G = C14.452- 1+4 order 448 = 26·7
45th non-split extension by C14 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.452- 1+4, (C7×Q8)⋊17D4, Q8⋊8(C7⋊D4), C7⋊6(Q8⋊5D4), (C22×Q8)⋊9D7, C28⋊7D4⋊38C2, (Q8×Dic7)⋊27C2, C28.262(C2×D4), D14⋊3Q8⋊42C2, (C2×Q8).189D14, C28.23D4⋊30C2, (C2×C14).309C24, (C2×C28).649C23, D14⋊C4.78C22, C22⋊3(Q8⋊2D7), (C22×C4).279D14, C14.157(C22×D4), (C2×D28).183C22, C4⋊Dic7.258C22, (Q8×C14).236C22, C23.240(C22×D7), C22.320(C23×D7), Dic7⋊C4.171C22, (C22×C28).287C22, (C22×C14).427C23, (C2×Dic7).290C23, (C4×Dic7).172C22, (C22×D7).135C23, C23.D7.146C22, C2.45(Q8.10D14), (Q8×C2×C14)⋊8C2, (C4×C7⋊D4)⋊27C2, C4.70(C2×C7⋊D4), (C2×C14)⋊18(C4○D4), (C2×Q8⋊2D7)⋊18C2, C14.129(C2×C4○D4), C2.36(C2×Q8⋊2D7), (C2×C4×D7).166C22, C2.30(C22×C7⋊D4), (C2×C4).244(C22×D7), (C2×C7⋊D4).139C22, SmallGroup(448,1270)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.452- 1+4
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=a7b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=a7b, be=eb, cd=dc, ce=ec, ede-1=a7b2d >
Subgroups: 1236 in 290 conjugacy classes, 115 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×C14, Q8⋊5D4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C2×C4×D7, C2×D28, Q8⋊2D7, C2×C7⋊D4, C22×C28, Q8×C14, Q8×C14, Q8×C14, C4×C7⋊D4, C28⋊7D4, Q8×Dic7, D14⋊3Q8, C28.23D4, C2×Q8⋊2D7, Q8×C2×C14, C14.452- 1+4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22×D4, C2×C4○D4, 2- 1+4, C7⋊D4, C22×D7, Q8⋊5D4, Q8⋊2D7, C2×C7⋊D4, C23×D7, C2×Q8⋊2D7, Q8.10D14, C22×C7⋊D4, C14.452- 1+4
►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)
(1 104 35 199)(2 103 36 198)(3 102 37 197)(4 101 38 210)(5 100 39 209)(6 99 40 208)(7 112 41 207)(8 111 42 206)(9 110 29 205)(10 109 30 204)(11 108 31 203)(12 107 32 202)(13 106 33 201)(14 105 34 200)(15 215 119 43)(16 214 120 56)(17 213 121 55)(18 212 122 54)(19 211 123 53)(20 224 124 52)(21 223 125 51)(22 222 126 50)(23 221 113 49)(24 220 114 48)(25 219 115 47)(26 218 116 46)(27 217 117 45)(28 216 118 44)(57 146 82 92)(58 145 83 91)(59 144 84 90)(60 143 71 89)(61 142 72 88)(62 141 73 87)(63 154 74 86)(64 153 75 85)(65 152 76 98)(66 151 77 97)(67 150 78 96)(68 149 79 95)(69 148 80 94)(70 147 81 93)(127 180 159 184)(128 179 160 183)(129 178 161 196)(130 177 162 195)(131 176 163 194)(132 175 164 193)(133 174 165 192)(134 173 166 191)(135 172 167 190)(136 171 168 189)(137 170 155 188)(138 169 156 187)(139 182 157 186)(140 181 158 185)
(15 119)(16 120)(17 121)(18 122)(19 123)(20 124)(21 125)(22 126)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(57 82)(58 83)(59 84)(60 71)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(99 208)(100 209)(101 210)(102 197)(103 198)(104 199)(105 200)(106 201)(107 202)(108 203)(109 204)(110 205)(111 206)(112 207)(127 159)(128 160)(129 161)(130 162)(131 163)(132 164)(133 165)(134 166)(135 167)(136 168)(137 155)(138 156)(139 157)(140 158)
