G = C14.522+ 1+4 order 448 = 26·7
52nd non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.522+ 1+4, C7⋊D4⋊3Q8, C4⋊C4.97D14, C7⋊6(D4⋊3Q8), C22⋊Q8⋊13D7, C22.2(Q8×D7), D14.10(C2×Q8), (C2×Q8).75D14, D14⋊3Q8⋊18C2, D14⋊Q8⋊22C2, Dic7.Q8⋊19C2, (C2×C28).59C23, C22⋊C4.61D14, Dic7.12(C2×Q8), Dic7⋊3Q8⋊27C2, Dic7⋊Q8⋊16C2, C14.38(C22×Q8), (C2×C14).180C24, D14⋊C4.25C22, Dic7⋊4D4.3C2, (C22×C4).242D14, C2.54(D4⋊6D14), Dic7.24(C4○D4), C22⋊Dic14⋊26C2, Dic7⋊C4.30C22, C4⋊Dic7.217C22, (Q8×C14).111C22, (C2×Dic7).91C23, C23.193(C22×D7), C22.201(C23×D7), (C22×C28).380C22, (C22×C14).208C23, (C4×Dic7).109C22, (C22×D7).202C23, C23.D7.120C22, (C2×Dic14).161C22, (C22×Dic7).121C22, (D7×C4⋊C4)⋊28C2, C2.21(C2×Q8×D7), C2.51(D7×C4○D4), (C2×C14).9(C2×Q8), (C4×C7⋊D4).18C2, (C7×C22⋊Q8)⋊16C2, (C2×Dic7⋊C4)⋊41C2, C14.163(C2×C4○D4), (C2×C4×D7).100C22, (C2×C4).50(C22×D7), (C7×C4⋊C4).162C22, (C2×C7⋊D4).127C22, (C7×C22⋊C4).35C22, SmallGroup(448,1089)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C14.522+ 1+4
G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc-1=b-1, dbd-1=a7b, be=eb, cd=dc, ce=ec, ede=a7b2d >
Subgroups: 956 in 228 conjugacy classes, 105 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, D7, C14, C14, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic14, C4×D7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×Q8, C22×D7, C22×C14, D4⋊3Q8, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, Q8×C14, C22⋊Dic14, Dic7⋊4D4, Dic7⋊3Q8, Dic7.Q8, D7×C4⋊C4, D14⋊Q8, C2×Dic7⋊C4, C4×C7⋊D4, Dic7⋊Q8, D14⋊3Q8, C7×C22⋊Q8, C14.522+ 1+4
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C4○D4, C24, D14, C22×Q8, C2×C4○D4, 2+ 1+4, C22×D7, D4⋊3Q8, Q8×D7, C23×D7, D4⋊6D14, C2×Q8×D7, D7×C4○D4, C14.522+ 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)
(1 207 185 223)(2 208 186 224)(3 209 187 211)(4 210 188 212)(5 197 189 213)(6 198 190 214)(7 199 191 215)(8 200 192 216)(9 201 193 217)(10 202 194 218)(11 203 195 219)(12 204 196 220)(13 205 183 221)(14 206 184 222)(15 54 176 76)(16 55 177 77)(17 56 178 78)(18 43 179 79)(19 44 180 80)(20 45 181 81)(21 46 182 82)(22 47 169 83)(23 48 170 84)(24 49 171 71)(25 50 172 72)(26 51 173 73)(27 52 174 74)(28 53 175 75)(29 98 138 61)(30 85 139 62)(31 86 140 63)(32 87 127 64)(33 88 128 65)(34 89 129 66)(35 90 130 67)(36 91 131 68)(37 92 132 69)(38 93 133 70)(39 94 134 57)(40 95 135 58)(41 96 136 59)(42 97 137 60)(99 144 157 119)(100 145 158 120)(101 146 159 121)(102 147 160 122)(103 148 161 123)(104 149 162 124)(105 150 163 125)(106 151 164 126)(107 152 165 113)(108 153 166 114)(109 154 167 115)(110 141 168 116)(111 142 155 117)(112 143 156 118)
