?
G = C14.222- (1+4) order 448 = 26·7
22nd non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.222- (1+4), C14.552+ (1+4), C22⋊Q8⋊17D7, C4⋊C4.195D14, (C2×Q8).77D14, D14⋊Q8⋊23C2, D14⋊D4.2C2, C22⋊C4.20D14, Dic7⋊3Q8⋊28C2, Dic7⋊Q8⋊17C2, D14.5D4⋊21C2, C28.23D4⋊15C2, (C2×C28).628C23, (C2×C14).184C24, D14⋊C4.26C22, Dic7.7(C4○D4), (C22×C4).246D14, C2.57(D4⋊6D14), Dic7.D4⋊26C2, (C2×D28).152C22, C23.D14⋊24C2, Dic7⋊C4.32C22, C4⋊Dic7.219C22, (Q8×C14).114C22, (C22×D7).75C23, C23.123(C22×D7), C22.205(C23×D7), C23.D7.35C22, C23.23D14⋊25C2, (C22×C14).212C23, (C22×C28).383C22, C7⋊5(C22.36C24), (C4×Dic7).112C22, (C2×Dic7).238C23, C2.23(Q8.10D14), (C2×Dic14).162C22, (C4×C7⋊D4)⋊59C2, C2.55(D7×C4○D4), C4⋊C4⋊D7⋊19C2, C4⋊C4⋊7D7⋊28C2, (C7×C22⋊Q8)⋊20C2, C14.167(C2×C4○D4), (C2×C4×D7).211C22, (C2×C4).54(C22×D7), (C7×C4⋊C4).165C22, (C2×C7⋊D4).131C22, (C7×C22⋊C4).39C22, SmallGroup(448,1093)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 988 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×13], C22, C22 [×9], C7, C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×2], D7 [×2], C14 [×3], C14, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C2×Q8 [×2], Dic7 [×2], Dic7 [×5], C28 [×6], D14 [×6], C2×C14, C2×C14 [×3], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×3], C42⋊2C2 [×2], C4⋊Q8, Dic14 [×3], C4×D7 [×3], D28, C2×Dic7 [×6], C7⋊D4 [×3], C2×C28 [×6], C2×C28, C7×Q8, C22×D7 [×2], C22×C14, C22.36C24, C4×Dic7 [×4], Dic7⋊C4 [×6], C4⋊Dic7, D14⋊C4 [×8], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4 [×3], C2×Dic14 [×2], C2×C4×D7 [×2], C2×D28, C2×C7⋊D4 [×2], C22×C28, Q8×C14, C23.D14, D14⋊D4, Dic7.D4 [×2], Dic7⋊3Q8, C4⋊C4⋊7D7, D14.5D4, D14⋊Q8 [×2], C4⋊C4⋊D7, C4×C7⋊D4, C23.23D14, Dic7⋊Q8, C28.23D4, C7×C22⋊Q8, C14.222- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×2], C24, D14 [×7], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×D7 [×7], C22.36C24, C23×D7, D4⋊6D14, Q8.10D14, D7×C4○D4, C14.222- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=1, c2=a7, d2=a7b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=a7b-1, dbd-1=a7b, be=eb, cd=dc, 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 81 24 68)(2 82 25 69)(3 83 26 70)(4 84 27 57)(5 71 28 58)(6 72 15 59)(7 73 16 60)(8 74 17 61)(9 75 18 62)(10 76 19 63)(11 77 20 64)(12 78 21 65)(13 79 22 66)(14 80 23 67)(29 99 54 85)(30 100 55 86)(31 101 56 87)(32 102 43 88)(33 103 44 89)(34 104 45 90)(35 105 46 91)(36 106 47 92)(37 107 48 93)(38 108 49 94)(39 109 50 95)(40 110 51 96)(41 111 52 97)(42 112 53 98)(113 169 139 185)(114 170 140 186)(115 171 127 187)(116 172 128 188)(117 173 129 189)(118 174 130 190)(119 175 131 191)(120 176 132 192)(121 177 133 193)(122 178 134 194)(123 179 135 195)(124 180 136 196)(125 181 137 183)(126 182 138 184)(141 201 168 215)(142 202 155 216)(143 203 156 217)(144 204 157 218)(145 205 158 219)(146 206 159 220)(147 207 160 221)(148 208 161 222)(149 209 162 223)(150 210 163 224)(151 197 164 211)(152 198 165 212)(153 199 166 213)(154 200 167 214)
