?
G = C14.162- (1+4) order 448 = 26·7
16th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.162- (1+4), C28⋊Q8⋊24C2, C22⋊Q8⋊5D7, C4⋊C4.95D14, (C4×D7).11D4, C4.187(D4×D7), D14.43(C2×D4), C28.232(C2×D4), Dic7.7(C2×D4), D14⋊3Q8⋊14C2, D14⋊2Q8⋊24C2, (C2×C28).52C23, (C2×Q8).124D14, C22⋊C4.14D14, C14.74(C22×D4), D14.D4⋊24C2, D14.5D4⋊16C2, C28.48D4⋊36C2, (C2×C14).172C24, D14⋊C4.21C22, (C22×C4).234D14, Dic7.D4⋊24C2, (C2×D28).221C22, Dic7⋊C4.25C22, C4⋊Dic7.213C22, (Q8×C14).105C22, (C2×Dic7).87C23, C22.193(C23×D7), C23.117(C22×D7), C23.D7.33C22, (C22×C28).252C22, (C22×C14).200C23, C7⋊2(C23.38C23), (C4×Dic7).104C22, (C22×D7).194C23, C2.35(D4.10D14), C2.17(Q8.10D14), (C2×Dic14).246C22, (C2×Q8×D7)⋊6C2, C2.47(C2×D4×D7), C4⋊C4⋊7D7⋊25C2, (C7×C22⋊Q8)⋊8C2, (C2×C4×D7).93C22, (C2×C4○D28).20C2, (C7×C4⋊C4).156C22, (C2×C4).590(C22×D7), (C2×C7⋊D4).120C22, (C7×C22⋊C4).27C22, SmallGroup(448,1081)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1244 in 270 conjugacy classes, 103 normal (43 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×18], D4 [×6], Q8 [×10], C23, C23 [×2], D7 [×3], C14 [×3], C14, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, C2×Q8 [×8], C4○D4 [×4], Dic7 [×2], Dic7 [×5], C28 [×2], C28 [×5], D14 [×2], D14 [×5], C2×C14, C2×C14 [×3], C42⋊C2, C22⋊Q8, C22⋊Q8 [×3], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic14 [×8], C4×D7 [×4], C4×D7 [×6], D28 [×2], C2×Dic7 [×2], C2×Dic7 [×4], C7⋊D4 [×4], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7 [×2], C22×C14, C23.38C23, C4×Dic7 [×2], Dic7⋊C4 [×4], C4⋊Dic7, C4⋊Dic7 [×2], D14⋊C4 [×6], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14 [×2], C2×Dic14 [×2], C2×C4×D7 [×2], C2×C4×D7 [×2], C2×D28, C4○D28 [×4], Q8×D7 [×4], C2×C7⋊D4 [×2], C22×C28, Q8×C14, D14.D4 [×2], Dic7.D4 [×2], C28⋊Q8 [×2], C4⋊C4⋊7D7, D14.5D4 [×2], D14⋊2Q8, C28.48D4, D14⋊3Q8, C7×C22⋊Q8, C2×C4○D28, C2×Q8×D7, C14.162- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C22×D4, 2- (1+4) [×2], C22×D7 [×7], C23.38C23, D4×D7 [×2], C23×D7, C2×D4×D7, Q8.10D14, D4.10D14, C14.162- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=b2, ab=ba, cac=dad-1=a-1, ae=ea, cbc=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)
(1 156 21 153)(2 157 22 154)(3 158 23 141)(4 159 24 142)(5 160 25 143)(6 161 26 144)(7 162 27 145)(8 163 28 146)(9 164 15 147)(10 165 16 148)(11 166 17 149)(12 167 18 150)(13 168 19 151)(14 155 20 152)(29 135 43 124)(30 136 44 125)(31 137 45 126)(32 138 46 113)(33 139 47 114)(34 140 48 115)(35 127 49 116)(36 128 50 117)(37 129 51 118)(38 130 52 119)(39 131 53 120)(40 132 54 121)(41 133 55 122)(42 134 56 123)(57 223 82 201)(58 224 83 202)(59 211 84 203)(60 212 71 204)(61 213 72 205)(62 214 73 206)(63 215 74 207)(64 216 75 208)(65 217 76 209)(66 218 77 210)(67 219 78 197)(68 220 79 198)(69 221 80 199)(70 222 81 200)(85 193 101 176)(86 194 102 177)(87 195 103 178)(88 196 104 179)(89 183 105 180)(90 184 106 181)(91 185 107 182)(92 186 108 169)(93 187 109 170)(94 188 110 171)(95 189 111 172)(96 190 112 173)(97 191 99 174)(98 192 100 175)
