?
G = C14.242- (1+4) order 448 = 26·7
24th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.242- (1+4), C14.1192+ (1+4), C28⋊Q8⋊28C2, C4⋊C4.99D14, C22⋊Q8⋊20D7, (Q8×Dic7)⋊17C2, D28⋊C4⋊30C2, D14⋊2Q8⋊29C2, C28⋊7D4.16C2, (C2×Q8).133D14, C22⋊C4.22D14, D14.D4⋊27C2, C28.214(C4○D4), C28.23D4⋊16C2, C4.73(D4⋊2D7), (C2×C14).187C24, (C2×C28).175C23, D14⋊C4.28C22, (C22×C4).249D14, C2.37(D4⋊8D14), Dic7.D4⋊27C2, (C2×D28).153C22, Dic7⋊C4.35C22, C4⋊Dic7.312C22, (Q8×C14).117C22, (C2×Dic7).94C23, (C22×D7).78C23, C22.208(C23×D7), C23.125(C22×D7), C23.21D14⋊31C2, (C22×C14).215C23, (C22×C28).262C22, C7⋊6(C22.36C24), (C4×Dic7).115C22, C23.D7.126C22, C2.25(Q8.10D14), (C2×Dic14).164C22, C4⋊C4⋊D7⋊22C2, C14.91(C2×C4○D4), (C7×C22⋊Q8)⋊23C2, C2.50(C2×D4⋊2D7), (C2×C4×D7).104C22, (C7×C4⋊C4).168C22, (C2×C4).593(C22×D7), (C2×C7⋊D4).39C22, (C7×C22⋊C4).42C22, SmallGroup(448,1096)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 988 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C7, C2×C4 [×2], C2×C4 [×4], 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, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C2×Q8 [×2], Dic7 [×6], C28 [×2], C28 [×5], 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 [×2], C4×D7 [×2], D28 [×2], C2×Dic7 [×4], C2×Dic7 [×2], C7⋊D4 [×2], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7 [×2], C22×C14, C22.36C24, C4×Dic7 [×2], C4×Dic7 [×2], Dic7⋊C4 [×2], C4⋊Dic7 [×3], C4⋊Dic7 [×2], D14⋊C4 [×8], C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14 [×2], C2×C4×D7 [×2], C2×D28, C2×C7⋊D4 [×2], C22×C28, Q8×C14, D14.D4 [×2], Dic7.D4 [×2], C28⋊Q8, D28⋊C4, D14⋊2Q8 [×2], C4⋊C4⋊D7 [×2], C23.21D14, C28⋊7D4, Q8×Dic7, C28.23D4, C7×C22⋊Q8, C14.242- (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, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, Q8.10D14, D4⋊8D14, C14.242- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=b2, e2=a7b2, bab-1=cac=a-1, ad=da, ae=ea, cbc=b-1, dbd-1=ebe-1=a7b, cd=dc, ece-1=a7c, 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 22 101 213)(2 21 102 212)(3 20 103 211)(4 19 104 224)(5 18 105 223)(6 17 106 222)(7 16 107 221)(8 15 108 220)(9 28 109 219)(10 27 110 218)(11 26 111 217)(12 25 112 216)(13 24 99 215)(14 23 100 214)(29 95 151 202)(30 94 152 201)(31 93 153 200)(32 92 154 199)(33 91 141 198)(34 90 142 197)(35 89 143 210)(36 88 144 209)(37 87 145 208)(38 86 146 207)(39 85 147 206)(40 98 148 205)(41 97 149 204)(42 96 150 203)(43 80 59 171)(44 79 60 170)(45 78 61 169)(46 77 62 182)(47 76 63 181)(48 75 64 180)(49 74 65 179)(50 73 66 178)(51 72 67 177)(52 71 68 176)(53 84 69 175)(54 83 70 174)(55 82 57 173)(56 81 58 172)(113 191 132 162)(114 190 133 161)(115 189 134 160)(116 188 135 159)(117 187 136 158)(118 186 137 157)(119 185 138 156)(120 184 139 155)(121 183 140 168)(122 196 127 167)(123 195 128 166)(124 194 129 165)(125 193 130 164)(126 192 131 163)