(1 220 42 55)(2 221 29 56)(3 222 30 43)(4 223 31 44)(5 224 32 45)(6 211 33 46)(7 212 34 47)(8 213 35 48)(9 214 36 49)(10 215 37 50)(11 216 38 51)(12 217 39 52)(13 218 40 53)(14 219 41 54)(15 109 126 197)(16 110 113 198)(17 111 114 199)(18 112 115 200)(19 99 116 201)(20 100 117 202)(21 101 118 203)(22 102 119 204)(23 103 120 205)(24 104 121 206)(25 105 122 207)(26 106 123 208)(27 107 124 209)(28 108 125 210)(57 167 75 128)(58 168 76 129)(59 155 77 130)(60 156 78 131)(61 157 79 132)(62 158 80 133)(63 159 81 134)(64 160 82 135)(65 161 83 136)(66 162 84 137)(67 163 71 138)(68 164 72 139)(69 165 73 140)(70 166 74 127)(85 172 146 183)(86 173 147 184)(87 174 148 185)(88 175 149 186)(89 176 150 187)(90 177 151 188)(91 178 152 189)(92 179 153 190)(93 180 154 191)(94 181 141 192)(95 182 142 193)(96 169 143 194)(97 170 144 195)(98 171 145 196)
(1 170 42 195)(2 171 29 196)(3 172 30 183)(4 173 31 184)(5 174 32 185)(6 175 33 186)(7 176 34 187)(8 177 35 188)(9 178 36 189)(10 179 37 190)(11 180 38 191)(12 181 39 192)(13 182 40 193)(14 169 41 194)(15 82 126 64)(16 83 113 65)(17 84 114 66)(18 71 115 67)(19 72 116 68)(20 73 117 69)(21 74 118 70)(22 75 119 57)(23 76 120 58)(24 77 121 59)(25 78 122 60)(26 79 123 61)(27 80 124 62)(28 81 125 63)(43 146 222 85)(44 147 223 86)(45 148 224 87)(46 149 211 88)(47 150 212 89)(48 151 213 90)(49 152 214 91)(50 153 215 92)(51 154 216 93)(52 141 217 94)(53 142 218 95)(54 143 219 96)(55 144 220 97)(56 145 221 98)(99 164 201 139)(100 165 202 140)(101 166 203 127)(102 167 204 128)(103 168 205 129)(104 155 206 130)(105 156 207 131)(106 157 208 132)(107 158 209 133)(108 159 210 134)(109 160 197 135)(110 161 198 136)(111 162 199 137)(112 163 200 138)
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), (1,104,35,199)(2,103,36,198)(3,102,37,197)(4,101,38,210)(5,100,39,209)(6,99,40,208)(7,112,41,207)(8,111,42,206)(9,110,29,205)(10,109,30,204)(11,108,31,203)(12,107,32,202)(13,106,33,201)(14,105,34,200)(15,215,119,43)(16,214,120,56)(17,213,121,55)(18,212,122,54)(19,211,123,53)(20,224,124,52)(21,223,125,51)(22,222,126,50)(23,221,113,49)(24,220,114,48)(25,219,115,47)(26,218,116,46)(27,217,117,45)(28,216,118,44)(57,146,82,92)(58,145,83,91)(59,144,84,90)(60,143,71,89)(61,142,72,88)(62,141,73,87)(63,154,74,86)(64,153,75,85)(65,152,76,98)(66,151,77,97)(67,150,78,96)(68,149,79,95)(69,148,80,94)(70,147,81,93)(127,180,159,184)(128,179,160,183)(129,178,161,196)(130,177,162,195)(131,176,163,194)(132,175,164,193)(133,174,165,192)(134,173,166,191)(135,172,167,190)(136,171,168,189)(137,170,155,188)(138,169,156,187)(139,182,157,186)(140,181,158,185), (15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(99,208)(100,209)(101,210)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,155)(138,156)(139,157)(140,158), (1,220,42,55)(2,221,29,56)(3,222,30,43)(4,223,31,44)(5,224,32,45)(6,211,33,46)(7,212,34,47)(8,213,35,48)(9,214,36,49)(10,215,37,50)(11,216,38,51)(12,217,39,52)(13,218,40,53)(14,219,41,54)(15,109,126,197)(16,110,113,198)(17,111,114,199)(18,112,115,200)(19,99,116,201)(20,100,117,202)(21,101,118,203)(22,102,119,204)(23,103,120,205)(24,104,121,206)(25,105,122,207)(26,106,123,208)(27,107,124,209)(28,108,125,210)(57,167,75,128)(58,168,76,129)(59,155,77,130)(60