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, C28.3Q8⋊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×C14).214C23, (C22×C28).261C22, C7⋊7(C22.46C24), C22.10(Q8⋊2D7), (C2×Dic7).240C23, (C4×Dic7).114C22, C2.37(D4.10D14), C23.D7.125C22, (C22×Dic7).123C22, C4⋊C4⋊7D7⋊30C2, C4⋊C4⋊D7⋊21C2, (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
Generators and relations for C14.772- 1+4
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 >
Subgroups: 828 in 214 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×Q8, C22×D7, C22×C14, C22.46C24, C4×Dic7, C4×Dic7, Dic7⋊C4, Dic7⋊C4, C4⋊Dic7, C4⋊Dic7, D14⋊C4, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, Q8×C14, C23.11D14, C22.D28, C28.3Q8, C28.3Q8, C4⋊C4⋊7D7, C4⋊C4⋊D7, C2×C4⋊Dic7, C4×C7⋊D4, Q8×Dic7, D14⋊3Q8, C7×C22⋊Q8, C14.772- 1+4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, 2- 1+4, C22×D7, C22.46C24, D4⋊2D7, Q8⋊2D7, C23×D7, C2×D4⋊2D7, C2×Q8⋊2D7, D4.10D14, C14.772- 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 23 90 117)(2 22 91 116)(3 21 92 115)(4 20 93 114)(5 19 94 113)(6 18 95 126)(7 17 96 125)(8 16 97 124)(9 15 98 123)(10 28 85 122)(11 27 86 121)(12 26 87 120)(13 25 88 119)(14 24 89 118)(29 181 110 75)(30 180 111 74)(31 179 112 73)(32 178 99 72)(33 177 100 71)(34 176 101 84)(35 175 102 83)(36 174 103 82)(37 173 104 81)(38 172 105 80)(39 171 106 79)(40 170 107 78)(41 169 108 77)(42 182 109 76)(43 186 164 201)(44 185 165 200)(45 184 166 199)(46 183 167 198)(47 196 168 197)(48 195 155 210)(49 194 156 209)(50 193 157 208)(51 192 158 207)(52 191 159 206)(53 190 160 205)(54 189 161 204)(55 188 162 203)(56 187 163 202)(57 127 213 152)(58 140 214 151)(59 139 215 150)(60 138 216 149)(61 137 217 148)(62 136 218 147)(63 135 219 146)(64 134 220 145)(65 133 221 144)(66 132 222 143)(67 131 223 142)(68 130 224 141)(69 129 211 154)(70 128 212 153)
(1 124)(2 123)(3 122)(4 121)(5 120)(6 119)(7 118)(8 117)(9 116)(10 115)(11 114)(12 113)(13 126)(14 125)(15 91)(16 90)(17 89)(18 88)(19 87)(20 86)(21 85)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 82)(30 81)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)(41 84)(42 83)(43 208)(44 207)(45 206)(46 205)(47 204)(48 203)(49 202)(50 201)(51 200)(52 199)(53 198)(54 197)(55 210)(56 209)(57 145)(58 144)(59 143)(60 142)(61 141)(62 154)(63 153)(64 152)(65 151)(66 150)(67 149)(68 148)(69 147)(70 146)(99 171)(100 170)(101 169)(102 182)(103 181)(104 180)(105 179)(106 178)(107 177)(108 176)(109 175)(110 174)(111 173)(112 172)(127 220)(128 219)(129 218)(130 217)(131 216)(132 215)(133 214)(134 213)(135 212)(136 211)(137 224)(138 223)(139 222)(140 221)(155 188)(156 187)(157 186)(158 185)(159 184)(160 183)(161 196)(162 195)(163 194)(164 193)(165 192)(166 191)(167 190)(168 189)