(1 195 8 188)(2 194 9 187)(3 193 10 186)(4 192 11 185)(5 191 12 184)(6 190 13 183)(7 189 14 196)(15 174 22 181)(16 173 23 180)(17 172 24 179)(18 171 25 178)(19 170 26 177)(20 169 27 176)(21 182 28 175)(29 224 36 217)(30 223 37 216)(31 222 38 215)(32 221 39 214)(33 220 40 213)(34 219 41 212)(35 218 42 211)(43 207 50 200)(44 206 51 199)(45 205 52 198)(46 204 53 197)(47 203 54 210)(48 202 55 209)(49 201 56 208)(57 113 64 120)(58 126 65 119)(59 125 66 118)(60 124 67 117)(61 123 68 116)(62 122 69 115)(63 121 70 114)(71 138 78 131)(72 137 79 130)(73 136 80 129)(74 135 81 128)(75 134 82 127)(76 133 83 140)(77 132 84 139)(85 143 92 150)(86 142 93 149)(87 141 94 148)(88 154 95 147)(89 153 96 146)(90 152 97 145)(91 151 98 144)(99 156 106 163)(100 155 107 162)(101 168 108 161)(102 167 109 160)(103 166 110 159)(104 165 111 158)(105 164 112 157)
(1 152 17 158)(2 151 18 157)(3 150 19 156)(4 149 20 155)(5 148 21 168)(6 147 22 167)(7 146 23 166)(8 145 24 165)(9 144 25 164)(10 143 26 163)(11 142 27 162)(12 141 28 161)(13 154 15 160)(14 153 16 159)(29 133 47 114)(30 132 48 113)(31 131 49 126)(32 130 50 125)(33 129 51 124)(34 128 52 123)(35 127 53 122)(36 140 54 121)(37 139 55 120)(38 138 56 119)(39 137 43 118)(40 136 44 117)(41 135 45 116)(42 134 46 115)(57 216 77 209)(58 215 78 208)(59 214 79 207)(60 213 80 206)(61 212 81 205)(62 211 82 204)(63 224 83 203)(64 223 84 202)(65 222 71 201)(66 221 72 200)(67 220 73 199)(68 219 74 198)(69 218 75 197)(70 217 76 210)(85 170 106 193)(86 169 107 192)(87 182 108 191)(88 181 109 190)(89 180 110 189)(90 179 111 188)(91 178 112 187)(92 177 99 186)(93 176 100 185)(94 175 101 184)(95 174 102 183)(96 173 103 196)(97 172 104 195)(98 171 105 194)
(1 74 24 61)(2 75 25 62)(3 76 26 63)(4 77 27 64)(5 78 28 65)(6 79 15 66)(7 80 16 67)(8 81 17 68)(9 82 18 69)(10 83 19 70)(11 84 20 57)(12 71 21 58)(13 72 22 59)(14 73 23 60)(29 99 54 85)(30 100 55 86)(31 101 56 87)(32 102 43 88)(33 103 44 89)(34 104 45 90)(35 105 46 91)(36 106 47 92)(37 107 48 93)(38 108 49 94)(39 109 50 95)(40 110 51 96)(41 111 52 97)(42 112 53 98)(113 185 139 169)(114 186 140 170)(115 187 127 171)(116 188 128 172)(117 189 129 173)(118 190 130 174)(119 191 131 175)(120 192 132 176)(121 193 133 177)(122 194 134 178)(123 195 135 179)(124 196 136 180)(125 183 137 181)(126 184 138 182)(141 222 168 208)(142 223 155 209)(143 224 156 210)(144 211 157 197)(145 212 158 198)(146 213 159 199)(147 214 160 200)(148 215 161 201)(149 216 162 202)(150 217 163 203)(151 218 164 204)(152 219 165 205)(153 220 166 206)(154 221 167 207)
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), (1,81,24,68)(2,82,25,69)(3,83,26,70)(4,84,27,57)(5,71,28,58)(6,72,15,59)(7,73,16,60)(8,74,17,61)(9,75,18,62)(10,76,19,63)(11,77,20,64)(12,78,21,65)(13,79,22,66)(14,80,23,67)(29,99,54,85)(30,100,55,86)(31,101,56,87)(32,102,43,88)(33,103,44,89)(34,104,45,90)(35,105,46,91)(36,106,47,92)(37,107,48,93)(38,108,49,94)(39,109,50,95)(40,110,51,96)(41,111,52,97)(42,112,53,98)(113,169,139,185)(114,170,140,186)(115,171,127,187)(116,172,128,188)(117,173,129,189)(118,174,130,190)(119,175,131,191)(120,176,132,192)(121,177,133,193)(122,178,134,194)(123,179,135,195)(124,180,136,196)(125,181,137,183)(126,182,138,184)(141,201,168,215)(142,202,155,216)(143