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G = C14.322+ 1+4order 448 = 26·7

32nd non-split extension by C14 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C14.322+ 1+4, C14.682- 1+4, C4⋊D45D7, C4⋊C4.89D14, (C2×Dic7)⋊9D4, C287D443C2, C22.2(D4×D7), (C2×D4).89D14, C22⋊C4.4D14, D14⋊D416C2, D14⋊Q813C2, Dic7⋊D48C2, Dic7.45(C2×D4), C14.60(C22×D4), C23.8(C22×D7), (C2×C28).172C23, (C2×C14).141C24, D14⋊C4.69C22, (C2×D28).30C22, (C22×C4).217D14, C4⋊Dic7.44C22, C2.34(D46D14), C22⋊Dic1415C2, (D4×C14).115C22, (C22×C14).12C23, (C22×D7).60C23, C22.162(C23×D7), C23.D7.19C22, Dic7⋊C4.158C22, (C22×C28).310C22, C72(C22.31C24), (C2×Dic7).224C23, C2.26(D4.10D14), (C2×Dic14).151C22, (C22×Dic7).102C22, C2.33(C2×D4×D7), (C7×C4⋊D4)⋊6C2, (C2×C14).4(C2×D4), (C2×D42D7)⋊9C2, (C2×C4×D7).80C22, (C2×Dic7⋊C4)⋊28C2, (C2×C4).35(C22×D7), (C7×C4⋊C4).137C22, (C2×C7⋊D4).24C22, (C7×C22⋊C4).6C22, SmallGroup(448,1050)

Series: Derived Chief Lower central Upper central

C1C2×C14 — C14.322+ 1+4
C1C7C14C2×C14C2×Dic7C22×Dic7C2×D42D7 — C14.322+ 1+4
C7C2×C14 — C14.322+ 1+4
C1C22C4⋊D4

Generators and relations for C14.322+ 1+4
 G = < a,b,c,d,e | a14=b4=e2=1, c2=a7, d2=b2, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=a7b-1, dbd-1=a7b, be=eb, dcd-1=ece=a7c, ede=a7b2d >

Subgroups: 1420 in 294 conjugacy classes, 103 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×12], C22, C22 [×2], C22 [×14], C7, C2×C4 [×2], C2×C4 [×2], C2×C4 [×20], D4 [×16], Q8 [×4], C23, C23 [×2], C23 [×2], D7 [×2], C14 [×3], C14 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4, C4⋊C4 [×7], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8 [×2], C4○D4 [×8], Dic7 [×4], Dic7 [×4], C28 [×4], D14 [×6], C2×C14, C2×C14 [×2], C2×C14 [×8], C2×C4⋊C4, C4⋊D4, C4⋊D4 [×7], C22⋊Q8 [×4], C2×C4○D4 [×2], Dic14 [×4], C4×D7 [×4], D28, C2×Dic7 [×10], C2×Dic7 [×5], C7⋊D4 [×10], C2×C28 [×2], C2×C28 [×2], C2×C28, C7×D4 [×5], C22×D7 [×2], C22×C14, C22×C14 [×2], C22.31C24, Dic7⋊C4 [×6], C4⋊Dic7, D14⋊C4 [×4], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C2×Dic14 [×2], C2×C4×D7 [×2], C2×D28, D42D7 [×8], C22×Dic7 [×2], C22×Dic7 [×2], C2×C7⋊D4 [×6], C22×C28, D4×C14, D4×C14 [×2], C22⋊Dic14 [×2], D14⋊D4 [×2], D14⋊Q8 [×2], C2×Dic7⋊C4, C287D4, Dic7⋊D4 [×4], C7×C4⋊D4, C2×D42D7 [×2], C14.322+ 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- 1+4, C22×D7 [×7], C22.31C24, D4×D7 [×2], C23×D7, C2×D4×D7, D46D14, D4.10D14, C14.322+ 1+4