(1 221 90 65)(2 220 91 64)(3 219 92 63)(4 218 93 62)(5 217 94 61)(6 216 95 60)(7 215 96 59)(8 214 97 58)(9 213 98 57)(10 212 85 70)(11 211 86 69)(12 224 87 68)(13 223 88 67)(14 222 89 66)(15 145 123 134)(16 144 124 133)(17 143 125 132)(18 142 126 131)(19 141 113 130)(20 154 114 129)(21 153 115 128)(22 152 116 127)(23 151 117 140)(24 150 118 139)(25 149 119 138)(26 148 120 137)(27 147 121 136)(28 146 122 135)(29 163 110 56)(30 162 111 55)(31 161 112 54)(32 160 99 53)(33 159 100 52)(34 158 101 51)(35 157 102 50)(36 156 103 49)(37 155 104 48)(38 168 105 47)(39 167 106 46)(40 166 107 45)(41 165 108 44)(42 164 109 43)(71 184 177 199)(72 183 178 198)(73 196 179 197)(74 195 180 210)(75 194 181 209)(76 193 182 208)(77 192 169 207)(78 191 170 206)(79 190 171 205)(80 189 172 204)(81 188 173 203)(82 187 174 202)(83 186 175 201)(84 185 176 200)
(1 31 97 105)(2 32 98 106)(3 33 85 107)(4 34 86 108)(5 35 87 109)(6 36 88 110)(7 37 89 111)(8 38 90 112)(9 39 91 99)(10 40 92 100)(11 41 93 101)(12 42 94 102)(13 29 95 103)(14 30 96 104)(15 171 116 72)(16 172 117 73)(17 173 118 74)(18 174 119 75)(19 175 120 76)(20 176 121 77)(21 177 122 78)(22 178 123 79)(23 179 124 80)(24 180 125 81)(25 181 126 82)(26 182 113 83)(27 169 114 84)(28 170 115 71)(43 68 157 217)(44 69 158 218)(45 70 159 219)(46 57 160 220)(47 58 161 221)(48 59 162 222)(49 60 163 223)(50 61 164 224)(51 62 165 211)(52 63 166 212)(53 64 167 213)(54 65 168 214)(55 66 155 215)(56 67 156 216)(127 205 145 183)(128 206 146 184)(129 207 147 185)(130 208 148 186)(131 209 149 187)(132 210 150 188)(133 197 151 189)(134 198 152 190)(135 199 153 191)(136 200 154 192)(137 201 141 193)(138 202 142 194)(139 203 143 195)(140 204 144 196)
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), (1,23,90,117)(2,22,91,116)(3,21,92,115)(4,20,93,114)(5,19,94,113)(6,18,95,126)(7,17,96,125)(8,16,97,124)(9,15,98,123)(10,28,85,122)(11,27,86,121)(12,26,87,120)(13,25,88,119)(14,24,89,118)(29,181,110,75)(30,180,111,74)(31,179,112,73)(32,178,99,72)(33,177,100,71)(34,176,101,84)(35,175,102,83)(36,174,103,82)(37,173,104,81)(38,172,105,80)(39,171,106,79)(40,170,107,78)(41,169,108,77)(42,182,109,76)(43,186,164,201)(44,185,165,200)(45,184,166,199)(46,183,167,198)(47,196,168,197)(48,195,155,210)(49,194,156,209)(50,193,157,208)(51,192,158,207)(52,191,159,206)(53,190,160,205)(54,189,161,204)(55,188,162,203)(56,187,163,202)(57,127,213,152)(58,140,214,151)(59,139,215,150)(60,138,216,149)(61,137,217,148)(62,136,218,147)(63,135,219,146)(64,134,220,145)(65,133,221,144)(66,132,222,143)(67,131,223,142)(68,130,224,141)(69,129,211,154)(70,128,212,153), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,126)(14,125)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,84)(42,83)(43,208)(44,207)(45,206)(46,205)(47,204)(48,203)(49,202)(50,201)(51,200)(52,199)(53,198)(54,197)(55,210)(56,209)(57,145)(58,144)(59,143)(60,142)(61,141)(62,154)(63,153)(64,152)(65,151)(66,150)(67,149)(68,148)(69,147)(70,146)(99,171)(100,170)(101,169)(102,182)(103,181)(104,180)(105,179)(106,178)(107,177)(108,176)(109,175)(110,174)(111,173)(112,172)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,224)(138,223)(139,222)(140,221)(155,188)(156,187)(157,186)(158,185)(159,184)(160,183)(161,196)(162,195)(163,194)(164,193)(165,192)(166,191)(167,190)(168,189), (1,221,90,65)(2,220,91,64)(3,219,92,63)(4,218,93,62)(5,217,94,61)(6,216,95,60)(7,215,96,