,203,156,217)(144,204,157,218)(145,205,158,219)(146,206,159,220)(147,207,160,221)(148,208,161,222)(149,209,162,223)(150,210,163,224)(151,197,164,211)(152,198,165,212)(153,199,166,213)(154,200,167,214), (1,195,8,188)(2,194,9,187)(3,193,10,186)(4,192,11,185)(5,191,12,184)(6,190,13,183)(7,189,14,196)(15,174,22,181)(16,173,23,180)(17,172,24,179)(18,171,25,178)(19,170,26,177)(20,169,27,176)(21,182,28,175)(29,224,36,217)(30,223,37,216)(31,222,38,215)(32,221,39,214)(33,220,40,213)(34,219,41,212)(35,218,42,211)(43,207,50,200)(44,206,51,199)(45,205,52,198)(46,204,53,197)(47,203,54,210)(48,202,55,209)(49,201,56,208)(57,113,64,120)(58,126,65,119)(59,125,66,118)(60,124,67,117)(61,123,68,116)(62,122,69,115)(63,121,70,114)(71,138,78,131)(72,137,79,130)(73,136,80,129)(74,135,81,128)(75,134,82,127)(76,133,83,140)(77,132,84,139)(85,143,92,150)(86,142,93,149)(87,141,94,148)(88,154,95,147)(89,153,96,146)(90,152,97,145)(91,151,98,144)(99,156,106,163)(100,155,107,162)(101,168,108,161)(102,167,109,160)(103,166,110,159)(104,165,111,158)(105,164,112,157), (1,152,17,158)(2,151,18,157)(3,150,19,156)(4,149,20,155)(5,148,21,168)(6,147,22,167)(7,146,23,166)(8,145,24,165)(9,144,25,164)(10,143,26,163)(11,142,27,162)(12,141,28,161)(13,154,15,160)(14,153,16,159)(29,133,47,114)(30,132,48,113)(31,131,49,126)(32,130,50,125)(33,129,51,124)(34,128,52,123)(35,127,53,122)(36,140,54,121)(37,139,55,120)(38,138,56,119)(39,137,43,118)(40,136,44,117)(41,135,45,116)(42,134,46,115)(57,216,77,209)(58,215,78,208)(59,214,79,207)(60,213,80,206)(61,212,81,205)(62,211,82,204)(63,224,83,203)(64,223,84,202)(65,222,71,201)(66,221,72,200)(67,220,73,199)(68,219,74,198)(69,218,75,197)(70,217,76,210)(85,170,106,193)(86,169,107,192)(87,182,108,191)(88,181,109,190)(89,180,110,189)(90,179,111,188)(91,178,112,187)(92,177,99,186)(93,176,100,185)(94,175,101,184)(95,174,102,183)(96,173,103,196)(97,172,104,195)(98,171,105,194), (1,74,24,61)(2,75,25,62)(3,76,26,63)(4,77,27,64)(5,78,28,65)(6,79,15,66)(7,80,16,67)(8,81,17,68)(9,82,18,69)(10,83,19,70)(11,84,20,57)(12,71,21,58)(13,72,22,59)(14,73,23,60)(29,99,54,85)(30,100,55,86)(31,101,56,87)(32,102,43,88)(33,103,44,89)(34,104,45,90)(35,105,46,91)(36,106,47,92)(37,107,48,93)(38,108,49,94)(39,109,50,95)(40,110,51,96)(41,111,52,97)(42,112,53,98)(113,185,139,169)(114,186,140,170)(115,187,127,171)(116,188,128,172)(117,189,129,173)(118,190,130,174)(119,191,131,175)(120,192,132,176)(121,193,133,177)(122,194,134,178)(123,195,135,179)(124,196,136,180)(125,183,137,181)(126,184,138,182)(141,222,168,208)(142,223,155,209)(143,224,156,210)(144,211,157,197)(145,212,158,198)(146,213,159,199)(147,214,160,200)(148,215,161,201)(149,216,162,202)(150,217,163,203)(151,218,164,204)(152,219,165,205)(153,220,166,206)(154,221,167,207)>;