,156,78,131)(61,157,79,132)(62,158,80,133)(63,159,81,134)(64,160,82,135)(65,161,83,136)(66,162,84,137)(67,163,71,138)(68,164,72,139)(69,165,73,140)(70,166,74,127)(85,172,146,183)(86,173,147,184)(87,174,148,185)(88,175,149,186)(89,176,150,187)(90,177,151,188)(91,178,152,189)(92,179,153,190)(93,180,154,191)(94,181,141,192)(95,182,142,193)(96,169,143,194)(97,170,144,195)(98,171,145,196), (1,170,42,195)(2,171,29,196)(3,172,30,183)(4,173,31,184)(5,174,32,185)(6,175,33,186)(7,176,34,187)(8,177,35,188)(9,178,36,189)(10,179,37,190)(11,180,38,191)(12,181,39,192)(13,182,40,193)(14,169,41,194)(15,82,126,64)(16,83,113,65)(17,84,114,66)(18,71,115,67)(19,72,116,68)(20,73,117,69)(21,74,118,70)(22,75,119,57)(23,76,120,58)(24,77,121,59)(25,78,122,60)(26,79,123,61)(27,80,124,62)(28,81,125,63)(43,146,222,85)(44,147,223,86)(45,148,224,87)(46,149,211,88)(47,150,212,89)(48,151,213,90)(49,152,214,91)(50,153,215,92)(51,154,216,93)(52,141,217,94)(53,142,218,95)(54,143,219,96)(55,144,220,97)(56,145,221,98)(99,164,201,139)(100,165,202,140)(101,166,203,127)(102,167,204,128)(103,168,205,129)(104,155,206,130)(105,156,207,131)(106,157,208,132)(107,158,209,133)(108,159,210,134)(109,160,197,135)(110,161,198,136)(111,162,199,137)(112,163,200,138)>;
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), (1,104,35,199)(2,103,36,198)(3,102,37,197)(4,101,38,210)(5,100,39,209)(6,99,40,208)(7,112,41,207)(8,111,42,206)(9,110,29,205)(10,109,30,204)(11,108,31,203)(12,107,32,202)(13,106,33,201)(14,105,34,200)(15,215,119,43)(16,214,120,56)(17,213,121,55)(18,212,122,54)(19,211,123,53)(20,224,124,52)(21,223,125,51)(22,222,126,50)(23,221,113,49)(24,220,114,48)(25,219,115,47)(26,218,116,46)(27,217,117,45)(28,216,118,44)(57,146,82,92)(58,145,83,91)(59,144,84,90)(60,143,71,89)(61,142,72,88)(62,141,73,87)(63,154,74,86)(64,153,75,85)(65,152,76,98)(66,151,77,97)(67,150,78,96)(68,149,79,95)(69,148,80,94)(70,147,81,93)(127,180,159,184)(128,179,160,183)(129,178,161,196)(130,177,162,195)(131,176,163,194)(132,175,164,193)(133,174,165,192)(134,173,166,191)(135,172,167,190)(136,171,168,189)(137,170,155,188)(138,169,156,187)(139,182,157,186)(140,181,158,185), (15,119)(16,120)(17,121)(18,122)(19,123)(20,124)(21,125)(22,126)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(57,82)(58,83)(59,84)(60,71)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,81)(99,208)(100,209)(101,210)(102,197)(103,198)(104,199)(105,200)(106,201)(107,202)(108,203)(109,204)(110,205)(111,206)(112,207)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(133,165)(134,166)(135,167)(136,168)(137,155)(138,156)(139,157)(140,158), (1,220,42,55)(2,221,29,56)(3,222,30,43)(4,223,31,44)(5,224,32,45)(6,211,33,46)(7,212,34,47)(8,213,35,48)(9,214,36,49)(10,215,37,50)(11,216,38,51)(12,217,39,52)(13,218,40,53)(14,219,41,54)(15,109,126,197)(16,110,113,198)(17,111,114,199)(18,112,115,200)(19,99,116,201)(20,100,117,202)(21,101,118,203)(22,102,119,204)(23,103,120,205)(24,104,121,206)(25,105,122,207)(26,106,123,208)(27,107,124,209)(28,108,125,210)(57,167,75,128)(58,168,76,129)(59,155,77,