(1 108)(2 107)(3 106)(4 105)(5 104)(6 103)(7 102)(8 101)(9 100)(10 99)(11 112)(12 111)(13 110)(14 109)(15 22)(16 21)(17 20)(18 19)(23 28)(24 27)(25 26)(29 143)(30 142)(31 141)(32 154)(33 153)(34 152)(35 151)(36 150)(37 149)(38 148)(39 147)(40 146)(41 145)(42 144)(43 49)(44 48)(45 47)(50 56)(51 55)(52 54)(57 67)(58 66)(59 65)(60 64)(61 63)(68 70)(71 174)(72 173)(73 172)(74 171)(75 170)(76 169)(77 182)(78 181)(79 180)(80 179)(81 178)(82 177)(83 176)(84 175)(86 98)(87 97)(88 96)(89 95)(90 94)(91 93)(113 135)(114 134)(115 133)(116 132)(117 131)(118 130)(119 129)(120 128)(121 127)(122 140)(123 139)(124 138)(125 137)(126 136)(155 166)(156 165)(157 164)(158 163)(159 162)(160 161)(167 168)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(197 201)(198 200)(202 210)(203 209)(204 208)(205 207)(211 222)(212 221)(213 220)(214 219)(215 218)(216 217)(223 224)
(1 137 101 118)(2 138 102 119)(3 139 103 120)(4 140 104 121)(5 127 105 122)(6 128 106 123)(7 129 107 124)(8 130 108 125)(9 131 109 126)(10 132 110 113)(11 133 111 114)(12 134 112 115)(13 135 99 116)(14 136 100 117)(15 157 220 186)(16 158 221 187)(17 159 222 188)(18 160 223 189)(19 161 224 190)(20 162 211 191)(21 163 212 192)(22 164 213 193)(23 165 214 194)(24 166 215 195)(25 167 216 196)(26 168 217 183)(27 155 218 184)(28 156 219 185)(29 179 151 74)(30 180 152 75)(31 181 153 76)(32 182 154 77)(33 169 141 78)(34 170 142 79)(35 171 143 80)(36 172 144 81)(37 173 145 82)(38 174 146 83)(39 175 147 84)(40 176 148 71)(41 177 149 72)(42 178 150 73)(43 203 59 96)(44 204 60 97)(45 205 61 98)(46 206 62 85)(47 207 63 86)(48 208 64 87)(49 209 65 88)(50 210 66 89)(51 197 67 90)(52 198 68 91)(53 199 69 92)(54 200 70 93)(55 201 57 94)(56 202 58 95)
(1 147 108 32)(2 148 109 33)(3 149 110 34)(4 150 111 35)(5 151 112 36)(6 152 99 37)(7 153 100 38)(8 154 101 39)(9 141 102 40)(10 142 103 41)(11 143 104 42)(12 144 105 29)(13 145 106 30)(14 146 107 31)(15 206 213 92)(16 207 214 93)(17 208 215 94)(18 209 216 95)(19 210 217 96)(20 197 218 97)(21 198 219 98)(22 199 220 85)(23 200 221 86)(24 201 222 87)(25 202 223 88)(26 203 224 89)(27 204 211 90)(28 205 212 91)(43 190 66 168)(44 191 67 155)(45 192 68 156)(46 193 69 157)(47 194 70 158)(48 195 57 159)(49 196 58 160)(50 183 59 161)(51 184 60 162)(52 185 61 163)(53 186 62 164)(54 187 63 165)(55 188 64 166)(56 189 65 167)(71 131 169 119)(72 132 170 120)(73 133 171 121)(74 134 172 122)(75 135 173 123)(76 136 174 124)(77 137 175 125)(78 138 176 126)(79 139 177 113)(80 140 178 114)(81 127 179 115)(82 128 180 116)(83 129 181 117)(84 130 182 118)
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), (1,22,101,213)(2,21,102,212)(3,20,103,211)(4,19,104,224)(5,18,105,223)(6,17,106,222)(7,16,107,221)(8,15,108,220)(9,28,109,219)(10,27,110,218)(11,26,111,217)(12,25,112,216)(13,24,99,215)(14,23,100,214)(29,95,151,202)(30,94,152,201)(31,93,153,200)(32,92,154,199)(33,91,141,198)(34,90,142,197)(35,89,143,210)(36,88,144,209)(37,87,145,208)(38,86,146,207)(39,85,147,206)(40,98,148,205)(41,97,149,204)(42,96,150,203)(43,80,59,171)(44,79,60,170)(45,78,61,169)(46,77,62,182)(47,76,63,181)(48,75,64,180)(49,74,65,179)(50,73,66,178)(51,72,67,177)(52,71,68,176)(53,84,69,175)(54,83,70,174)(55,82,57,173)(56,81,58,172)(113,191,132,162)(114,190,133,161)(115,189,134,160)(116,188,135,159)(117,187,136,158)(118,186,137,157)(119,185,138,156)(120,184,139,155)(121,183,140,168)(122,196,127,167)(123,195,128,166)(