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), (1,81,24,68)(2,82,25,69)(3,83,26,70)(4,84,27,57)(5,71,28,58)(6,72,15,59)(7,73,16,60)(8,74,17,61)(9,75,18,62)(10,76,19,63)(11,77,20,64)(12,78,21,65)(13,79,22,66)(14,80,23,67)(29,99,54,85)(30,100,55,86)(31,101,56,87)(32,102,43,88)(33,103,44,89)(34,104,45,90)(35,105,46,91)(36,106,47,92)(37,107,48,93)(38,108,49,94)(39,109,50,95)(40,110,51,96)(41,111,52,97)(42,112,53,98)(113,169,139,185)(114,170,140,186)(115,171,127,187)(116,172,128,188)(117,173,129,189)(118,174,130,190)(119,175,131,191)(120,176,132,192)(121,177,133,193)(122,178,134,194)(123,179,135,195)(124,180,136,196)(125,181,137,183)(126,182,138,184)(141,201,168,215)(142,202,155,216)(143,203,156,217)(144,204,157,218)(145,205,158,219)(146,206,159,220)(147,207,160,221)(148,208,161,222)(149,209,162,223)(150,210,163,224)(151,197,164,211)(152,198,165,212)(153,199,166,213)(154,200,167,214), (1,195,8,188)(2,194,9,187)(3,193,10,186)(4,192,11,185)(5,191,12,184)(6,190,13,183)(7,189,14,196)(15,174,22,181)(16,173,23,180)(17,172,24,179)(18,171,25,178)(19,170,26,177)(20,169,27,176)(21,182,28,175)(29,224,36,217)(30,223,37,216)(31,222,38,215)(32,221,39,214)(33,220,40,213)(34,219,41,212)(35,218,42,211)(43,207,50,200)(44,206,51,199)(45,205,52,198)(46,204,53,197)(47,203,54,210)(48,202,55,209)(49,201,56,208)(57,113,64,120)(58,126,65,119)(59,125,66,118)(60,124,67,117)(61,123,68,116)(62,122,69,115)(63,121,70,114)(71,138,78,131)(72,137,79,130)(73,136,80,129)(74,135,81,128)(75,134,82,127)(76,133,83,140)(77,132,84,139)(85,143,92,150)(86,142,93,149)(87,141,94,148)(88,154,95,147)(89,153,96,146)(90,152,97,145)(91,151,98,144)(99,156,106,163)(100,155,107,162)(101,168,108,161)(102,167,109,160)(103,166,110,159)(104,165,111,158)(105,164,112,157), (1,152,17,158)(2,151,18,157)(3,150,19,156)(4,149,20,155)(5,148,21,168)(6,147,22,167)(7,146,23,166)(8,145,24,165)(9,144,25,164)(10,143,26,163)(11,142,27,162)(12,141,28,161)(13,154,15,160)(14,153,16,159)(29,133,47,114)(30,132,48,113)(31,131,49,126)(32,130,50,125)(33,129,51,124)(34,128,52,123)(35,127,53,122)(36,140,54,121)(37,139,55,120)(38,138,56,119)(39,137,43,118)(40,136,44,117)(41,135,45,116)(42,134,46,115)(57,216,77,209)(58,215,78,208)(59,214,79,207)(60,213,80,206)(61,212,81,205)(62,211,82,204)(63,224,83,203)(64,223,84,202)(65,222,71,201)(66,221,72,200)(67,220,73,199)(68,219,74,198)(69,218,75,197)(70,217,76,210)(85,170,106,193)(86,169,107,192)(87,182,108,191)(88,181,109,190)(89,180,110,189)(90,179,111,188)(91,178,112,187)(92,177,99,186)(93,176,100,185)(94,175,101,184)(95,174,102,183)(96,173,103,196)(97,172,104,195)(98,171,105,194), (1,74,24,61)(2,75,25,62)(3,76,26,63)(4,77,27,64)(5,78,28,65)(6,79,15,66)(7,80,16,67)(8,81,17,68)(9,82,18,69)(10,83,19,70)(11,84,20,57)(12,71,21,58)(13,72,22,59)(14,73,23,60)(29,99,54,85)(30,100,55,86)(31,101,56,87)(32,102,43,88)(33,103,44,89)(34,104,45,90)(35,105,46,91)(36,106,47,92)(37,107,48,93)(38,108,49,94)(39,109,50,95)(40,110,51,96)(41,111,52,97)(42,112,53,98)(113,185,139,169)(114,186,140,170)(115,187,127,171)(116,188,128,172)(117,189,129,173)(118,190,130,174)(119,191,131,175)(120,192,132,176)(121,193,133,177)(122,194,134,178)(123,195,135,179)(124,196,136,180)(125,183,137,181)(126,184,138,182)(141,222,168,208)(142,223,155,209)(143,224,156,210)(144,211,157,197)(145,212,158,198)(146,213,159,199)(147,214,160,200)(148,215,161,201)(149,216,162,202)(150,217,163,203)(151,218,164,204)(152,219,165,205)(153,220,166,206)(154,221,167,207) );