(1 153)(2 152)(3 151)(4 150)(5 149)(6 148)(7 147)(8 146)(9 145)(10 144)(11 143)(12 142)(13 141)(14 154)(15 162)(16 161)(17 160)(18 159)(19 158)(20 157)(21 156)(22 155)(23 168)(24 167)(25 166)(26 165)(27 164)(28 163)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 140)(37 139)(38 138)(39 137)(40 136)(41 135)(42 134)(43 122)(44 121)(45 120)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 126)(54 125)(55 124)(56 123)(57 203)(58 202)(59 201)(60 200)(61 199)(62 198)(63 197)(64 210)(65 209)(66 208)(67 207)(68 206)(69 205)(70 204)(71 222)(72 221)(73 220)(74 219)(75 218)(76 217)(77 216)(78 215)(79 214)(80 213)(81 212)(82 211)(83 224)(84 223)(85 189)(86 188)(87 187)(88 186)(89 185)(90 184)(91 183)(92 196)(93 195)(94 194)(95 193)(96 192)(97 191)(98 190)(99 174)(100 173)(101 172)(102 171)(103 170)(104 169)(105 182)(106 181)(107 180)(108 179)(109 178)(110 177)(111 176)(112 175)
(1 58 21 83)(2 57 22 82)(3 70 23 81)(4 69 24 80)(5 68 25 79)(6 67 26 78)(7 66 27 77)(8 65 28 76)(9 64 15 75)(10 63 16 74)(11 62 17 73)(12 61 18 72)(13 60 19 71)(14 59 20 84)(29 105 43 89)(30 104 44 88)(31 103 45 87)(32 102 46 86)(33 101 47 85)(34 100 48 98)(35 99 49 97)(36 112 50 96)(37 111 51 95)(38 110 52 94)(39 109 53 93)(40 108 54 92)(41 107 55 91)(42 106 56 90)(113 187 138 170)(114 186 139 169)(115 185 140 182)(116 184 127 181)(117 183 128 180)(118 196 129 179)(119 195 130 178)(120 194 131 177)(121 193 132 176)(122 192 133 175)(123 191 134 174)(124 190 135 173)(125 189 136 172)(126 188 137 171)(141 207 158 215)(142 206 159 214)(143 205 160 213)(144 204 161 212)(145 203 162 211)(146 202 163 224)(147 201 164 223)(148 200 165 222)(149 199 166 221)(150 198 167 220)(151 197 168 219)(152 210 155 218)(153 209 156 217)(154 208 157 216)
(1 127 21 116)(2 128 22 117)(3 129 23 118)(4 130 24 119)(5 131 25 120)(6 132 26 121)(7 133 27 122)(8 134 28 123)(9 135 15 124)(10 136 16 125)(11 137 17 126)(12 138 18 113)(13 139 19 114)(14 140 20 115)(29 164 43 147)(30 165 44 148)(31 166 45 149)(32 167 46 150)(33 168 47 151)(34 155 48 152)(35 156 49 153)(36 157 50 154)(37 158 51 141)(38 159 52 142)(39 160 53 143)(40 161 54 144)(41 162 55 145)(42 163 56 146)(57 183 82 180)(58 184 83 181)(59 185 84 182)(60 186 71 169)(61 187 72 170)(62 188 73 171)(63 189 74 172)(64 190 75 173)(65 191 76 174)(66 192 77 175)(67 193 78 176)(68 194 79 177)(69 195 80 178)(70 196 81 179)(85 219 101 197)(86 220 102 198)(87 221 103 199)(88 222 104 200)(89 223 105 201)(90 224 106 202)(91 211 107 203)(92 212 108 204)(93 213 109 205)(94 214 110 206)(95 215 111 207)(96 216 112 208)(97 217 99 209)(98 218 100 210)