124,194,129,165)(125,193,130,164)(126,192,131,163), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,112)(12,111)(13,110)(14,109)(15,22)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,143)(30,142)(31,141)(32,154)(33,153)(34,152)(35,151)(36,150)(37,149)(38,148)(39,147)(40,146)(41,145)(42,144)(43,49)(44,48)(45,47)(50,56)(51,55)(52,54)(57,67)(58,66)(59,65)(60,64)(61,63)(68,70)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,182)(78,181)(79,180)(80,179)(81,178)(82,177)(83,176)(84,175)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,140)(123,139)(124,138)(125,137)(126,136)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161)(167,168)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(197,201)(198,200)(202,210)(203,209)(204,208)(205,207)(211,222)(212,221)(213,220)(214,219)(215,218)(216,217)(223,224), (1,137,101,118)(2,138,102,119)(3,139,103,120)(4,140,104,121)(5,127,105,122)(6,128,106,123)(7,129,107,124)(8,130,108,125)(9,131,109,126)(10,132,110,113)(11,133,111,114)(12,134,112,115)(13,135,99,116)(14,136,100,117)(15,157,220,186)(16,158,221,187)(17,159,222,188)(18,160,223,189)(19,161,224,190)(20,162,211,191)(21,163,212,192)(22,164,213,193)(23,165,214,194)(24,166,215,195)(25,167,216,196)(26,168,217,183)(27,155,218,184)(28,156,219,185)(29,179,151,74)(30,180,152,75)(31,181,153,76)(32,182,154,77)(33,169,141,78)(34,170,142,79)(35,171,143,80)(36,172,144,81)(37,173,145,82)(38,174,146,83)(39,175,147,84)(40,176,148,71)(41,177,149,72)(42,178,150,73)(43,203,59,96)(44,204,60,97)(45,205,61,98)(46,206,62,85)(47,207,63,86)(48,208,64,87)(49,209,65,88)(50,210,66,89)(51,197,67,90)(52,198,68,91)(53,199,69,92)(54,200,70,93)(55,201,57,94)(56,202,58,95), (1,147,108,32)(2,148,109,33)(3,149,110,34)(4,150,111,35)(5,151,112,36)(6,152,99,37)(7,153,100,38)(8,154,101,39)(9,141,102,40)(10,142,103,41)(11,143,104,42)(12,144,105,29)(13,145,106,30)(14,146,107,31)(15,206,213,92)(16,207,214,93)(17,208,215,94)(18,209,216,95)(19,210,217,96)(20,197,218,97)(21,198,219,98)(22,199,220,85)(23,200,221,86)(24,201,222,87)(25,202,223,88)(26,203,224,89)(27,204,211,90)(28,205,212,91)(43,190,66,168)(44,191,67,155)(45,192,68,156)(46,193,69,157)(47,194,70,158)(48,195,57,159)(49,196,58,160)(50,183,59,161)(51,184,60,162)(52,185,61,163)(53,186,62,164)(54,187,63,165)(55,188,64,166)(56,189,65,167)(71,131,169,119)(72,132,170,120)(73,133,171,121)(74,134,172,122)(75,135,173,123)(76,136,174,124)(77,137,175,125)(78,138,176,126)(79,139,177,113)(80,140,178,114)(81,127,179,115)(82,128,180,116)(83,129,181,117)(84,130,182,118)>;
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), (1,22,101,213)(2,21,102,212)(3,20,103,211)(4,19,104,224)(5,18,105,223)(6,17,106,222)(7,16,107,221)(8,15,108,220)(9,28,109,219)(10,27,110,218)(11,26,111,217)(12,25,112,216)(13,24,99,215)(14,23,100,214)(29,95,151,202)(30,94,152,201)(31,93,153,200)(32,92,154,199)(33,91,141,198)(34,90,142,197)(35,89,143,210)(36,88,144,209)(37,87,145,208)(38,86,146,207)(39,85,147,206)(40,98,148,205)(41,97,149,204)(42,96,150,203)(43,80,59,171)(44,79,60,170)(45,78,61,169)(46,77,62,182)(47,76,63,181)(48,75,64,180)(49,74,65,179)(50,73,66,178)(51,72,67,177)(52,71,68,176