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)], [(1,81,24,68),(2,82,25,69),(3,83,26,70),(4,84,27,57),(5,71,28,58),(6,72,15,59),(7,73,16,60),(8,74,17,61),(9,75,18,62),(10,76,19,63),(11,77,20,64),(12,78,21,65),(13,79,22,66),(14,80,23,67),(29,99,54,85),(30,100,55,86),(31,101,56,87),(32,102,43,88),(33,103,44,89),(34,104,45,90),(35,105,46,91),(36,106,47,92),(37,107,48,93),(38,108,49,94),(39,109,50,95),(40,110,51,96),(41,111,52,97),(42,112,53,98),(113,169,139,185),(114,170,140,186),(115,171,127,187),(116,172,128,188),(117,173,129,189),(118,174,130,190),(119,175,131,191),(120,176,132,192),(121,177,133,193),(122,178,134,194),(123,179,135,195),(124,180,136,196),(125,181,137,183),(126,182,138,184),(141,201,168,215),(142,202,155,216),(143,203,156,217),(144,204,157,218),(145,205,158,219),(146,206,159,220),(147,207,160,221),(148,208,161,222),(149,209,162,223),(150,210,163,224),(151,197,164,211),(152,198,165,212),(153,199,166,213),(154,200,167,214)], [(1,195,8,188),(2,194,9,187),(3,193,10,186),(4,192,11,185),(5,191,12,184),(6,190,13,183),(7,189,14,196),(15,174,22,181),(16,173,23,180),(17,172,24,179),(18,171,25,178),(19,170,26,177),(20,169,27,176),(21,182,28,175),(29,224,36,217),(30,223,37,216),(31,222,38,215),(32,221,39,214),(33,220,40,213),(34,219,41,212),(35,218,42,211),(43,207,50,200),(44,206,51,199),(45,205,52,198),(46,204,53,197),(47,203,54,210),(48,202,55,209),(49,201,56,208),(57,113,64,120),(58,126,65,119),(59,125,66,118),(60,124,67,117),(61,123,68,116),(62,122,69,115),(63,121,70,114),(71,138,78,131),(72,137,79,130),(73,136,80,129),(74,135,81,128),(75,134,82,127),(76,133,83,140),(77,132,84,139),(85,143,92,150),(86,142,93,149),(87,141,94,148),(88,154,95,147),(89,153,96,146),(90,152,97,145),(91,151,98,144),(99,156,106,163),(100,155,107,162),(101,168,108,161),(102,167,109,160),(103,166,110,159),(104,165,111,158),(105,164,112,157)], [(1,152,17,158),(2,151,18,157),(3,150,19,156),(4,149,20,155),(5,148,21,168),(6,147,22,167),(7,146,23,166),(8,145,24,165),(9,144,25,164),(10,143,26,163),(11,142,27,162),(12,141,28,161),(13,154,15,160),(14,153,16,159),(29,133,47,114),(30,132,48,113),(31,131,49,126),(32,130,50,125),(33,129,51,124),(34,128,52,123),(35,127,53,122),(36,140,54,121),(37,139,55,120),(38,138,56,119),(39,137,43,118),(40,136,44,117),(41,135,45,116),(42,134,46,115),(57,216,77,209),(58,215,78,208),(59,214,79,207),(60,213,80,206),(61,212,81,205),(62,211,82,204),(63,224,83,203),(64,223,84,202),(65,222,71,201),(66,221,72,200),(67,220,73,199),(68,219,74,198),(69,218,75,197),(70,217,76,210),(85,170,106,193),(86,169,107,192),(87,182,108,191),(88,181,109,190),(89,180,110,189),(90,179,111,188),(91,178,112,187),(92,177,99,186),(93,176,100,185),(94,175,101,184),(95,174,102,183),(96,173,103,196),(97,172,104,195),(98,171,105,194)], [(1,74,24,61),(2,75,25,62),(3,76,26,63),(4,77,27,64),(5,78,28,65),(6,79,15,66),(7,80,16,67),(8,81,17,68),(9,82,18,69),(10,83,19,70),(11,84,20,57),(12,71,21,58),(13,72,22,59),(14,73,23,60),(29,99,54,85),(30,100,55,86),(31,101,56,87),(32,102,43,88),(33,103,44,89),(34,104,45,90),(35,105,46,91),(36,106,47,92),(37,107,48,93),(38,108,49,94),(39,109,50,95),(40,110,51,96),(41,111,52,97