?
G = C14.772- (1+4) order 448 = 26·7
32nd non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.772- (1+4), C4⋊C4.98D14, C22⋊Q8⋊19D7, (Q8×Dic7)⋊16C2, D14⋊3Q8⋊23C2, (C2×C28).63C23, (C2×Q8).132D14, C22⋊C4.63D14, C4.Dic14⋊25C2, C28.213(C4○D4), (C2×C14).186C24, D14⋊C4.27C22, (C22×C4).248D14, C4.102(D4⋊2D7), C23.11D14⋊9C2, Dic7⋊C4.34C22, C4⋊Dic7.220C22, (Q8×C14).116C22, C22.D28.2C2, (C22×D7).77C23, C22.207(C23×D7), C23.195(C22×D7), (C22×C28).261C22, (C22×C14).214C23, C7⋊7(C22.46C24), C22.10(Q8⋊2D7), (C2×Dic7).240C23, (C4×Dic7).114C22, C23.D7.125C22, C2.37(D4.10D14), (C22×Dic7).123C22, C4⋊C4⋊D7⋊21C2, C4⋊C4⋊7D7⋊30C2, (C2×C4⋊Dic7)⋊43C2, (C4×C7⋊D4).11C2, (C7×C22⋊Q8)⋊22C2, C14.115(C2×C4○D4), C2.49(C2×D4⋊2D7), C2.19(C2×Q8⋊2D7), (C2×C4×D7).103C22, (C2×C4).56(C22×D7), (C2×C14).27(C4○D4), (C7×C4⋊C4).167C22, (C2×C7⋊D4).133C22, (C7×C22⋊C4).41C22, SmallGroup(448,1095)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 828 in 214 conjugacy classes, 99 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×5], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], D4 [×2], Q8 [×2], C23, C23, D7, C14 [×3], C14 [×2], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×13], C22×C4, C22×C4 [×3], C2×D4, C2×Q8, Dic7 [×7], C28 [×2], C28 [×5], D14 [×3], C2×C14, C2×C14 [×2], C2×C14 [×2], C2×C4⋊C4, C42⋊C2 [×3], C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C22.D4 [×2], C42.C2 [×3], C42⋊2C2 [×2], C4×D7 [×2], C2×Dic7 [×3], C2×Dic7 [×4], C2×Dic7 [×4], C7⋊D4 [×2], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7, C22×C14, C22.46C24, C4×Dic7, C4×Dic7 [×4], Dic7⋊C4, Dic7⋊C4 [×4], C4⋊Dic7 [×2], C4⋊Dic7 [×6], D14⋊C4, D14⋊C4 [×4], C23.D7, C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×C4×D7, C22×Dic7 [×2], C2×C7⋊D4, C22×C28, Q8×C14, C23.11D14 [×2], C22.D28 [×2], C4.Dic14, C4.Dic14 [×2], C4⋊C4⋊7D7, C4⋊C4⋊D7 [×2], C2×C4⋊Dic7, C4×C7⋊D4, Q8×Dic7, D14⋊3Q8, C7×C22⋊Q8, C14.772- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D7 [×7], C22.46C24, D4⋊2D7 [×2], Q8⋊2D7 [×2], C23×D7, C2×D4⋊2D7, C2×Q8⋊2D7, D4.10D14, C14.772- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7b2, bab-1=cac=dad-1=a-1, 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 36 188 44)(2 35 189 43)(3 34 190 56)(4 33 191 55)(5 32 192 54)(6 31 193 53)(7 30 194 52)(8 29 195 51)(9 42 196 50)(10 41 183 49)(11 40 184 48)(12 39 185 47)(13 38 186 46)(14 37 187 45)(15 159 205 131)(16 158 206 130)(17 157 207 129)(18 156 208 128)(19 155 209 127)(20 168 210 140)(21 167 197 139)(22 166 198 138)(23 165 199 137)(24 164 200 136)(25 163 201 135)(26 162 202 134)(27 161 203 133)(28 160 204 132)(57 106 213 125)(58 105 214 124)(59 104 215 123)(60 103 216 122)(61 102 217 121)(62 101 218 120)(63 100 219 119)(64 99 220 118)(65 112 221 117)(66 111 222 116)(67 110 223 115)(68 109 224 114)(69 108 211 113)(70 107 212 126)(71 175 151 95)(72 174 152 94)(73 173 153 93)(74 172 154 92)(75 171 141 91)(76 170 142 90)(77 169 143 89)(78 182 144 88)(79 181 145 87)(80 180 146 86)(81 179 147 85)(82 178 148 98)(83 177 149 97)(84 176 150 96)