(1 182 8 175)(2 169 9 176)(3 170 10 177)(4 171 11 178)(5 172 12 179)(6 173 13 180)(7 174 14 181)(15 186 22 193)(16 187 23 194)(17 188 24 195)(18 189 25 196)(19 190 26 183)(20 191 27 184)(21 192 28 185)(29 151 36 144)(30 152 37 145)(31 153 38 146)(32 154 39 147)(33 141 40 148)(34 142 41 149)(35 143 42 150)(43 197 50 204)(44 198 51 205)(45 199 52 206)(46 200 53 207)(47 201 54 208)(48 202 55 209)(49 203 56 210)(57 160 64 167)(58 161 65 168)(59 162 66 155)(60 163 67 156)(61 164 68 157)(62 165 69 158)(63 166 70 159)(71 219 78 212)(72 220 79 213)(73 221 80 214)(74 222 81 215)(75 223 82 216)(76 224 83 217)(77 211 84 218)(85 107 92 100)(86 108 93 101)(87 109 94 102)(88 110 95 103)(89 111 96 104)(90 112 97 105)(91 99 98 106)(113 132 120 139)(114 133 121 140)(115 134 122 127)(116 135 123 128)(117 136 124 129)(118 137 125 130)(119 138 126 131)
(1 93 185 70)(2 92 186 69)(3 91 187 68)(4 90 188 67)(5 89 189 66)(6 88 190 65)(7 87 191 64)(8 86 192 63)(9 85 193 62)(10 98 194 61)(11 97 195 60)(12 96 196 59)(13 95 183 58)(14 94 184 57)(15 165 176 107)(16 164 177 106)(17 163 178 105)(18 162 179 104)(19 161 180 103)(20 160 181 102)(21 159 182 101)(22 158 169 100)(23 157 170 99)(24 156 171 112)(25 155 172 111)(26 168 173 110)(27 167 174 109)(28 166 175 108)(29 209 138 211)(30 208 139 224)(31 207 140 223)(32 206 127 222)(33 205 128 221)(34 204 129 220)(35 203 130 219)(36 202 131 218)(37 201 132 217)(38 200 133 216)(39 199 134 215)(40 198 135 214)(41 197 136 213)(42 210 137 212)(43 117 79 142)(44 116 80 141)(45 115 81 154)(46 114 82 153)(47 113 83 152)(48 126 84 151)(49 125 71 150)(50 124 72 149)(51 123 73 148)(52 122 74 147)(53 121 75 146)(54 120 76 145)(55 119 77 144)(56 118 78 143)
(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 27)(16 26)(17 25)(18 24)(19 23)(20 22)(29 135)(30 134)(31 133)(32 132)(33 131)(34 130)(35 129)(36 128)(37 127)(38 140)(39 139)(40 138)(41 137)(42 136)(43 49)(44 48)(45 47)(50 56)(51 55)(52 54)(57 85)(58 98)(59 97)(60 96)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 79)(72 78)(73 77)(74 76)(80 84)(81 83)(99 168)(100 167)(101 166)(102 165)(103 164)(104 163)(105 162)(106 161)(107 160)(108 159)(109 158)(110 157)(111 156)(112 155)(113 147)(114 146)(115 145)(116 144)(117 143)(118 142)(119 141)(120 154)(121 153)(122 152)(123 151)(124 150)(125 149)(126 148)(169 181)(170 180)(171 179)(172 178)(173 177)(174 176)(183 187)(184 186)(188 196)(189 195)(190 194)(191 193)(197 203)(198 202)(199 201)(204 210)(205 209)(206 208)(211 221)(212 220)(213 219)(214 218)(215 217)(222 224)
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), (1,207,185,223)(2,208,186,224)(3,209,187,211)(4,210,188,212)(5,197,189,213)(6,198,190,214)(7,199,191,215)(8,200,192,216)(9,201,193,217)(10,202,194,218)(11,203,195,219)(12,204,196,220)(13,205,183,221)(14,206,184,222)(15,54,176,76)(16,55,177,77)(17,56,178,78)(18,43,179,79)(19,44,180,80)(20,45,181,81)(21,46,182,82)(22,47,169,83)(23,48,170,84)(24,49,171,71)(25,50,172,72)(26,51,173,73)(27,52,174,74)(28,53,175,75)(29,98,138,61)(30,85,139,62)(31,86,140,63)(32,87,127,64)(33,88,128,65)(34,89,129,66)(35,90,130,67)(36,91,131,68)(37,92,132,69)(38,93,133,70)(39,94,134,57)(40,95,135,58)(41,96,136,59)(42,97,137,60)(99,144,157,119)(100,145,158,120)(101,146,159,121)(102,147,160,122)(103,148,161,123)(104,149,162,124)(105,150,163,125)(106,151,164,126)(107,