),(42,112,53,98),(113,185,139,169),(114,186,140,170),(115,187,127,171),(116,188,128,172),(117,189,129,173),(118,190,130,174),(119,191,131,175),(120,192,132,176),(121,193,133,177),(122,194,134,178),(123,195,135,179),(124,196,136,180),(125,183,137,181),(126,184,138,182),(141,222,168,208),(142,223,155,209),(143,224,156,210),(144,211,157,197),(145,212,158,198),(146,213,159,199),(147,214,160,200),(148,215,161,201),(149,216,162,202),(150,217,163,203),(151,218,164,204),(152,219,165,205),(153,220,166,206),(154,221,167,207)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 18 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 25 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 13 | 2 |
| 12 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 16 |
| 0 | 0 | 0 | 0 | 0 | 0 | 26 | 22 |
| 19 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 6 | 6 |
| 0 | 0 | 0 | 0 | 7 | 26 | 14 | 23 |
| 0 | 0 | 0 | 0 | 26 | 26 | 26 | 26 |
| 0 | 0 | 0 | 0 | 22 | 3 | 22 | 3 |
| 25 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 21 | 8 | 4 | 25 | 0 | 0 | 0 | 0 |
| 7 | 8 | 11 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 2 | 17 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 |
| 17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 16 | 7 | 16 |
| 0 | 0 | 0 | 0 | 26 | 22 | 26 | 22 |
G:=sub<GL(8,GF(29))| [18,25,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,18,25,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,8,2,0,0,0,0,0,0,0,0,9,13,0,0,0,0,0,0,8,2],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,17,0,17,0,0,0,0,0,0,17,0,17,0,0,0,0,0,0,0,0,7,26,0,0,0,0,0,0,16,22,0,0,0,0,0,0,0,0,7,26,0,0,0,0,0,0,16,22],[19,16,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,19,16,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,3,7,26,22,0,0,0,0,3,26,26,3,0,0,0,0,6,14,26,22,0,0,0,0,6,23,26,3],[25,18,21,7,0,0,0,0,4,4,8,8,0,0,0,0,0,0,4,11,0,0,0,0,0,0,25,25,0,0,0,0,0,0,0,0,12,0,17,0,0,0,0,0,27,17,2,12,0,0,0,0,0,0,17,0,0,0,0,0,0,0,2,12],[17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,22,3,7,26,0,0,0,0,13,7,16,22,0,0,0,0,0,0,7,26,0,0,0,0,0,0,16,22] >;
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 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 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 4 | 28 | 28 | 2 | 2 | 4 | ··· | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊6D14 | Q8.10D14 | D7×C4○D4 |
| kernel | C14.222- (1+4) | C23.D14 | D14⋊D4 | Dic7.D4 | Dic7⋊3Q8 | C4⋊C4⋊7D7 | D14.5D4 | D14⋊Q8 | C4⋊C4⋊D7 | C4×C7⋊D4 | C23.23D14 | Dic7⋊Q8 | C28.23D4 | C7×C22⋊Q8 | C22⋊Q8 | Dic7 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{22}2_-^{(1+4)} % in TeX
G:=Group("C14.22ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1093);
// by ID
G=gap.SmallGroup(448,1093);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,555,100,1571,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=1,c^2=a^7,d^2=a^7*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=a^7*b^-1,d*b*d^-1=a^7*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^7*b^2*d>;
// generators/relations
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