(1 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 56)(11 55)(12 54)(13 53)(14 52)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 132)(22 131)(23 130)(24 129)(25 128)(26 127)(27 140)(28 139)(29 188)(30 187)(31 186)(32 185)(33 184)(34 183)(35 196)(36 195)(37 194)(38 193)(39 192)(40 191)(41 190)(42 189)(57 118)(58 117)(59 116)(60 115)(61 114)(62 113)(63 126)(64 125)(65 124)(66 123)(67 122)(68 121)(69 120)(70 119)(71 182)(72 181)(73 180)(74 179)(75 178)(76 177)(77 176)(78 175)(79 174)(80 173)(81 172)(82 171)(83 170)(84 169)(85 154)(86 153)(87 152)(88 151)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 144)(96 143)(97 142)(98 141)(99 213)(100 212)(101 211)(102 224)(103 223)(104 222)(105 221)(106 220)(107 219)(108 218)(109 217)(110 216)(111 215)(112 214)(155 202)(156 201)(157 200)(158 199)(159 198)(160 197)(161 210)(162 209)(163 208)(164 207)(165 206)(166 205)(167 204)(168 203)
(1 179 188 85)(2 178 189 98)(3 177 190 97)(4 176 191 96)(5 175 192 95)(6 174 193 94)(7 173 194 93)(8 172 195 92)(9 171 196 91)(10 170 183 90)(11 169 184 89)(12 182 185 88)(13 181 186 87)(14 180 187 86)(15 214 205 58)(16 213 206 57)(17 212 207 70)(18 211 208 69)(19 224 209 68)(20 223 210 67)(21 222 197 66)(22 221 198 65)(23 220 199 64)(24 219 200 63)(25 218 201 62)(26 217 202 61)(27 216 203 60)(28 215 204 59)(29 147 51 81)(30 146 52 80)(31 145 53 79)(32 144 54 78)(33 143 55 77)(34 142 56 76)(35 141 43 75)(36 154 44 74)(37 153 45 73)(38 152 46 72)(39 151 47 71)(40 150 48 84)(41 149 49 83)(42 148 50 82)(99 158 118 130)(100 157 119 129)(101 156 120 128)(102 155 121 127)(103 168 122 140)(104 167 123 139)(105 166 124 138)(106 165 125 137)(107 164 126 136)(108 163 113 135)(109 162 114 134)(110 161 115 133)(111 160 116 132)(112 159 117 131)
(1 208 195 25)(2 209 196 26)(3 210 183 27)(4 197 184 28)(5 198 185 15)(6 199 186 16)(7 200 187 17)(8 201 188 18)(9 202 189 19)(10 203 190 20)(11 204 191 21)(12 205 192 22)(13 206 193 23)(14 207 194 24)(29 135 44 156)(30 136 45 157)(31 137 46 158)(32 138 47 159)(33 139 48 160)(34 140 49 161)(35 127 50 162)(36 128 51 163)(37 129 52 164)(38 130 53 165)(39 131 54 166)(40 132 55 167)(41 133 56 168)(42 134 43 155)(57 181 220 94)(58 182 221 95)(59 169 222 96)(60 170 223 97)(61 171 224 98)(62 172 211 85)(63 173 212 86)(64 174 213 87)(65 175 214 88)(66 176 215 89)(67 177 216 90)(68 178 217 91)(69 179 218 92)(70 180 219 93)(71 105 144 117)(72 106 145 118)(73 107 146 119)(74 108 147 120)(75 109 148 121)(76 110 149 122)(77 111 150 123)(78 112 151 124)(79 99 152 125)(80 100 153 126)(81 101 154 113)(82 102 141 114)(83 103 142 115)(84 104 143 116)