59)(8,214,97,58)(9,213,98,57)(10,212,85,70)(11,211,86,69)(12,224,87,68)(13,223,88,67)(14,222,89,66)(15,145,123,134)(16,144,124,133)(17,143,125,132)(18,142,126,131)(19,141,113,130)(20,154,114,129)(21,153,115,128)(22,152,116,127)(23,151,117,140)(24,150,118,139)(25,149,119,138)(26,148,120,137)(27,147,121,136)(28,146,122,135)(29,163,110,56)(30,162,111,55)(31,161,112,54)(32,160,99,53)(33,159,100,52)(34,158,101,51)(35,157,102,50)(36,156,103,49)(37,155,104,48)(38,168,105,47)(39,167,106,46)(40,166,107,45)(41,165,108,44)(42,164,109,43)(71,184,177,199)(72,183,178,198)(73,196,179,197)(74,195,180,210)(75,194,181,209)(76,193,182,208)(77,192,169,207)(78,191,170,206)(79,190,171,205)(80,189,172,204)(81,188,173,203)(82,187,174,202)(83,186,175,201)(84,185,176,200), (1,31,97,105)(2,32,98,106)(3,33,85,107)(4,34,86,108)(5,35,87,109)(6,36,88,110)(7,37,89,111)(8,38,90,112)(9,39,91,99)(10,40,92,100)(11,41,93,101)(12,42,94,102)(13,29,95,103)(14,30,96,104)(15,171,116,72)(16,172,117,73)(17,173,118,74)(18,174,119,75)(19,175,120,76)(20,176,121,77)(21,177,122,78)(22,178,123,79)(23,179,124,80)(24,180,125,81)(25,181,126,82)(26,182,113,83)(27,169,114,84)(28,170,115,71)(43,68,157,217)(44,69,158,218)(45,70,159,219)(46,57,160,220)(47,58,161,221)(48,59,162,222)(49,60,163,223)(50,61,164,224)(51,62,165,211)(52,63,166,212)(53,64,167,213)(54,65,168,214)(55,66,155,215)(56,67,156,216)(127,205,145,183)(128,206,146,184)(129,207,147,185)(130,208,148,186)(131,209,149,187)(132,210,150,188)(133,197,151,189)(134,198,152,190)(135,199,153,191)(136,200,154,192)(137,201,141,193)(138,202,142,194)(139,203,143,195)(140,204,144,196)>;
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), (1,23,90,117)(2,22,91,116)(3,21,92,115)(4,20,93,114)(5,19,94,113)(6,18,95,126)(7,17,96,125)(8,16,97,124)(9,15,98,123)(10,28,85,122)(11,27,86,121)(12,26,87,120)(13,25,88,119)(14,24,89,118)(29,181,110,75)(30,180,111,74)(31,179,112,73)(32,178,99,72)(33,177,100,71)(34,176,101,84)(35,175,102,83)(36,174,103,82)(37,173,104,81)(38,172,105,80)(39,171,106,79)(40,170,107,78)(41,169,108,77)(42,182,109,76)(43,186,164,201)(44,185,165,200)(45,184,166,199)(46,183,167,198)(47,196,168,197)(48,195,155,210)(49,194,156,209)(50,193,157,208)(51,192,158,207)(52,191,159,206)(53,190,160,205)(54,189,161,204)(55,188,162,203)(56,187,163,202)(57,127,213,152)(58,140,214,151)(59,139,215,150)(60,138,216,149)(61,137,217,148)(62,136,218,147)(63,135,219,146)(64,134,220,145)(65,133,221,144)(66,132,222,143)(67,131,223,142)(68,130,224,141)(69,129,211,154)(70,128,212,153), (1,124)(2,123)(3,122)(4,121)(5,120)(6,119)(7,118)(8,117)(9,116)(10,115)(11,114)(12,113)(13,126)(14,125)(15,91)(16,90)(17,89)(18,88)(19,87)(20,86)(21,85)(22,98)(23,97)(24,96)(25,95)(26,94)(27,93)(28,92)(29,82)(30,81)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)(41,84)(42,83)(43,208)(44,207)(45,206)(46,205)(47,204)(48,203)(49,202)(50,201)(51,200)(52,199)(53,198)(54,197)(55,210)(56,209)(57,145)(58,144)(59,143)(60,142)(61,141)(62,154)(63,153)(64,152)(65,151)(66,150)(67,149)(68,148)(69,147)(70,146)(99,171)(100,170)(101,169)(102,182)(103