152,165,113)(108,153,166,114)(109,154,167,115)(110,141,168,116)(111,142,155,117)(112,143,156,118), (1,182,8,175)(2,169,9,176)(3,170,10,177)(4,171,11,178)(5,172,12,179)(6,173,13,180)(7,174,14,181)(15,186,22,193)(16,187,23,194)(17,188,24,195)(18,189,25,196)(19,190,26,183)(20,191,27,184)(21,192,28,185)(29,151,36,144)(30,152,37,145)(31,153,38,146)(32,154,39,147)(33,141,40,148)(34,142,41,149)(35,143,42,150)(43,197,50,204)(44,198,51,205)(45,199,52,206)(46,200,53,207)(47,201,54,208)(48,202,55,209)(49,203,56,210)(57,160,64,167)(58,161,65,168)(59,162,66,155)(60,163,67,156)(61,164,68,157)(62,165,69,158)(63,166,70,159)(71,219,78,212)(72,220,79,213)(73,221,80,214)(74,222,81,215)(75,223,82,216)(76,224,83,217)(77,211,84,218)(85,107,92,100)(86,108,93,101)(87,109,94,102)(88,110,95,103)(89,111,96,104)(90,112,97,105)(91,99,98,106)(113,132,120,139)(114,133,121,140)(115,134,122,127)(116,135,123,128)(117,136,124,129)(118,137,125,130)(119,138,126,131), (1,93,185,70)(2,92,186,69)(3,91,187,68)(4,90,188,67)(5,89,189,66)(6,88,190,65)(7,87,191,64)(8,86,192,63)(9,85,193,62)(10,98,194,61)(11,97,195,60)(12,96,196,59)(13,95,183,58)(14,94,184,57)(15,165,176,107)(16,164,177,106)(17,163,178,105)(18,162,179,104)(19,161,180,103)(20,160,181,102)(21,159,182,101)(22,158,169,100)(23,157,170,99)(24,156,171,112)(25,155,172,111)(26,168,173,110)(27,167,174,109)(28,166,175,108)(29,209,138,211)(30,208,139,224)(31,207,140,223)(32,206,127,222)(33,205,128,221)(34,204,129,220)(35,203,130,219)(36,202,131,218)(37,201,132,217)(38,200,133,216)(39,199,134,215)(40,198,135,214)(41,197,136,213)(42,210,137,212)(43,117,79,142)(44,116,80,141)(45,115,81,154)(46,114,82,153)(47,113,83,152)(48,126,84,151)(49,125,71,150)(50,124,72,149)(51,123,73,148)(52,122,74,147)(53,121,75,146)(54,120,76,145)(55,119,77,144)(56,118,78,143), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(29,135)(30,134)(31,133)(32,132)(33,131)(34,130)(35,129)(36,128)(37,127)(38,140)(39,139)(40,138)(41,137)(42,136)(43,49)(44,48)(45,47)(50,56)(51,55)(52,54)(57,85)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(99,168)(100,167)(101,166)(102,165)(103,164)(104,163)(105,162)(106,161)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,154)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(169,181)(170,180)(171,179)(172,178)(173,177)(174,176)(183,187)(184,186)(188,196)(189,195)(190,194)(191,193)(197,203)(198,202)(199,201)(204,210)(205,209)(206,208)(211,221)(212,220)(213,219)(214,218)(215,217)(222,224)>;
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), (1,207,185,223)(2,208,186,224)(3,209,187,211)(4,210,188,212)(5,197,189,213)(6,198,190,214)(7,199,191,215)(8,200,192,216)(9,201,193,217)(10,202,194,218)(11,203,195,219)(12,204,196,220)(13,205,183,221)(14,206,184,222)(15,54,176,76)(16,55,177,77)(17,56,178,78)(18,43,179,79)(19,44,180,80)(20,45,181,81)(21,46,182,82)(22,47,169,83)(23,48,170,84)(24,49,171,71)(25,50,172,72)(26,51,173,73)(27,52,174,74)(28,53,175,75)(29,98,138,61)(30,85,139,62)(31,86,140,63)