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), (1,156,21,153)(2,157,22,154)(3,158,23,141)(4,159,24,142)(5,160,25,143)(6,161,26,144)(7,162,27,145)(8,163,28,146)(9,164,15,147)(10,165,16,148)(11,166,17,149)(12,167,18,150)(13,168,19,151)(14,155,20,152)(29,135,43,124)(30,136,44,125)(31,137,45,126)(32,138,46,113)(33,139,47,114)(34,140,48,115)(35,127,49,116)(36,128,50,117)(37,129,51,118)(38,130,52,119)(39,131,53,120)(40,132,54,121)(41,133,55,122)(42,134,56,123)(57,223,82,201)(58,224,83,202)(59,211,84,203)(60,212,71,204)(61,213,72,205)(62,214,73,206)(63,215,74,207)(64,216,75,208)(65,217,76,209)(66,218,77,210)(67,219,78,197)(68,220,79,198)(69,221,80,199)(70,222,81,200)(85,193,101,176)(86,194,102,177)(87,195,103,178)(88,196,104,179)(89,183,105,180)(90,184,106,181)(91,185,107,182)(92,186,108,169)(93,187,109,170)(94,188,110,171)(95,189,111,172)(96,190,112,173)(97,191,99,174)(98,192,100,175), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,154)(15,162)(16,161)(17,160)(18,159)(19,158)(20,157)(21,156)(22,155)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,126)(54,125)(55,124)(56,123)(57,203)(58,202)(59,201)(60,200)(61,199)(62,198)(63,197)(64,210)(65,209)(66,208)(67,207)(68,206)(69,205)(70,204)(71,222)(72,221)(73,220)(74,219)(75,218)(76,217)(77,216)(78,215)(79,214)(80,213)(81,212)(82,211)(83,224)(84,223)(85,189)(86,188)(87,187)(88,186)(89,185)(90,184)(91,183)(92,196)(93,195)(94,194)(95,193)(96,192)(97,191)(98,190)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,182)(106,181)(107,180)(108,179)(109,178)(110,177)(111,176)(112,175), (1,58,21,83)(2,57,22,82)(3,70,23,81)(4,69,24,80)(5,68,25,79)(6,67,26,78)(7,66,27,77)(8,65,28,76)(9,64,15,75)(10,63,16,74)(11,62,17,73)(12,61,18,72)(13,60,19,71)(14,59,20,84)(29,105,43,89)(30,104,44,88)(31,103,45,87)(32,102,46,86)(33,101,47,85)(34,100,48,98)(35,99,49,97)(36,112,50,96)(37,111,51,95)(38,110,52,94)(39,109,53,93)(40,108,54,92)(41,107,55,91)(42,106,56,90)(113,187,138,170)(114,186,139,169)(115,185,140,182)(116,184,127,181)(117,183,128,180)(118,196,129,179)(119,195,130,178)(120,194,131,177)(121,193,132,176)(122,192,133,175)(123,191,134,174)(124,190,135,173)(125,189,136,172)(126,188,137,171)(141,207,158,215)(142,206,159,214)(143,205,160,213)(144,204,161,212)(145,203,162,211)(146,202,163,224)(147,201,164,223)(148,200,165,222)(149,199,166,221)(150,198,167,220)(151,197,168,219)(152,210,155,218)(153,209,156,217)(154,208,157,216), (1,127,21,116)(2,128,22,117)(3,129,23,118)(4,130,24,119)(5,131,25,120)(6,132,26,121)(7,133,27,122)(8,134,28,123)(9,135,15,124)(10,136,16,125)(11,137,17,126)(12,138,18,113)(13,139,19,114)(14,140,20,115)(29,164,43,147)(30,165,44,148)(31,166,45,149)(32,167,46,150)(33,168,47,151)(34,155,48,152)(35,156,49,153)(36,157,50,154)(37,158,51,141)(38,159,52,142)(39,160,53,143)(40,161,54,144)(41,162,55,145)(42,163,56,146)(57,183,82,180)(58,184,83,181)(59,185,84,182)(60,186,71,169)(61,187,72,170)(62,188,73,171)(63,189,74,172)(64,190,75,173)(65,191,76,174)(66,192,77,175)(67,193,78,176)(68,194,79,177)(69,195,80,178)(70,196,81,179)(85,219,101,197)(86,220,102,198)(87,221,103,199)(88,222,104,200)(89,223,105,201)(90,224,106,202)(91,211,107,203)(92,212,108,204)(93,213,109,205)(94,214,110,206)(95,215,111,207)(96,216,112,208)(97,217,99,209)(98,218,100,210)>;