Smallest permutation representation of C14.322+ 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 57 218 135)(2 58 219 136)(3 59 220 137)(4 60 221 138)(5 61 222 139)(6 62 223 140)(7 63 224 127)(8 64 211 128)(9 65 212 129)(10 66 213 130)(11 67 214 131)(12 68 215 132)(13 69 216 133)(14 70 217 134)(15 55 85 100)(16 56 86 101)(17 43 87 102)(18 44 88 103)(19 45 89 104)(20 46 90 105)(21 47 91 106)(22 48 92 107)(23 49 93 108)(24 50 94 109)(25 51 95 110)(26 52 96 111)(27 53 97 112)(28 54 98 99)(29 180 155 72)(30 181 156 73)(31 182 157 74)(32 169 158 75)(33 170 159 76)(34 171 160 77)(35 172 161 78)(36 173 162 79)(37 174 163 80)(38 175 164 81)(39 176 165 82)(40 177 166 83)(41 178 167 84)(42 179 168 71)(113 143 184 198)(114 144 185 199)(115 145 186 200)(116 146 187 201)(117 147 188 202)(118 148 189 203)(119 149 190 204)(120 150 191 205)(121 151 192 206)(122 152 193 207)(123 153 194 208)(124 154 195 209)(125 141 196 210)(126 142 183 197)
(1 208 8 201)(2 207 9 200)(3 206 10 199)(4 205 11 198)(5 204 12 197)(6 203 13 210)(7 202 14 209)(15 41 22 34)(16 40 23 33)(17 39 24 32)(18 38 25 31)(19 37 26 30)(20 36 27 29)(21 35 28 42)(43 75 50 82)(44 74 51 81)(45 73 52 80)(46 72 53 79)(47 71 54 78)(48 84 55 77)(49 83 56 76)(57 187 64 194)(58 186 65 193)(59 185 66 192)(60 184 67 191)(61 183 68 190)(62 196 69 189)(63 195 70 188)(85 167 92 160)(86 166 93 159)(87 165 94 158)(88 164 95 157)(89 163 96 156)(90 162 97 155)(91 161 98 168)(99 172 106 179)(100 171 107 178)(101 170 108 177)(102 169 109 176)(103 182 110 175)(104 181 111 174)(105 180 112 173)(113 131 120 138)(114 130 121 137)(115 129 122 136)(116 128 123 135)(117 127 124 134)(118 140 125 133)(119 139 126 132)(141 223 148 216)(142 222 149 215)(143 221 150 214)(144 220 151 213)(145 219 152 212)(146 218 153 211)(147 217 154 224)
(1 44 218 103)(2 45 219 104)(3 46 220 105)(4 47 221 106)(5 48 222 107)(6 49 223 108)(7 50 224 109)(8 51 211 110)(9 52 212 111)(10 53 213 112)(11 54 214 99)(12 55 215 100)(13 56 216 101)(14 43 217 102)(15 61 85 139)(16 62 86 140)(17 63 87 127)(18 64 88 128)(19 65 89 129)(20 66 90 130)(21 67 91 131)(22 68 92 132)(23 69 93 133)(24 70 94 134)(25 57 95 135)(26 58 96 136)(27 59 97 137)(28 60 98 138)(29 192 155 121)(30 193 156 122)(31 194 157 123)(32 195 158 124)(33 196 159 125)(34 183 160 126)(35 184 161 113)(36 185 162 114)(37 186 163 115)(38 187 164 116)(39 188 165 117)(40 189 166 118)(41 190 167 119)(42 191 168 120)(71 143 179 198)(72 144 180 199)(73 145 181 200)(74 146 182 201)(75 147 169 202)(76 148 170 203)(77 149 171 204)(78 150 172 205)(79 151 173 206)(80 152 174 207)(81 153 175 208)(82 154 176 209)(83 141 177 210)(84 142 178 197)
(1 74)(2 73)(3 72)(4 71)(5 84)(6 83)(7 82)(8 81)(9 80)(10 79)(11 78)(12 77)(13 76)(14 75)(15 183)(16 196)(17 195)(18 194)(19 193)(20 192)(21 191)(22 190)(23 189)(24 188)(25 187)(26 186)(27 185)(28 184)(29 59)(30 58)(31 57)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 60)(43 209)(44 208)(45 207)(46 206)(47 205)(48 204)(49 203)(50 202)(51 201)(52 200)(53 199)(54 198)(55 197)(56 210)(85 126)(86 125)(87 124)(88 123)(89 122)(90 121)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 143)(100 142)(101 141)(102 154)(103 153)(104 152)(105 151)(106 150)(107 149)(108 148)(109 147)(110 146)(111 145)(112 144)(127 165)(128 164)(129 163)(130 162)(131 161)(132 160)(133 159)(134 158)(135 157)(136 156)(137 155)(138 168)(139 167)(140 166)(169 217)(170 216)(171 215)(172 214)(173 213)(174 212)(175 211)(176 224)(177 223)(178 222)(179 221)(180 220)(181 219)(182 218)

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

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

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

64 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L7A7B7C14A···14I14J···14O14P···14U28A···28L28M···28R
order122222222244444444444477714···1414···1414···1428···2828···28
size111122442828444414141414282828282222···24···48···84···48···8

64 irreducible representations

dim11111111122222244444
type++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2D4D7D14D14D14D142+ 1+42- 1+4D4×D7D46D14D4.10D14
kernelC14.322+ 1+4C22⋊Dic14D14⋊D4D14⋊Q8C2×Dic7⋊C4C287D4Dic7⋊D4C7×C4⋊D4C2×D42D7C2×Dic7C4⋊D4C22⋊C4C4⋊C4C22×C4C2×D4C14C14C22C2C2
# reps12221141243633911666

Matrix representation of C14.322+ 1+4 in GL6(𝔽29)

2800000
0280000
003800
0021800
000018
00001210
,
8110000
18210000
00271878
00112314
00002111
000028
,
1180000
21180000
00241600
002500
002812152
00224314
,
010000
100000
00280114
00028250
0001510
00142401
,
21180000
1180000
002416320
00252721
002812152
00224314

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,3,21,0,0,0,0,8,8,0,0,0,0,0,0,1,12,0,0,0,0,8,10],[8,18,0,0,0,0,11,21,0,0,0,0,0,0,27,11,0,0,0,0,18,2,0,0,0,0,7,3,21,2,0,0,8,14,11,8],[11,21,0,0,0,0,8,18,0,0,0,0,0,0,24,2,28,22,0,0,16,5,12,4,0,0,0,0,15,3,0,0,0,0,2,14],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,14,0,0,0,28,15,24,0,0,11,25,1,0,0,0,4,0,0,1],[21,11,0,0,0,0,18,8,0,0,0,0,0,0,24,2,28,22,0,0,16,5,12,4,0,0,3,27,15,3,0,0,20,21,2,14] >;

C14.322+ 1+4 in GAP, Magma, Sage, TeX

C_{14}._{32}2_+^{1+4}
% in TeX

G:=Group("C14.32ES+(2,2)");
// GroupNames label

G:=SmallGroup(448,1050);
// by ID

G=gap.SmallGroup(448,1050);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,387,1123,570,185,18822]);
// 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,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*b,d*c*d^-1=e*c*e=a^7*c,e*d*e=a^7*b^2*d>;
// generators/relations

׿
×
𝔽