(32,87,127,64)(33,88,128,65)(34,89,129,66)(35,90,130,67)(36,91,131,68)(37,92,132,69)(38,93,133,70)(39,94,134,57)(40,95,135,58)(41,96,136,59)(42,97,137,60)(99,144,157,119)(100,145,158,120)(101,146,159,121)(102,147,160,122)(103,148,161,123)(104,149,162,124)(105,150,163,125)(106,151,164,126)(107,152,165,113)(108,153,166,114)(109,154,167,115)(110,141,168,116)(111,142,155,117)(112,143,156,118), (1,182,8,175)(2,169,9,176)(3,170,10,177)(4,171,11,178)(5,172,12,179)(6,173,13,180)(7,174,14,181)(15,186,22,193)(16,187,23,194)(17,188,24,195)(18,189,25,196)(19,190,26,183)(20,191,27,184)(21,192,28,185)(29,151,36,144)(30,152,37,145)(31,153,38,146)(32,154,39,147)(33,141,40,148)(34,142,41,149)(35,143,42,150)(43,197,50,204)(44,198,51,205)(45,199,52,206)(46,200,53,207)(47,201,54,208)(48,202,55,209)(49,203,56,210)(57,160,64,167)(58,161,65,168)(59,162,66,155)(60,163,67,156)(61,164,68,157)(62,165,69,158)(63,166,70,159)(71,219,78,212)(72,220,79,213)(73,221,80,214)(74,222,81,215)(75,223,82,216)(76,224,83,217)(77,211,84,218)(85,107,92,100)(86,108,93,101)(87,109,94,102)(88,110,95,103)(89,111,96,104)(90,112,97,105)(91,99,98,106)(113,132,120,139)(114,133,121,140)(115,134,122,127)(116,135,123,128)(117,136,124,129)(118,137,125,130)(119,138,126,131), (1,93,185,70)(2,92,186,69)(3,91,187,68)(4,90,188,67)(5,89,189,66)(6,88,190,65)(7,87,191,64)(8,86,192,63)(9,85,193,62)(10,98,194,61)(11,97,195,60)(12,96,196,59)(13,95,183,58)(14,94,184,57)(15,165,176,107)(16,164,177,106)(17,163,178,105)(18,162,179,104)(19,161,180,103)(20,160,181,102)(21,159,182,101)(22,158,169,100)(23,157,170,99)(24,156,171,112)(25,155,172,111)(26,168,173,110)(27,167,174,109)(28,166,175,108)(29,209,138,211)(30,208,139,224)(31,207,140,223)(32,206,127,222)(33,205,128,221)(34,204,129,220)(35,203,130,219)(36,202,131,218)(37,201,132,217)(38,200,133,216)(39,199,134,215)(40,198,135,214)(41,197,136,213)(42,210,137,212)(43,117,79,142)(44,116,80,141)(45,115,81,154)(46,114,82,153)(47,113,83,152)(48,126,84,151)(49,125,71,150)(50,124,72,149)(51,123,73,148)(52,122,74,147)(53,121,75,146)(54,120,76,145)(55,119,77,144)(56,118,78,143), (2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)(29,135)(30,134)(31,133)(32,132)(33,131)(34,130)(35,129)(36,128)(37,127)(38,140)(39,139)(40,138)(41,137)(42,136)(43,49)(44,48)(45,47)(50,56)(51,55)(52,54)(57,85)(58,98)(59,97)(60,96)(61,95)(62,94)(63,93)(64,92)(65,91)(66,90)(67,89)(68,88)(69,87)(70,86)(71,79)(72,78)(73,77)(74,76)(80,84)(81,83)(99,168)(100,167)(101,166)(102,165)(103,164)(104,163)(105,162)(106,161)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,147)(114,146)(115,145)(116,144)(117,143)(118,142)(119,141)(120,154)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(169,181)(170,180)(171,179)(172,178)(173,177)(174,176)(183,187)(184,186)(188,196)(189,195)(190,194)(191,193)(197,203)(198,202)(199,201)(204,210)(205,209)(206,208)(211,221)(212,220)(213,219)(214,218)(215,217)(222,224) );