)(53,84,69,175)(54,83,70,174)(55,82,57,173)(56,81,58,172)(113,191,132,162)(114,190,133,161)(115,189,134,160)(116,188,135,159)(117,187,136,158)(118,186,137,157)(119,185,138,156)(120,184,139,155)(121,183,140,168)(122,196,127,167)(123,195,128,166)(124,194,129,165)(125,193,130,164)(126,192,131,163), (1,108)(2,107)(3,106)(4,105)(5,104)(6,103)(7,102)(8,101)(9,100)(10,99)(11,112)(12,111)(13,110)(14,109)(15,22)(16,21)(17,20)(18,19)(23,28)(24,27)(25,26)(29,143)(30,142)(31,141)(32,154)(33,153)(34,152)(35,151)(36,150)(37,149)(38,148)(39,147)(40,146)(41,145)(42,144)(43,49)(44,48)(45,47)(50,56)(51,55)(52,54)(57,67)(58,66)(59,65)(60,64)(61,63)(68,70)(71,174)(72,173)(73,172)(74,171)(75,170)(76,169)(77,182)(78,181)(79,180)(80,179)(81,178)(82,177)(83,176)(84,175)(86,98)(87,97)(88,96)(89,95)(90,94)(91,93)(113,135)(114,134)(115,133)(116,132)(117,131)(118,130)(119,129)(120,128)(121,127)(122,140)(123,139)(124,138)(125,137)(126,136)(155,166)(156,165)(157,164)(158,163)(159,162)(160,161)(167,168)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(197,201)(198,200)(202,210)(203,209)(204,208)(205,207)(211,222)(212,221)(213,220)(214,219)(215,218)(216,217)(223,224), (1,137,101,118)(2,138,102,119)(3,139,103,120)(4,140,104,121)(5,127,105,122)(6,128,106,123)(7,129,107,124)(8,130,108,125)(9,131,109,126)(10,132,110,113)(11,133,111,114)(12,134,112,115)(13,135,99,116)(14,136,100,117)(15,157,220,186)(16,158,221,187)(17,159,222,188)(18,160,223,189)(19,161,224,190)(20,162,211,191)(21,163,212,192)(22,164,213,193)(23,165,214,194)(24,166,215,195)(25,167,216,196)(26,168,217,183)(27,155,218,184)(28,156,219,185)(29,179,151,74)(30,180,152,75)(31,181,153,76)(32,182,154,77)(33,169,141,78)(34,170,142,79)(35,171,143,80)(36,172,144,81)(37,173,145,82)(38,174,146,83)(39,175,147,84)(40,176,148,71)(41,177,149,72)(42,178,150,73)(43,203,59,96)(44,204,60,97)(45,205,61,98)(46,206,62,85)(47,207,63,86)(48,208,64,87)(49,209,65,88)(50,210,66,89)(51,197,67,90)(52,198,68,91)(53,199,69,92)(54,200,70,93)(55,201,57,94)(56,202,58,95), (1,147,108,32)(2,148,109,33)(3,149,110,34)(4,150,111,35)(5,151,112,36)(6,152,99,37)(7,153,100,38)(8,154,101,39)(9,141,102,40)(10,142,103,41)(11,143,104,42)(12,144,105,29)(13,145,106,30)(14,146,107,31)(15,206,213,92)(16,207,214,93)(17,208,215,94)(18,209,216,95)(19,210,217,96)(20,197,218,97)(21,198,219,98)(22,199,220,85)(23,200,221,86)(24,201,222,87)(25,202,223,88)(26,203,224,89)(27,204,211,90)(28,205,212,91)(43,190,66,168)(44,191,67,155)(45,192,68,156)(46,193,69,157)(47,194,70,158)(48,195,57,159)(49,196,58,160)(50,183,59,161)(51,184,60,162)(52,185,61,163)(53,186,62,164)(54,187,63,165)(55,188,64,166)(56,189,65,167)(71,131,169,119)(72,132,170,120)(73,133,171,121)(74,134,172,122)(75,135,173,123)(76,136,174,124)(77,137,175,125)(78,138,176,126)(79,139,177,113)(80,140,178,114)(81,127,179,115)(82,128,180,116)(83,129,181,117)(84,130,182,118) );