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), (1,36,188,44)(2,35,189,43)(3,34,190,56)(4,33,191,55)(5,32,192,54)(6,31,193,53)(7,30,194,52)(8,29,195,51)(9,42,196,50)(10,41,183,49)(11,40,184,48)(12,39,185,47)(13,38,186,46)(14,37,187,45)(15,159,205,131)(16,158,206,130)(17,157,207,129)(18,156,208,128)(19,155,209,127)(20,168,210,140)(21,167,197,139)(22,166,198,138)(23,165,199,137)(24,164,200,136)(25,163,201,135)(26,162,202,134)(27,161,203,133)(28,160,204,132)(57,106,213,125)(58,105,214,124)(59,104,215,123)(60,103,216,122)(61,102,217,121)(62,101,218,120)(63,100,219,119)(64,99,220,118)(65,112,221,117)(66,111,222,116)(67,110,223,115)(68,109,224,114)(69,108,211,113)(70,107,212,126)(71,175,151,95)(72,174,152,94)(73,173,153,93)(74,172,154,92)(75,171,141,91)(76,170,142,90)(77,169,143,89)(78,182,144,88)(79,181,145,87)(80,180,146,86)(81,179,147,85)(82,178,148,98)(83,177,149,97)(84,176,150,96), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,56)(11,55)(12,54)(13,53)(14,52)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,140)(28,139)(29,188)(30,187)(31,186)(32,185)(33,184)(34,183)(35,196)(36,195)(37,194)(38,193)(39,192)(40,191)(41,190)(42,189)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,182)(72,181)(73,180)(74,179)(75,178)(76,177)(77,176)(78,175)(79,174)(80,173)(81,172)(82,171)(83,170)(84,169)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,213)(100,212)(101,211)(102,224)(103,223)(104,222)(105,221)(106,220)(107,219)(108,218)(109,217)(110,216)(111,215)(112,214)(155,202)(156,201)(157,200)(158,199)(159,198)(160,197)(161,210)(162,209)(163,208)(164,207)(165,206)(166,205)(167,204)(168,203), (1,179,188,85)(2,178,189,98)(3,177,190,97)(4,176,191,96)(5,175,192,95)(6,174,193,94)(7,173,194,93)(8,172,195,92)(9,171,196,91)(10,170,183,90)(11,169,184,89)(12,182,185,88)(13,181,186,87)(14,180,187,86)(15,214,205,58)(16,213,206,57)(17,212,207,70)(18,211,208,69)(19,224,209,68)(20,223,210,67)(21,222,197,66)(22,221,198,65)(23,220,199,64)(24,219,200,63)(25,218,201,62)(26,217,202,61)(27,216,203,60)(28,215,204,59)(29,147,51,81)(30,146,52,80)(31,145,53,79)(32,144,54,78)(33,143,55,77)(34,142,56,76)(35,141,43,75)(36,154,44,74)(37,153,45,73)(38,152,46,72)(39,151,47,71)(40,150,48,84)(41,149,49,83)(42,148,50,82)(99,158,118,130)(100,157,119,129)(101,156,120,128)(102,155,121,127)(103,168,122,140)(104,167,123,139)(105,166,124,138)(106,165,125,137)(107,164,126,136)(108,163,113,135)(109,162,114,134)(110,161,115,133)(111,160,116,132)(112,159,117,131), (1,208,195,25)(2,209,196,26)(3,210,183,27)(4,197,184,28)(5,198,185,15)(6,199,186,16)(7,200,187,17)(8,201,188,18)(9,202,189,19)(10,203,190,20)(11,204,191,21)(12,205,192,22)(13,206,193,23)(14,207,194,24)(29,135,44,156)(30,136,45,157)(31,137,46,158)(32,138,47,159)(33,139,48,160)(34,140,49,161)(35,127,50,162)(36,128,51,163)(37,129,52,164)(38,130,53,165)(39,131,54,166)(40,132,55,167)(41,133,56,168)(42,134,43,155)(57,181,220,94)(58,182,221,95)(59,169,222,96)(60,170,223,97)(61,171,224,98)(62,172,211,85)(63,173,212,86)(64,174,213,87)(65,175,214,88)(66,176,215,89)(67,177,216,90)(68,178,217,91)(69,179,218,92)(70,180,219,93)(71,105,144,117)(72,106,145,118)(73,107,146,119)(74,108,147,120)(75,109,148,121)(76,110,149,122)(77,111,150,123)(78,112,151,124)(79,99,152,125)(80,100,153,126)(81,101,154,113)(82,102,141,114)(83,103,142,115)(84,104,143,116)>;