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), (1,156,21,153)(2,157,22,154)(3,158,23,141)(4,159,24,142)(5,160,25,143)(6,161,26,144)(7,162,27,145)(8,163,28,146)(9,164,15,147)(10,165,16,148)(11,166,17,149)(12,167,18,150)(13,168,19,151)(14,155,20,152)(29,135,43,124)(30,136,44,125)(31,137,45,126)(32,138,46,113)(33,139,47,114)(34,140,48,115)(35,127,49,116)(36,128,50,117)(37,129,51,118)(38,130,52,119)(39,131,53,120)(40,132,54,121)(41,133,55,122)(42,134,56,123)(57,223,82,201)(58,224,83,202)(59,211,84,203)(60,212,71,204)(61,213,72,205)(62,214,73,206)(63,215,74,207)(64,216,75,208)(65,217,76,209)(66,218,77,210)(67,219,78,197)(68,220,79,198)(69,221,80,199)(70,222,81,200)(85,193,101,176)(86,194,102,177)(87,195,103,178)(88,196,104,179)(89,183,105,180)(90,184,106,181)(91,185,107,182)(92,186,108,169)(93,187,109,170)(94,188,110,171)(95,189,111,172)(96,190,112,173)(97,191,99,174)(98,192,100,175), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,154)(15,162)(16,161)(17,160)(18,159)(19,158)(20,157)(21,156)(22,155)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,140)(37,139)(38,138)(39,137)(40,136)(41,135)(42,134)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,126)(54,125)(55,124)(56,123)(57,203)(58,202)(59,201)(60,200)(61,199)(62,198)(63,197)(64,210)(65,209)(66,208)(67,207)(68,206)(69,205)(70,204)(71,222)(72,221)(73,220)(74,219)(75,218)(76,217)(77,216)(78,215)(79,214)(80,213)(81,212)(82,211)(83,224)(84,223)(85,189)(86,188)(87,187)(88,186)(89,185)(90,184)(91,183)(92,196)(93,195)(94,194)(95,193)(96,192)(97,191)(98,190)(99,174)(100,173)(101,172)(102,171)(103,170)(104,169)(105,182)(106,181)(107,180)(108,179)(109,178)(110,177)(111,176)(112,175), (1,58,21,83)(2,57,22,82)(3,70,23,81)(4,69,24,80)(5,68,25,79)(6,67,26,78)(7,66,27,77)(8,65,28,76)(9,64,15,75)(10,63,16,74)(11,62,17,73)(12,61,18,72)(13,60,19,71)(14,59,20,84)(29,105,43,89)(30,104,44,88)(31,103,45,87)(32,102,46,86)(33,101,47,85)(34,100,48,98)(35,99,49,97)(36,112,50,96)(37,111,51,95)(38,110,52,94)(39,109,53,93)(40,108,54,92)(41,107,55,91)(42,106,56,90)(113,187,138,170)(114,186,139,169)(115,185,140,182)(116,184,127,181)(117,183,128,180)(118,196,129,179)(119,195,130,178)(120,194,131,177)(121,193,132,176)(122,192,133,175)(123,191,134,174)(124,190,135,173)(125,189,136,172)(126,188,137,171)(141,207,158,215)(142,206,159,214)(143,205,160,213)(144,204,161,212)(145,203,162,211)(146,202,163,224)(147,201,164,223)(148,200,165,222)(149,199,166,221)(150,198,167,220)(151,197,168,219)(152,210,155,218)(153,209,156,217)(154,208,157,216), (1,127,21,116)(2,128,22,117)(3,129,23,118)(4,130,24,119)(5,131,25,120)(6,132,26,121)(7,133,27,122)(8,134,28,123)(9,135,15,124)(10,136,16,125)(11,137,17,126)(12,138,18,113)(13,139,19,114)(14,140,20,115)(29,164,43,147)(30,165,44,148)(31,166,45,149)(32,167,46,150)(33,168,47,151)(34,155,48,152)(35,156,49,153)(36,157,50,154)(37,158,51,141)(38,159,52,142)(39,160,53,143)(40,161,54,144)(41,162,55,145)(42,163,56,146)(57,183,82,180)(58,184,83,181)(59,185,84,182)(60,186,71,169)(61,187,72,170)(62,188,73,171)(63,189,74,172)(64,190,75,173)(65,191,76,174)(66,192,77,175)(67,193,78,176)(68,194,79,177)(69,195,80,178)(70,196,81,179)(85,219,101,197)(86,220,102,198)(87,221,103,199)(88,222,104,200)(89,223,105,201)(90,224,106,202)(91,211,107,203)(92,212,108,204)(93,213,109,205)(94,214,110,206)(95,215,111,207)(96,216,112,208)(97,217,99,209)(98,218,100,210) );