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)], [(1,22,101,213),(2,21,102,212),(3,20,103,211),(4,19,104,224),(5,18,105,223),(6,17,106,222),(7,16,107,221),(8,15,108,220),(9,28,109,219),(10,27,110,218),(11,26,111,217),(12,25,112,216),(13,24,99,215),(14,23,100,214),(29,95,151,202),(30,94,152,201),(31,93,153,200),(32,92,154,199),(33,91,141,198),(34,90,142,197),(35,89,143,210),(36,88,144,209),(37,87,145,208),(38,86,146,207),(39,85,147,206),(40,98,148,205),(41,97,149,204),(42,96,150,203),(43,80,59,171),(44,79,60,170),(45,78,61,169),(46,77,62,182),(47,76,63,181),(48,75,64,180),(49,74,65,179),(50,73,66,178),(51,72,67,177),(52,71,68,176),(53,84,69,175),(54,83,70,174),(55,82,57,173),(56,81,58,172),(113,191,132,162),(114,190,133,161),(115,189,134,160),(116,188,135,159),(117,187,136,158),(118,186,137,157),(119,185,138,156),(120,184,139,155),(121,183,140,168),(122,196,127,167),(123,195,128,166),(124,194,129,165),(125,193,130,164),(126,192,131,163)], [(1,108),(2,107),(3,106),(4,105),(5,104),(6,103),(7,102),(8,101),(9,100),(10,99),(11,112),(12,111),(13,110),(14,109),(15,22),(16,21),(17,20),(18,19),(23,28),(24,27),(25,26),(29,143),(30,142),(31,141),(32,154),(33,153),(34,152),(35,151),(36,150),(37,149),(38,148),(39,147),(40,146),(41,145),(42,144),(43,49),(44,48),(45,47),(50,56),(51,55),(52,54),(57,67),(58,66),(59,65),(60,64),(61,63),(68,70),(71,174),(72,173),(73,172),(74,171),(75,170),(76,169),(77,182),(78,181),(79,180),(80,179),(81,178),(82,177),(83,176),(84,175),(86,98),(87,97),(88,96),(89,95),(90,94),(91,93),(113,135),(114,134),(115,133),(116,132),(117,131),(118,130),(119,129),(120,128),(121,127),(122,140),(123,139),(124,138),(125,137),(126,136),(155,166),(156,165),(157,164),(158,163),(159,162),(160,161),(167,168),(183,196),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190),(197,201),(198,200),(202,210),(203,209),(204,208),(205,207),(211,222),(212,221),(213,220),(214,219),(215,218),(216,217),(223,224)], [(1,137,101,118),(2,138,102,119),(3,139,103,120),(4,140,104,121),(5,127,105,122),(6,128,106,123),(7,129,107,124),(8,130,108,125),(9,131,109,126),(10,132,110,113),(11,133,111,114),(12,134,112,115),(13,135,99,116),(14,136,100,117),(15,157,220,186),(16,158,221,187),(17,159,222,188),(18,160,223,189),(19,161,224,190),(20,162,211,191),(21,163,212,192),(22,164,213,193),(23,165,214,194),(24,166,215,195),(25,167,216,196),(26,168,217,183),(27,155,218,184),(28,156,219,185),(29,179,151,74),(30,180,152,75),(31,181,153,76),(32,182,154,77),(33,169,141,78),(34,170,142,79),(35,171,143,80),(36,172,144,81),(37,173,145,82),(38,174,146,83),(39,175,147,84),(40,176,148,71),(41,177,149,72),(42,178,150,73),(43,203,59,96),(44,204,60,97),(45,205,61,98),(46,206,62,85),(47,207,63,86),(48,208,64,87),(49,209,65,88),(50,210,66,89),(51,197,67,90),(52,198,68,91),(53,199,69,92),(54,200,70,93),(55,201,57,94),(56,202,58,95)], [(1,147,108,32),(2,148,109,33),(3,149,110,34),(4,150,111,35),(5,151,112,36),(6,152,99,37),(7,153,100,38),(8,154,101,39),(9,141,102,40),(10,142,103,41),(11,143,104,42),(12,144,105,29),(13,145,106,30),(14,146,107,31),(15,206,213,92),(16,207,214,93),(17,208,215,94),(18,209,216,95),(19,210,217,96),(20,197,218,97),(21,198,219,98),(22,199,220,85),(23,200,221,86),(24,201,222,87),(25,202,223,88),(26,203,224,89),(27,204,211,90),(28,205,212,91),(43,190,66,168),(44,191,67,155),(45,192,68,156),(46,193,69,157),(47,194,70,158),(48,195,57,159),(49,196,58,160),(50,183,59,161),(51,184,60,162),(52,185,61,163),(53,186,62,164),(54,187,63,165),(55,188,64,166),(56,189,65,167),(71,131,169,119),(72,132,170,120),(73,133,171,121),(74,134,172,122),(75,135,173,123),(76,136,174,124),(77,137,175,125),(78,138,176,126),(79,139,177,113),(80,140,178,114),(81,127,179,115),(82,128,180,116),(83,129,181,117),(84,130,182,118)])