,181)(104,180)(105,179)(106,178)(107,177)(108,176)(109,175)(110,174)(111,173)(112,172)(127,220)(128,219)(129,218)(130,217)(131,216)(132,215)(133,214)(134,213)(135,212)(136,211)(137,224)(138,223)(139,222)(140,221)(155,188)(156,187)(157,186)(158,185)(159,184)(160,183)(161,196)(162,195)(163,194)(164,193)(165,192)(166,191)(167,190)(168,189), (1,221,90,65)(2,220,91,64)(3,219,92,63)(4,218,93,62)(5,217,94,61)(6,216,95,60)(7,215,96,59)(8,214,97,58)(9,213,98,57)(10,212,85,70)(11,211,86,69)(12,224,87,68)(13,223,88,67)(14,222,89,66)(15,145,123,134)(16,144,124,133)(17,143,125,132)(18,142,126,131)(19,141,113,130)(20,154,114,129)(21,153,115,128)(22,152,116,127)(23,151,117,140)(24,150,118,139)(25,149,119,138)(26,148,120,137)(27,147,121,136)(28,146,122,135)(29,163,110,56)(30,162,111,55)(31,161,112,54)(32,160,99,53)(33,159,100,52)(34,158,101,51)(35,157,102,50)(36,156,103,49)(37,155,104,48)(38,168,105,47)(39,167,106,46)(40,166,107,45)(41,165,108,44)(42,164,109,43)(71,184,177,199)(72,183,178,198)(73,196,179,197)(74,195,180,210)(75,194,181,209)(76,193,182,208)(77,192,169,207)(78,191,170,206)(79,190,171,205)(80,189,172,204)(81,188,173,203)(82,187,174,202)(83,186,175,201)(84,185,176,200), (1,31,97,105)(2,32,98,106)(3,33,85,107)(4,34,86,108)(5,35,87,109)(6,36,88,110)(7,37,89,111)(8,38,90,112)(9,39,91,99)(10,40,92,100)(11,41,93,101)(12,42,94,102)(13,29,95,103)(14,30,96,104)(15,171,116,72)(16,172,117,73)(17,173,118,74)(18,174,119,75)(19,175,120,76)(20,176,121,77)(21,177,122,78)(22,178,123,79)(23,179,124,80)(24,180,125,81)(25,181,126,82)(26,182,113,83)(27,169,114,84)(28,170,115,71)(43,68,157,217)(44,69,158,218)(45,70,159,219)(46,57,160,220)(47,58,161,221)(48,59,162,222)(49,60,163,223)(50,61,164,224)(51,62,165,211)(52,63,166,212)(53,64,167,213)(54,65,168,214)(55,66,155,215)(56,67,156,216)(127,205,145,183)(128,206,146,184)(129,207,147,185)(130,208,148,186)(131,209,149,187)(132,210,150,188)(133,197,151,189)(134,198,152,190)(135,199,153,191)(136,200,154,192)(137,201,141,193)(138,202,142,194)(139,203,143,195)(140,204,144,196) );
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)], [(1,23,90,117),(2,22,91,116),(3,21,92,115),(4,20,93,114),(5,19,94,113),(6,18,95,126),(7,17,96,125),(8,16,97,124),(9,15,98,123),(10,28,85,122),(11,27,86,121),(12,26,87,120),(13,25,88,119),(14,24,89,118),(29,181,110,75),(30,180,111,74),(31,179,112,73),(32,178,99,72),(33,177,100,71),(34,176,101,84),(35,175,102,83),(36,174,103,82),(37,173,104,81),(38,172,105,80),(39,171,106,79),(40,170,107,78),(41,169,108,77),(42,182,109,76),(43,186,164,201),(44,185,165,200),(45,184,166,199),(46,183,167,198),(47,196,168,197),(48,195,155,210),(49,194,156,209),(50,193,157,208),(51,192,158,207),(52,191,159,206),(53,190,160,205),(54,189,161,204),(55,188,162,203),(56,187,163,202),(57,127,213,152),(58,140,214,151),(59,139,215,150),(60,138,216,149),(61,137,217,148),(62,136,218,147),(63,135,219,146),(64,134,220,145),(65,133,221,144),(66,132,222,143),(67,131,223,142),(68,130,224,141),(69,129,211,154),(70,128,212,153)], [(1,124),(2,123),(3,122),(4,121),(5,120),(6,119),(7,118),(8