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)], [(1,156,21,153),(2,157,22,154),(3,158,23,141),(4,159,24,142),(5,160,25,143),(6,161,26,144),(7,162,27,145),(8,163,28,146),(9,164,15,147),(10,165,16,148),(11,166,17,149),(12,167,18,150),(13,168,19,151),(14,155,20,152),(29,135,43,124),(30,136,44,125),(31,137,45,126),(32,138,46,113),(33,139,47,114),(34,140,48,115),(35,127,49,116),(36,128,50,117),(37,129,51,118),(38,130,52,119),(39,131,53,120),(40,132,54,121),(41,133,55,122),(42,134,56,123),(57,223,82,201),(58,224,83,202),(59,211,84,203),(60,212,71,204),(61,213,72,205),(62,214,73,206),(63,215,74,207),(64,216,75,208),(65,217,76,209),(66,218,77,210),(67,219,78,197),(68,220,79,198),(69,221,80,199),(70,222,81,200),(85,193,101,176),(86,194,102,177),(87,195,103,178),(88,196,104,179),(89,183,105,180),(90,184,106,181),(91,185,107,182),(92,186,108,169),(93,187,109,170),(94,188,110,171),(95,189,111,172),(96,190,112,173),(97,191,99,174),(98,192,100,175)], [(1,153),(2,152),(3,151),(4,150),(5,149),(6,148),(7,147),(8,146),(9,145),(10,144),(11,143),(12,142),(13,141),(14,154),(15,162),(16,161),(17,160),(18,159),(19,158),(20,157),(21,156),(22,155),(23,168),(24,167),(25,166),(26,165),(27,164),(28,163),(29,133),(30,132),(31,131),(32,130),(33,129),(34,128),(35,127),(36,140),(37,139),(38,138),(39,137),(40,136),(41,135),(42,134),(43,122),(44,121),(45,120),(46,119),(47,118),(48,117),(49,116),(50,115),(51,114),(52,113),(53,126),(54,125),(55,124),(56,123),(57,203),(58,202),(59,201),(60,200),(61,199),(62,198),(63,197),(64,210),(65,209),(66,208),(67,207),(68,206),(69,205),(70,204),(71,222),(72,221),(73,220),(74,219),(75,218),(76,217),(77,216),(78,215),(79,214),(80,213),(81,212),(82,211),(83,224),(84,223),(85,189),(86,188),(87,187),(88,186),(89,185),(90,184),(91,183),(92,196),(93,195),(94,194),(95,193),(96,192),(97,191),(98,190),(99,174),(100,173),(101,172),(102,171),(103,170),(104,169),(105,182),(106,181),(107,180),(108,179),(109,178),(110,177),(111,176),(112,175)], [(1,58,21,83),(2,57,22,82),(3,70,23,81),(4,69,24,80),(5,68,25,79),(6,67,26,78),(7,66,27,77),(8,65,28,76),(9,64,15,75),(10,63,16,74),(11,62,17,73),(12,61,18,72),(13,60,19,71),(14,59,20,84),(29,105,43,89),(30,104,44,88),(31,103,45,87),(32,102,46,86),(33,101,47,85),(34,100,48,98),(35,99,49,97),(36,112,50,96),(37,111,51,95),(38,110,52,94),(39,109,53,93),(40,108,54,92),(41,107,55,91),(42,106,56,90),(113,187,138,170),(114,186,139,169),(115,185,140,182),(116,184,127,181),(117,183,128,180),(118,196,129,179),(119,195,130,178),(120,194,131,177),(121,193,132,176),(122,192,133,175),(123,191,134,174),(124,190,135,173),(125,189,136,172),(126,188,137,171),(141,207,158,215),(142,206,159,214),(143,205,160,213),(144,204,161,212),(145,203,162,211),(146,202,163,224),(147,201,164,223),(148,200,165,222),(149,199,166,221),(150,198,167,220),(151,197,168,219),(152,210,155,218