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)], [(1,207,185,223),(2,208,186,224),(3,209,187,211),(4,210,188,212),(5,197,189,213),(6,198,190,214),(7,199,191,215),(8,200,192,216),(9,201,193,217),(10,202,194,218),(11,203,195,219),(12,204,196,220),(13,205,183,221),(14,206,184,222),(15,54,176,76),(16,55,177,77),(17,56,178,78),(18,43,179,79),(19,44,180,80),(20,45,181,81),(21,46,182,82),(22,47,169,83),(23,48,170,84),(24,49,171,71),(25,50,172,72),(26,51,173,73),(27,52,174,74),(28,53,175,75),(29,98,138,61),(30,85,139,62),(31,86,140,63),(32,87,127,64),(33,88,128,65),(34,89,129,66),(35,90,130,67),(36,91,131,68),(37,92,132,69),(38,93,133,70),(39,94,134,57),(40,95,135,58),(41,96,136,59),(42,97,137,60),(99,144,157,119),(100,145,158,120),(101,146,159,121),(102,147,160,122),(103,148,161,123),(104,149,162,124),(105,150,163,125),(106,151,164,126),(107,152,165,113),(108,153,166,114),(109,154,167,115),(110,141,168,116),(111,142,155,117),(112,143,156,118)], [(1,182,8,175),(2,169,9,176),(3,170,10,177),(4,171,11,178),(5,172,12,179),(6,173,13,180),(7,174,14,181),(15,186,22,193),(16,187,23,194),(17,188,24,195),(18,189,25,196),(19,190,26,183),(20,191,27,184),(21,192,28,185),(29,151,36,144),(30,152,37,145),(31,153,38,146),(32,154,39,147),(33,141,40,148),(34,142,41,149),(35,143,42,150),(43,197,50,204),(44,198,51,205),(45,199,52,206),(46,200,53,207),(47,201,54,208),(48,202,55,209),(49,203,56,210),(57,160,64,167),(58,161,65,168),(59,162,66,155),(60,163,67,156),(61,164,68,157),(62,165,69,158),(63,166,70,159),(71,219,78,212),(72,220,79,213),(73,221,80,214),(74,222,81,215),(75,223,82,216),(76,224,83,217),(77,211,84,218),(85,107,92,100),(86,108,93,101),(87,109,94,102),(88,110,95,103),(89,111,96,104),(90,112,97,105),(91,99,98,106),(113,132,120,139),(114,133,121,140),(115,134,122,127),(116,135,123,128),(117,136,124,129),(118,137,125,130),(119,138,126,131)], [(1,93,185,70),(2,92,186,69),(3,91,187,68),(4,90,188,67),(5,89,189,66),(6,88,190,65),(7,87,191,64),(8,86,192,63),(9,85,193,62),(10,98,194,61),(11,97,195,60),(12,96,196,59),(13,95,183,58),(14,94,184,57),(15,165,176,107),(16,164,177,106),(17,163,178,105),(18,162,179,104),(19,161,180,103),(20,160,181,102),(21,159,182,101),(22,158,169,100),(23,157,170,99),(24,156,171,112),(25,155,172,111),(26,168,173,110),(27,167,174,109),(28,166,175,108),(29,209,138,211),(30,208,139,224),(31,207,140,223),(32,206,127,222),(33,205,128,221),(34,204,129,220),(35,203,130,219),(36,202,131,218),(37,201,132,217),(38,200,133,216),(39,199,134,215),(40,198,135,214),(41,197,136,213),(42,210,137,212),(43,117,79,142),(44,116,80,141),(45,115,81,154),(46,114,82,153),(47,113,83,152),(48,126,84,151),(49,125,71,150),(50,124,72,149),(51,123,73,148),(52,122,74,147),(53,121,75,146),(54,120,76,145),(55,119,77,144),(56,118,78,143)], [(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(15,27),(16,26),(17,25),(18,24),(19,23),(20,22),(29,135),(30,134),(31,133),(32,132),(33,131),(34,130),(35,129),(36,128),(37,127),(38,140),(39,139),(40,138),(41,137),(42,136),(43,49),(44,48),(45,47),(50,56),(51,55),(