),(13,76,16,63),(14,77,17,64),(29,109,53,87),(30,110,54,88),(31,111,55,89),(32,112,56,90),(33,99,43,91),(34,100,44,92),(35,101,45,93),(36,102,46,94),(37,103,47,95),(38,104,48,96),(39,105,49,97),(40,106,50,98),(41,107,51,85),(42,108,52,86),(113,177,130,196),(114,178,131,183),(115,179,132,184),(116,180,133,185),(117,181,134,186),(118,182,135,187),(119,169,136,188),(120,170,137,189),(121,171,138,190),(122,172,139,191),(123,173,140,192),(124,174,127,193),(125,175,128,194),(126,176,129,195),(141,197,156,222),(142,198,157,223),(143,199,158,224),(144,200,159,211),(145,201,160,212),(146,202,161,213),(147,203,162,214),(148,204,163,215),(149,205,164,216),(150,206,165,217),(151,207,166,218),(152,208,167,219),(153,209,168,220),(154,210,155,221)], [(1,190,8,183),(2,189,9,196),(3,188,10,195),(4,187,11,194),(5,186,12,193),(6,185,13,192),(7,184,14,191),(15,174,22,181),(16,173,23,180),(17,172,24,179),(18,171,25,178),(19,170,26,177),(20,169,27,176),(21,182,28,175),(29,223,36,216),(30,222,37,215),(31,221,38,214),(32,220,39,213),(33,219,40,212),(34,218,41,211),(35,217,42,224),(43,208,50,201),(44,207,51,200),(45,206,52,199),(46,205,53,198),(47,204,54,197),(48,203,55,210),(49,202,56,209),(57,115,64,122),(58,114,65,121),(59,113,66,120),(60,126,67,119),(61,125,68,118),(62,124,69,117),(63,123,70,116),(71,131,78,138),(72,130,79,137),(73,129,80,136),(74,128,81,135),(75,127,82,134),(76,140,83,133),(77,139,84,132),(85,144,92,151),(86,143,93,150),(87,142,94,149),(88,141,95,148),(89,154,96,147),(90,153,97,146),(91,152,98,145),(99,167,106,160),(100,166,107,159),(101,165,108,158),(102,164,109,157),(103,163,110,156),(104,162,111,155),(105,161,112,168)], [(1,152,18,167),(2,151,19,166),(3,150,20,165),(4,149,21,164),(5,148,22,163),(6,147,23,162),(7,146,24,161),(8,145,25,160),(9,144,26,159),(10,143,27,158),(11,142,28,157),(12,141,15,156),(13,154,16,155),(14,153,17,168),(29,135,53,118),(30,134,54,117),(31,133,55,116),(32,132,56,115),(33,131,43,114),(34,130,44,113),(35,129,45,126),(36,128,46,125),(37,127,47,124),(38,140,48,123),(39,139,49,122),(40,138,50,121),(41,137,51,120),(42,136,52,119),(57,220,84,209),(58,219,71,208),(59,218,72,207),(60,217,73,206),(61,216,74,205),(62,215,75,204),(63,214,76,203),(64,213,77,202),(65,212,78,201),(66,211,79,200),(67,224,80,199),(68,223,81,198),(69,222,82,197),(70,221,83,210),(85,177,107,196),(86,176,108,195),(87,175,109,194),(88,174,110,193),(89,173,111,192),(90,172,112,191),(91,171,99,190),(92,170,100,189),(93,169,101,188),(94,182,102,187),(95,181,103,186),(96,180,104,185),(97,179,105,184),(98,178,106,183)], [(1,71,18,58),(2,72,19,59),(3,73,20,60),(4,74,21,61),(5,75,22,62),(6,76,23,63),(7,77,24,64),(8,78,25,65),(9,79,26,66),(10,80,27,67),(11,81,28,68),(12,82,15,69),(13,83,16,70),(14,84,17,57),(29,109,53,87),(30,110,54,88),(31,111,55,89),(32,112,56,90),(33,99,43,91),(34,100,44,92),(35,101,45,93),(36,102,46,94),(37,103,47,95),(38,104,48,96),(39,105,49,97),(40,106,50,98),(41,107,51,85),(42,108,52,86),(113,189,130,170),(114,190,131,171),(115,191,132,172),(116,192,133,173),(117,193,134,174),(118,194,135,175),(119,195,136,176),(120,196,137,177),(121,183,138,178),(122,184,139,179),(123,185,140,180),(124,186,127,181),(125,187,128,182),(126,188,129,169),(141,222,156,197),(142,223,157,198),(143,224,158,199),(144,211,159,200),(145,212,160,201),(146,213,161,202),(147,214,162,203),(148,215,163,204),(149,216,164,205),(150,217,165,206),(151,218,166,207),(152,219,167,208),(153,220,168,209),(154,221,155,210)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 4 | 0 | 0 |
| 0 | 0 | 5 | 21 | 0 | 0 |
| 0 | 0 | 25 | 0 | 0 | 4 |
| 0 | 0 | 8 | 4 | 7 | 22 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 5 | 0 | 0 |
| 0 | 0 | 28 | 27 | 0 | 0 |
| 0 | 0 | 24 | 0 | 8 | 5 |
| 0 | 0 | 10 | 5 | 16 | 21 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 17 | 11 | 26 |
| 0 | 0 | 12 | 7 | 5 | 26 |
| 0 | 0 | 3 | 0 | 21 | 26 |
| 0 | 0 | 17 | 17 | 23 | 5 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 11 | 0 | 0 |
| 0 | 0 | 18 | 27 | 0 | 0 |
| 0 | 0 | 16 | 0 | 4 | 13 |
| 0 | 0 | 6 | 11 | 21 | 25 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 24 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 10 | 15 | 8 | 5 |
| 0 | 0 | 14 | 9 | 16 | 21 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,5,25,8,0,0,4,21,0,4,0,0,0,0,0,7,0,0,0,0,4,22],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,2,28,24,10,0,0,5,27,0,5,0,0,0,0,8,16,0,0,0,0,5,21],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,25,12,3,17,0,0,17,7,0,17,0,0,11,5,21,23,0,0,26,26,26,5],[0,12,0,0,0,0,17,0,0,0,0,0,0,0,2,18,16,6,0,0,11,27,0,11,0,0,0,0,4,21,0,0,0,0,13,25],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,27,1,10,14,0,0,24,2,15,9,0,0,0,0,8,16,0,0,0,0,5,21] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 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 | 4 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2- (1+4) | D4⋊2D7 | Q8.10D14 | D4.10D14 |
| kernel | C14.152- (1+4) | C23.D14 | Dic7⋊3Q8 | Dic7.Q8 | C4.Dic14 | C28.48D4 | C23.21D14 | Dic7⋊Q8 | Q8×Dic7 | C7×C22⋊Q8 | C22⋊Q8 | C28 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C2 | C2 |
| # reps | 1 | 4 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 6 | 9 | 3 | 3 | 2 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{15}2_-^{(1+4)} % in TeX
G:=Group("C14.15ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1078);
// by ID
G=gap.SmallGroup(448,1078);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,120,219,268,675,297,18822]);
// 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=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=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=b^2*d>;
// generators/relations
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