),(153,209,156,217),(154,208,157,216)], [(1,127,21,116),(2,128,22,117),(3,129,23,118),(4,130,24,119),(5,131,25,120),(6,132,26,121),(7,133,27,122),(8,134,28,123),(9,135,15,124),(10,136,16,125),(11,137,17,126),(12,138,18,113),(13,139,19,114),(14,140,20,115),(29,164,43,147),(30,165,44,148),(31,166,45,149),(32,167,46,150),(33,168,47,151),(34,155,48,152),(35,156,49,153),(36,157,50,154),(37,158,51,141),(38,159,52,142),(39,160,53,143),(40,161,54,144),(41,162,55,145),(42,163,56,146),(57,183,82,180),(58,184,83,181),(59,185,84,182),(60,186,71,169),(61,187,72,170),(62,188,73,171),(63,189,74,172),(64,190,75,173),(65,191,76,174),(66,192,77,175),(67,193,78,176),(68,194,79,177),(69,195,80,178),(70,196,81,179),(85,219,101,197),(86,220,102,198),(87,221,103,199),(88,222,104,200),(89,223,105,201),(90,224,106,202),(91,211,107,203),(92,212,108,204),(93,213,109,205),(94,214,110,206),(95,215,111,207),(96,216,112,208),(97,217,99,209),(98,218,100,210)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 19 | 0 | 0 |
| 0 | 0 | 15 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 19 |
| 0 | 0 | 0 | 0 | 15 | 1 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 6 | 0 | 0 |
| 0 | 0 | 20 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 6 |
| 0 | 0 | 0 | 0 | 20 | 13 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 4 | 0 | 0 |
| 0 | 0 | 23 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 4 |
| 0 | 0 | 0 | 0 | 23 | 5 |
| 23 | 21 | 0 | 0 | 0 | 0 |
| 8 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 22 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 25 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,15,0,0,0,0,19,1,0,0,0,0,0,0,25,15,0,0,0,0,19,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,16,20,0,0,0,0,6,13,0,0,0,0,0,0,16,20,0,0,0,0,6,13],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,24,23,0,0,0,0,4,5,0,0,0,0,0,0,24,23,0,0,0,0,4,5],[23,8,0,0,0,0,21,6,0,0,0,0,0,0,0,0,0,25,0,0,0,0,7,0,0,0,0,4,0,0,0,0,22,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,0,0,0,0,0,17] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | ··· | 4N | 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 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 28 | 2 | 2 | 4 | ··· | 4 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | D14 | D14 | D14 | D14 | 2- (1+4) | D4×D7 | Q8.10D14 | D4.10D14 |
| kernel | C14.162- (1+4) | D14.D4 | Dic7.D4 | C28⋊Q8 | C4⋊C4⋊7D7 | D14.5D4 | D14⋊2Q8 | C28.48D4 | D14⋊3Q8 | C7×C22⋊Q8 | C2×C4○D28 | C2×Q8×D7 | C4×D7 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 3 | 6 | 9 | 3 | 3 | 2 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{16}2_-^{(1+4)} % in TeX
G:=Group("C14.16ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1081);
// by ID
G=gap.SmallGroup(448,1081);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,219,100,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,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=b^2*d>;
// generators/relations
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