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), (1,36,188,44)(2,35,189,43)(3,34,190,56)(4,33,191,55)(5,32,192,54)(6,31,193,53)(7,30,194,52)(8,29,195,51)(9,42,196,50)(10,41,183,49)(11,40,184,48)(12,39,185,47)(13,38,186,46)(14,37,187,45)(15,159,205,131)(16,158,206,130)(17,157,207,129)(18,156,208,128)(19,155,209,127)(20,168,210,140)(21,167,197,139)(22,166,198,138)(23,165,199,137)(24,164,200,136)(25,163,201,135)(26,162,202,134)(27,161,203,133)(28,160,204,132)(57,106,213,125)(58,105,214,124)(59,104,215,123)(60,103,216,122)(61,102,217,121)(62,101,218,120)(63,100,219,119)(64,99,220,118)(65,112,221,117)(66,111,222,116)(67,110,223,115)(68,109,224,114)(69,108,211,113)(70,107,212,126)(71,175,151,95)(72,174,152,94)(73,173,153,93)(74,172,154,92)(75,171,141,91)(76,170,142,90)(77,169,143,89)(78,182,144,88)(79,181,145,87)(80,180,146,86)(81,179,147,85)(82,178,148,98)(83,177,149,97)(84,176,150,96), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,56)(11,55)(12,54)(13,53)(14,52)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,129)(25,128)(26,127)(27,140)(28,139)(29,188)(30,187)(31,186)(32,185)(33,184)(34,183)(35,196)(36,195)(37,194)(38,193)(39,192)(40,191)(41,190)(42,189)(57,118)(58,117)(59,116)(60,115)(61,114)(62,113)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,182)(72,181)(73,180)(74,179)(75,178)(76,177)(77,176)(78,175)(79,174)(80,173)(81,172)(82,171)(83,170)(84,169)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,144)(96,143)(97,142)(98,141)(99,213)(100,212)(101,211)(102,224)(103,223)(104,222)(105,221)(106,220)(107,219)(108,218)(109,217)(110,216)(111,215)(112,214)(155,202)(156,201)(157,200)(158,199)(159,198)(160,197)(161,210)(162,209)(163,208)(164,207)(165,206)(166,205)(167,204)(168,203), (1,179,188,85)(2,178,189,98)(3,177,190,97)(4,176,191,96)(5,175,192,95)(6,174,193,94)(7,173,194,93)(8,172,195,92)(9,171,196,91)(10,170,183,90)(11,169,184,89)(12,182,185,88)(13,181,186,87)(14,180,187,86)(15,214,205,58)(16,213,206,57)(17,212,207,70)(18,211,208,69)(19,224,209,68)(20,223,210,67)(21,222,197,66)(22,221,198,65)(23,220,199,64)(24,219,200,63)(25,218,201,62)(26,217,202,61)(27,216,203,60)(28,215,204,59)(29,147,51,81)(30,146,52,80)(31,145,53,79)(32,144,54,78)(33,143,55,77)(34,142,56,76)(35,141,43,75)(36,154,44,74)(37,153,45,73)(38,152,46,72)(39,151,47,71)(40,150,48,84)(41,149,49,83)(42,148,50,82)(99,158,118,130)(100,157,119,129)(101,156,120,128)(102,155,121,127)(103,168,122,140)(104,167,123,139)(105,166,124,138)(106,165,125,137)(107,164,126,136)(108,163,113,135)(109,162,114,134)(110,161,115,133)(111,160,116,132)(112,159,117,131), (1,208,195,25)(2,209,196,26)(3,210,183,27)(4,197,184,28)(5,198,185,15)(6,199,186,16)(7,200,187,17)(8,201,188,18)(9,202,189,19)(10,203,190,20)(11,204,191,21)(12,205,192,22)(13,206,193,23)(14,207,194,24)(29,135,44,156)(30,136,45,157)(31,137,46,158)(32,138,47,159)(33,139,48,160)(34,140,49,161)(35,127,50,162)(36,128,51,163)(37,129,52,164)(38,130,53,165)(39,131,54,166)(40,132,55,167)(41,133,56,168)(42,134,43,155)(57,181,220,94)(58,182,221,95)(59,169,222,96)(60,170,223,97)(61,171,224,98)(62,172,211,85)(63,173,212,86)(64,174,213,87)(65,175,214,88)(66,176,215,89)(67,177,216,90)(68,178,217,91)(69,179,218,92)(70,180,219,93)(71,105,144,117)(72,106,145,118)(73,107,146,119)(74,108,147,120)(75,109,148,121)(76,110,149,122)(77,111,150,123)(78,112,151,124)(79,99,152,125)(80,100,153,126)(81,101,154,113)(82,102,141,114)(83,103,142,115)(84,104,143,116) );