52,54),(57,85),(58,98),(59,97),(60,96),(61,95),(62,94),(63,93),(64,92),(65,91),(66,90),(67,89),(68,88),(69,87),(70,86),(71,79),(72,78),(73,77),(74,76),(80,84),(81,83),(99,168),(100,167),(101,166),(102,165),(103,164),(104,163),(105,162),(106,161),(107,160),(108,159),(109,158),(110,157),(111,156),(112,155),(113,147),(114,146),(115,145),(116,144),(117,143),(118,142),(119,141),(120,154),(121,153),(122,152),(123,151),(124,150),(125,149),(126,148),(169,181),(170,180),(171,179),(172,178),(173,177),(174,176),(183,187),(184,186),(188,196),(189,195),(190,194),(191,193),(197,203),(198,202),(199,201),(204,210),(205,209),(206,208),(211,221),(212,220),(213,219),(214,218),(215,217),(222,224)]])
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 4Q | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 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 | 2 | 2 | 14 | 14 | 2 | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 28 | 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 | 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 | C2 | Q8 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ 1+4 | Q8×D7 | D4⋊6D14 | D7×C4○D4 |
| kernel | C14.522+ 1+4 | C22⋊Dic14 | Dic7⋊4D4 | Dic7⋊3Q8 | Dic7.Q8 | D7×C4⋊C4 | D14⋊Q8 | C2×Dic7⋊C4 | C4×C7⋊D4 | Dic7⋊Q8 | D14⋊3Q8 | C7×C22⋊Q8 | C7⋊D4 | C22⋊Q8 | Dic7 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
Matrix representation of C14.522+ 1+4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 21 | 0 | 0 |
| 0 | 0 | 8 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 23 | 8 | 0 | 0 | 0 | 0 |
| 28 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 22 | 1 |
| 7 | 15 | 0 | 0 | 0 | 0 |
| 16 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 7 | 15 | 0 | 0 | 0 | 0 |
| 16 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 3 | 0 | 0 |
| 0 | 0 | 8 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 7 |
| 0 | 0 | 0 | 0 | 19 | 10 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 26 | 0 | 0 |
| 0 | 0 | 21 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 7 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,26,8,0,0,0,0,21,21,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[23,28,0,0,0,0,8,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,22,0,0,0,0,0,1],[7,16,0,0,0,0,15,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[7,16,0,0,0,0,15,22,0,0,0,0,0,0,8,8,0,0,0,0,3,21,0,0,0,0,0,0,19,19,0,0,0,0,7,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,21,0,0,0,0,26,8,0,0,0,0,0,0,1,7,0,0,0,0,0,28] >;
C14.522+ 1+4 in GAP, Magma, Sage, TeX
C_{14}._{52}2_+^{1+4} % in TeX
G:=Group("C14.52ES+(2,2)"); // GroupNames label
G:=SmallGroup(448,1089);
// by ID
G=gap.SmallGroup(448,1089);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,100,570,409,80,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=e^2=1,c^2=a^7,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^7*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a^7*b^2*d>;
// generators/relations