Matrix representation ►G ⊆ GL8(𝔽29)
| 20 | 19 | 0 | 0 | 0 | 0 | 0 | 0 |
| 20 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 22 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 19 | 10 |
| 16 | 3 | 28 | 25 | 0 | 0 | 0 | 0 |
| 11 | 13 | 15 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 20 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 15 | 9 |
| 11 | 22 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 27 | 6 | 28 | 0 | 0 | 0 | 0 | 0 |
| 21 | 13 | 22 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 28 | 22 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 25 | 19 | 13 | 27 | 0 | 0 | 0 | 0 |
| 20 | 11 | 12 | 22 | 0 | 0 | 0 | 0 |
| 22 | 21 | 5 | 10 | 0 | 0 | 0 | 0 |
| 8 | 0 | 19 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 21 | 0 |
| 0 | 0 | 0 | 0 | 0 | 15 | 0 | 21 |
| 0 | 0 | 0 | 0 | 21 | 0 | 14 | 0 |
| 0 | 0 | 0 | 0 | 0 | 21 | 0 | 14 |
| 17 | 0 | 0 | 25 | 0 | 0 | 0 | 0 |
| 0 | 17 | 4 | 18 | 0 | 0 | 0 | 0 |
| 24 | 15 | 12 | 0 | 0 | 0 | 0 | 0 |
| 14 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 0 | 0 |
G:=sub<GL(8,GF(29))| [20,20,0,0,0,0,0,0,19,6,0,0,0,0,0,0,0,0,19,10,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,22,19,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,22,19,0,0,0,0,0,0,10,10],[16,11,0,0,0,0,0,0,3,13,0,0,0,0,0,0,28,15,17,3,0,0,0,0,25,18,0,12,0,0,0,0,0,0,0,0,20,15,0,0,0,0,0,0,10,9,0,0,0,0,0,0,0,0,20,15,0,0,0,0,0,0,10,9],[11,13,27,21,0,0,0,0,22,18,6,13,0,0,0,0,0,0,28,22,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,22,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,22,1],[25,20,22,8,0,0,0,0,19,11,21,0,0,0,0,0,13,12,5,19,0,0,0,0,27,22,10,17,0,0,0,0,0,0,0,0,15,0,21,0,0,0,0,0,0,15,0,21,0,0,0,0,21,0,14,0,0,0,0,0,0,21,0,14],[17,0,24,14,0,0,0,0,0,17,15,0,0,0,0,0,0,4,12,0,0,0,0,0,25,18,0,12,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
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 | 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 | D7 | C4○D4 | D14 | D14 | D14 | D14 | 2+ (1+4) | 2- (1+4) | D4⋊2D7 | Q8.10D14 | D4⋊8D14 |
| kernel | C14.242- (1+4) | D14.D4 | Dic7.D4 | C28⋊Q8 | D28⋊C4 | D14⋊2Q8 | C4⋊C4⋊D7 | C23.21D14 | C28⋊7D4 | Q8×Dic7 | C28.23D4 | C7×C22⋊Q8 | C22⋊Q8 | C28 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 6 | 9 | 3 | 3 | 1 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{24}2_-^{(1+4)} % in TeX
G:=Group("C14.24ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1096);
// by ID
G=gap.SmallGroup(448,1096);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,219,675,570,192,18822]);
// 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=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^7*b,c*d=d*c,e*c*e^-1=a^7*c,e*d*e^-1=b^2*d>;
// generators/relations
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