130)(60,156,78,131)(61,157,79,132)(62,158,80,133)(63,159,81,134)(64,160,82,135)(65,161,83,136)(66,162,84,137)(67,163,71,138)(68,164,72,139)(69,165,73,140)(70,166,74,127)(85,172,146,183)(86,173,147,184)(87,174,148,185)(88,175,149,186)(89,176,150,187)(90,177,151,188)(91,178,152,189)(92,179,153,190)(93,180,154,191)(94,181,141,192)(95,182,142,193)(96,169,143,194)(97,170,144,195)(98,171,145,196), (1,170,42,195)(2,171,29,196)(3,172,30,183)(4,173,31,184)(5,174,32,185)(6,175,33,186)(7,176,34,187)(8,177,35,188)(9,178,36,189)(10,179,37,190)(11,180,38,191)(12,181,39,192)(13,182,40,193)(14,169,41,194)(15,82,126,64)(16,83,113,65)(17,84,114,66)(18,71,115,67)(19,72,116,68)(20,73,117,69)(21,74,118,70)(22,75,119,57)(23,76,120,58)(24,77,121,59)(25,78,122,60)(26,79,123,61)(27,80,124,62)(28,81,125,63)(43,146,222,85)(44,147,223,86)(45,148,224,87)(46,149,211,88)(47,150,212,89)(48,151,213,90)(49,152,214,91)(50,153,215,92)(51,154,216,93)(52,141,217,94)(53,142,218,95)(54,143,219,96)(55,144,220,97)(56,145,221,98)(99,164,201,139)(100,165,202,140)(101,166,203,127)(102,167,204,128)(103,168,205,129)(104,155,206,130)(105,156,207,131)(106,157,208,132)(107,158,209,133)(108,159,210,134)(109,160,197,135)(110,161,198,136)(111,162,199,137)(112,163,200,138) );
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)], [(1,104,35,199),(2,103,36,198),(3,102,37,197),(4,101,38,210),(5,100,39,209),(6,99,40,208),(7,112,41,207),(8,111,42,206),(9,110,29,205),(10,109,30,204),(11,108,31,203),(12,107,32,202),(13,106,33,201),(14,105,34,200),(15,215,119,43),(16,214,120,56),(17,213,121,55),(18,212,122,54),(19,211,123,53),(20,224,124,52),(21,223,125,51),(22,222,126,50),(23,221,113,49),(24,220,114,48),(25,219,115,47),(26,218,116,46),(27,217,117,45),(28,216,118,44),(57,146,82,92),(58,145,83,91),(59,144,84,90),(60,143,71,89),(61,142,72,88),(62,141,73,87),(63,154,74,86),(64,153,75,85),(65,152,76,98),(66,151,77,97),(67,150,78,96),(68,149,79,95),(69,148,80,94),(70,147,81,93),(127,180,159,184),(128,179,160,183),(129,178,161,196),(130,177,162,195),(131,176,163,194),(132,175,164,193),(133,174,165,192),(134,173,166,191),(135,172,167,190),(136,171,168,189),(137,170,155,188),(138,169,156,187),(139,182,157,186),(140,181,158,185)], [(15,119),(16,120),(17,121),(18,122),(19,123),(20,124),(21,125),(22,126),(23,113),(24,114),(25,115),(26,116),(27,117),(28,118),(57,82),(58,83),(59,84),(60,71),(61,72),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,81),(99,208),(100,209),(101,210),(102,197),(103,198),(104,199),(105,200),(106,201),(107,202),(108,203),(109,204),(110,205),(111,206),(112,207),(127,159),(128,160),(129,161),(130,162),(131,163),(132,164),(133,165),(134,166),(135,167),(136,168),(137,155),(138,156),(139,157),(140,158)], [(1,220,42,55),(2,221,29,56),(3,222,30,43),(4,223,31,44),(5,224,32,45),(6,211,33,46),(7,212,34,47),(8,213,35,48),(9,214,36,49),(10,215,37,50),(11,216,38,51),(12,217,39,52),(13,218,40,53),(14,219,41,54),(15,109,126,197),(16,110,113,198),(17,111,114,199),(18,112,115,200),(19,99,116,201),(20,100,117,202),(21,101,118,203),(22,102,119,204