,117),(9,116),(10,115),(11,114),(12,113),(13,126),(14,125),(15,91),(16,90),(17,89),(18,88),(19,87),(20,86),(21,85),(22,98),(23,97),(24,96),(25,95),(26,94),(27,93),(28,92),(29,82),(30,81),(31,80),(32,79),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,72),(40,71),(41,84),(42,83),(43,208),(44,207),(45,206),(46,205),(47,204),(48,203),(49,202),(50,201),(51,200),(52,199),(53,198),(54,197),(55,210),(56,209),(57,145),(58,144),(59,143),(60,142),(61,141),(62,154),(63,153),(64,152),(65,151),(66,150),(67,149),(68,148),(69,147),(70,146),(99,171),(100,170),(101,169),(102,182),(103,181),(104,180),(105,179),(106,178),(107,177),(108,176),(109,175),(110,174),(111,173),(112,172),(127,220),(128,219),(129,218),(130,217),(131,216),(132,215),(133,214),(134,213),(135,212),(136,211),(137,224),(138,223),(139,222),(140,221),(155,188),(156,187),(157,186),(158,185),(159,184),(160,183),(161,196),(162,195),(163,194),(164,193),(165,192),(166,191),(167,190),(168,189)], [(1,221,90,65),(2,220,91,64),(3,219,92,63),(4,218,93,62),(5,217,94,61),(6,216,95,60),(7,215,96,59),(8,214,97,58),(9,213,98,57),(10,212,85,70),(11,211,86,69),(12,224,87,68),(13,223,88,67),(14,222,89,66),(15,145,123,134),(16,144,124,133),(17,143,125,132),(18,142,126,131),(19,141,113,130),(20,154,114,129),(21,153,115,128),(22,152,116,127),(23,151,117,140),(24,150,118,139),(25,149,119,138),(26,148,120,137),(27,147,121,136),(28,146,122,135),(29,163,110,56),(30,162,111,55),(31,161,112,54),(32,160,99,53),(33,159,100,52),(34,158,101,51),(35,157,102,50),(36,156,103,49),(37,155,104,48),(38,168,105,47),(39,167,106,46),(40,166,107,45),(41,165,108,44),(42,164,109,43),(71,184,177,199),(72,183,178,198),(73,196,179,197),(74,195,180,210),(75,194,181,209),(76,193,182,208),(77,192,169,207),(78,191,170,206),(79,190,171,205),(80,189,172,204),(81,188,173,203),(82,187,174,202),(83,186,175,201),(84,185,176,200)], [(1,31,97,105),(2,32,98,106),(3,33,85,107),(4,34,86,108),(5,35,87,109),(6,36,88,110),(7,37,89,111),(8,38,90,112),(9,39,91,99),(10,40,92,100),(11,41,93,101),(12,42,94,102),(13,29,95,103),(14,30,96,104),(15,171,116,72),(16,172,117,73),(17,173,118,74),(18,174,119,75),(19,175,120,76),(20,176,121,77),(21,177,122,78),(22,178,123,79),(23,179,124,80),(24,180,125,81),(25,181,126,82),(26,182,113,83),(27,169,114,84),(28,170,115,71),(43,68,157,217),(44,69,158,218),(45,70,159,219),(46,57,160,220),(47,58,161,221),(48,59,162,222),(49,60,163,223),(50,61,164,224),(51,62,165,211),(52,63,166,212),(53,64,167,213),(54,65,168,214),(55,66,155,215),(56,67,156,216),(127,205,145,183),(128,206,146,184),(129,207,147,185),(130,208,148,186),(131,209,149,187),(132,210,150,188),(133,197,151,189),(134,198,152,190),(135,199,153,191),(136,200,154,192),(137,201,141,193),(138,202,142,194),(139,203,143,195),(140,204,144,196)]])
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 | C28.3Q8 | 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 |
Matrix representation of C14.772- 1+4 ►in 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] >;
C14.772- 1+4 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