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)], [(1,36,188,44),(2,35,189,43),(3,34,190,56),(4,33,191,55),(5,32,192,54),(6,31,193,53),(7,30,194,52),(8,29,195,51),(9,42,196,50),(10,41,183,49),(11,40,184,48),(12,39,185,47),(13,38,186,46),(14,37,187,45),(15,159,205,131),(16,158,206,130),(17,157,207,129),(18,156,208,128),(19,155,209,127),(20,168,210,140),(21,167,197,139),(22,166,198,138),(23,165,199,137),(24,164,200,136),(25,163,201,135),(26,162,202,134),(27,161,203,133),(28,160,204,132),(57,106,213,125),(58,105,214,124),(59,104,215,123),(60,103,216,122),(61,102,217,121),(62,101,218,120),(63,100,219,119),(64,99,220,118),(65,112,221,117),(66,111,222,116),(67,110,223,115),(68,109,224,114),(69,108,211,113),(70,107,212,126),(71,175,151,95),(72,174,152,94),(73,173,153,93),(74,172,154,92),(75,171,141,91),(76,170,142,90),(77,169,143,89),(78,182,144,88),(79,181,145,87),(80,180,146,86),(81,179,147,85),(82,178,148,98),(83,177,149,97),(84,176,150,96)], [(1,51),(2,50),(3,49),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,56),(11,55),(12,54),(13,53),(14,52),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(21,132),(22,131),(23,130),(24,129),(25,128),(26,127),(27,140),(28,139),(29,188),(30,187),(31,186),(32,185),(33,184),(34,183),(35,196),(36,195),(37,194),(38,193),(39,192),(40,191),(41,190),(42,189),(57,118),(58,117),(59,116),(60,115),(61,114),(62,113),(63,126),(64,125),(65,124),(66,123),(67,122),(68,121),(69,120),(70,119),(71,182),(72,181),(73,180),(74,179),(75,178),(76,177),(77,176),(78,175),(79,174),(80,173),(81,172),(82,171),(83,170),(84,169),(85,154),(86,153),(87,152),(88,151),(89,150),(90,149),(91,148),(92,147),(93,146),(94,145),(95,144),(96,143),(97,142),(98,141),(99,213),(100,212),(101,211),(102,224),(103,223),(104,222),(105,221),(106,220),(107,219),(108,218),(109,217),(110,216),(111,215),(112,214),(155,202),(156,201),(157,200),(158,199),(159,198),(160,197),(161,210),(162,209),(163,208),(164,207),(165,206),(166,205),(167,204),(168,203)], [(1,179,188,85),(2,178,189,98),(3,177,190,97),(4,176,191,96),(5,175,192,95),(6,174,193,94),(7,173,194,93),(8,172,195,92),(9,171,196,91),(10,170,183,90),(11,169,184,89),(12,182,185,88),(13,181,186,87),(14,180,187,86),(15,214,205,58),(16,213,206,57),(17,212,207,70),(18,211,208,69),(19,224,209,68),(20,223,210,67),(21,222,197,66),(22,221,198,65),(23,220,199,64),(24,219,200,63),(25,218,201,62),(26,217,202,61),(27,216,203,60),(28,215,204,59),(29,147,51,81),(30,146,52,80),(31,145,53,79),(32,144,54,78),(33,143,55,77),(34,142,56,76),(35,141,43,75),(36,154,44,74),(37,153,45,73),(38,152,46,72),(39,151,47,71),(40,150,48,84),(41,149,49,83),(42,148,50,82),(99,158,118,130),(100,157,119,129),(101,156,120,128),(102,155,121,127),(103,168,122,140),(104,167,123,139),(105,166,124,138),(106,165,125,137),(107,164,126,136),(108,163,113,135),(109,162,114,134),(110,161,115,133),(111,160,116,132),(112,159,117,131)], [(1,208,195,25),(2,209,196,26),(3,210,183,27),(4,197,184,28),(5,198,185,15),(6,199,186,16),(7,200,187,17),(8,201,188,18),(9,202,189,19),(10,203,190,20),(11,204,191,21),(12,205,192,22),(13,206,193,23),(14,207,194,24),(29,135,44,156),(30,136,45,157),(31,137,46,158),(32,138,47,159),(33,139,48,160),(34,140,49,161),(35,127,50,162),(36,128,51,163),(37,129,52,164),(38,130,53,165),(39,131,54,166),(40,132,55,167),(41,133,56,168),(42,134,43,155),(57,181,220,94),(58,182,221,95),(59,169,222,96),(60,170,223,97),(61,171,224,98),(62,172,211,85),(63,173,212,86),(64,174,213,87),(65,175,214,88),(66,176,215,89),(67,177,216,90),(68,178,217,91),(69,179,218,92),(70,180,219,93),(71,105,144,117),(72,106,145,118),(73,107,146,119),(74,108,147,120),(75,109,148,121),(76,110,149,122),(77,111,150,123),(78,112,151,124),(79,99,152,125),(80,100,153,126),(81,101,154,113),(82,102,141,114),(83,103,142,115),(84,104,143,116)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 10 | 0 | 0 |
| 0 | 0 | 14 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 8 | 0 | 0 |
| 0 | 0 | 23 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 18 |
| 0 | 0 | 0 | 0 | 10 | 16 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 21 | 0 | 0 |
| 0 | 0 | 6 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 11 |
| 0 | 0 | 0 | 0 | 19 | 13 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 21 | 0 | 0 |
| 0 | 0 | 6 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 5 |
| 0 | 0 | 0 | 0 | 22 | 6 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 13 |
| 0 | 0 | 0 | 0 | 4 | 18 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,14,0,0,0,0,10,4,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,17,0,0,0,0,17,0,0,0,0,0,0,0,22,23,0,0,0,0,8,7,0,0,0,0,0,0,13,10,0,0,0,0,18,16],[0,17,0,0,0,0,12,0,0,0,0,0,0,0,7,6,0,0,0,0,21,22,0,0,0,0,0,0,16,19,0,0,0,0,11,13],[12,0,0,0,0,0,0,17,0,0,0,0,0,0,7,6,0,0,0,0,21,22,0,0,0,0,0,0,23,22,0,0,0,0,5,6],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,11,4,0,0,0,0,13,18] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4O | 4P | 4Q | 4R | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
67 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2- (1+4) | D4⋊2D7 | Q8⋊2D7 | D4.10D14 |
| kernel | C14.772- (1+4) | C23.11D14 | C22.D28 | C4.Dic14 | C4⋊C4⋊7D7 | C4⋊C4⋊D7 | C2×C4⋊Dic7 | C4×C7⋊D4 | Q8×Dic7 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | C28 | C2×C14 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 3 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{77}2_-^{(1+4)} % in TeX
G:=Group("C14.77ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1095);
// by ID
G=gap.SmallGroup(448,1095);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,100,675,570,185,192,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=b^2,e^2=a^7*b^2,b*a*b^-1=c*a*c=d*a*d^-1=a^-1,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
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