),(23,103,120,205),(24,104,121,206),(25,105,122,207),(26,106,123,208),(27,107,124,209),(28,108,125,210),(57,167,75,128),(58,168,76,129),(59,155,77,130),(60,156,78,131),(61,157,79,132),(62,158,80,133),(63,159,81,134),(64,160,82,135),(65,161,83,136),(66,162,84,137),(67,163,71,138),(68,164,72,139),(69,165,73,140),(70,166,74,127),(85,172,146,183),(86,173,147,184),(87,174,148,185),(88,175,149,186),(89,176,150,187),(90,177,151,188),(91,178,152,189),(92,179,153,190),(93,180,154,191),(94,181,141,192),(95,182,142,193),(96,169,143,194),(97,170,144,195),(98,171,145,196)], [(1,170,42,195),(2,171,29,196),(3,172,30,183),(4,173,31,184),(5,174,32,185),(6,175,33,186),(7,176,34,187),(8,177,35,188),(9,178,36,189),(10,179,37,190),(11,180,38,191),(12,181,39,192),(13,182,40,193),(14,169,41,194),(15,82,126,64),(16,83,113,65),(17,84,114,66),(18,71,115,67),(19,72,116,68),(20,73,117,69),(21,74,118,70),(22,75,119,57),(23,76,120,58),(24,77,121,59),(25,78,122,60),(26,79,123,61),(27,80,124,62),(28,81,125,63),(43,146,222,85),(44,147,223,86),(45,148,224,87),(46,149,211,88),(47,150,212,89),(48,151,213,90),(49,152,214,91),(50,153,215,92),(51,154,216,93),(52,141,217,94),(53,142,218,95),(54,143,219,96),(55,144,220,97),(56,145,221,98),(99,164,201,139),(100,165,202,140),(101,166,203,127),(102,167,204,128),(103,168,205,129),(104,155,206,130),(105,156,207,131),(106,157,208,132),(107,158,209,133),(108,159,210,134),(109,160,197,135),(110,161,198,136),(111,162,199,137),(112,163,200,138)]])
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 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 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C7⋊D4 | 2- 1+4 | Q8⋊2D7 | Q8.10D14 |
| kernel | C14.452- 1+4 | C4×C7⋊D4 | C28⋊7D4 | Q8×Dic7 | D14⋊3Q8 | C28.23D4 | C2×Q8⋊2D7 | Q8×C2×C14 | C7×Q8 | C22×Q8 | C2×C14 | C22×C4 | C2×Q8 | Q8 | C14 | C22 | C2 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 4 | 3 | 4 | 9 | 12 | 24 | 1 | 6 | 6 |
Matrix representation of C14.452- 1+4 ►in GL4(𝔽29) generated by
| 4 | 0 | 0 | 0 |
| 2 | 22 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 12 | 21 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 17 | 24 |
| 0 | 0 | 17 | 12 |
| 1 | 0 | 0 | 0 |
| 3 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 3 | 28 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 17 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 2 |
| 0 | 0 | 28 | 1 |
G:=sub<GL(4,GF(29))| [4,2,0,0,0,22,0,0,0,0,28,0,0,0,0,28],[12,0,0,0,21,17,0,0,0,0,17,17,0,0,24,12],[1,3,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,3,0,0,0,28,0,0,0,0,12,0,0,0,5,17],[28,0,0,0,0,28,0,0,0,0,28,28,0,0,2,1] >;
C14.452- 1+4 in GAP, Magma, Sage, TeX
C_{14}._{45}2_-^{1+4} % in TeX
G:=Group("C14.45ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1270);
// by ID
G=gap.SmallGroup(448,1270);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,387,184,675,18822]);
// 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=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^7*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations