?
G = C14.232- (1+4) order 448 = 26·7
23rd non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.232- (1+4), C28⋊Q8⋊27C2, C22⋊Q8⋊18D7, C4⋊C4.196D14, (Q8×Dic7)⋊15C2, D14⋊3Q8⋊22C2, (C2×C28).62C23, (C2×Q8).131D14, C22⋊C4.21D14, Dic7⋊3Q8⋊29C2, C28.212(C4○D4), (C2×C14).185C24, (C22×C4).247D14, C4.101(D4⋊2D7), D14⋊C4.130C22, Dic7.37(C4○D4), C23.D14⋊25C2, Dic7⋊C4.33C22, C4⋊Dic7.376C22, Dic7.D4.2C2, (Q8×C14).115C22, (C22×D7).76C23, C23.124(C22×D7), C22.206(C23×D7), C23.21D14⋊30C2, (C22×C14).213C23, (C22×C28).260C22, C7⋊5(C22.50C24), (C2×Dic7).239C23, (C4×Dic7).113C22, C23.D7.124C22, C2.24(Q8.10D14), (C2×Dic14).163C22, C2.56(D7×C4○D4), C4⋊C4⋊7D7⋊29C2, C4⋊C4⋊D7⋊20C2, (C4×C7⋊D4).10C2, (C7×C22⋊Q8)⋊21C2, C14.168(C2×C4○D4), C2.48(C2×D4⋊2D7), (C2×C4×D7).102C22, (C2×C4).55(C22×D7), (C7×C4⋊C4).166C22, (C2×C7⋊D4).132C22, (C7×C22⋊C4).40C22, SmallGroup(448,1094)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 828 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], Q8 [×6], C23, C23, D7, C14 [×3], C14, C42 [×7], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×9], C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8 [×2], Dic7 [×2], Dic7 [×6], C28 [×2], C28 [×5], D14 [×3], C2×C14, C2×C14 [×3], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8, C22⋊Q8, C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, Dic14 [×4], C4×D7 [×2], C2×Dic7 [×3], C2×Dic7 [×4], C7⋊D4 [×2], C2×C28 [×2], C2×C28 [×4], C2×C28 [×2], C7×Q8 [×2], C22×D7, C22×C14, C22.50C24, C4×Dic7, C4×Dic7 [×6], Dic7⋊C4, Dic7⋊C4 [×4], C4⋊Dic7 [×2], C4⋊Dic7 [×2], D14⋊C4, D14⋊C4 [×4], C23.D7, C23.D7 [×2], C7×C22⋊C4 [×2], C7×C4⋊C4, C7×C4⋊C4 [×2], C2×Dic14 [×2], C2×C4×D7, C2×C7⋊D4, C22×C28, Q8×C14, C23.D14 [×2], Dic7.D4 [×2], Dic7⋊3Q8 [×2], C28⋊Q8, C4⋊C4⋊7D7, C4⋊C4⋊D7 [×2], C23.21D14, C4×C7⋊D4, Q8×Dic7, D14⋊3Q8, C7×C22⋊Q8, C14.232- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D7, C4○D4 [×4], C24, D14 [×7], C2×C4○D4 [×2], 2- (1+4), C22×D7 [×7], C22.50C24, D4⋊2D7 [×2], C23×D7, C2×D4⋊2D7, Q8.10D14, D7×C4○D4, C14.232- (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, bd=db, 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)
(1 168 27 154)(2 155 28 141)(3 156 15 142)(4 157 16 143)(5 158 17 144)(6 159 18 145)(7 160 19 146)(8 161 20 147)(9 162 21 148)(10 163 22 149)(11 164 23 150)(12 165 24 151)(13 166 25 152)(14 167 26 153)(29 128 55 126)(30 129 56 113)(31 130 43 114)(32 131 44 115)(33 132 45 116)(34 133 46 117)(35 134 47 118)(36 135 48 119)(37 136 49 120)(38 137 50 121)(39 138 51 122)(40 139 52 123)(41 140 53 124)(42 127 54 125)(57 220 74 200)(58 221 75 201)(59 222 76 202)(60 223 77 203)(61 224 78 204)(62 211 79 205)(63 212 80 206)(64 213 81 207)(65 214 82 208)(66 215 83 209)(67 216 84 210)(68 217 71 197)(69 218 72 198)(70 219 73 199)(85 186 110 181)(86 187 111 182)(87 188 112 169)(88 189 99 170)(89 190 100 171)(90 191 101 172)(91 192 102 173)(92 193 103 174)(93 194 104 175)(94 195 105 176)(95 196 106 177)(96 183 107 178)(97 184 108 179)(98 185 109 180)
(1 154 8 147)(2 153 9 146)(3 152 10 145)(4 151 11 144)(5 150 12 143)(6 149 13 142)(7 148 14 141)(15 166 22 159)(16 165 23 158)(17 164 24 157)(18 163 25 156)(19 162 26 155)(20 161 27 168)(21 160 28 167)(29 133 36 140)(30 132 37 139)(31 131 38 138)(32 130 39 137)(33 129 40 136)(34 128 41 135)(35 127 42 134)(43 115 50 122)(44 114 51 121)(45 113 52 120)(46 126 53 119)(47 125 54 118)(48 124 55 117)(49 123 56 116)(57 199 64 206)(58 198 65 205)(59 197 66 204)(60 210 67 203)(61 209 68 202)(62 208 69 201)(63 207 70 200)(71 222 78 215)(72 221 79 214)(73 220 80 213)(74 219 81 212)(75 218 82 211)(76 217 83 224)(77 216 84 223)(85 192 92 185)(86 191 93 184)(87 190 94 183)(88 189 95 196)(89 188 96 195)(90 187 97 194)(91 186 98 193)(99 170 106 177)(100 169 107 176)(101 182 108 175)(102 181 109 174)(103 180 110 173)(104 179 111 172)(105 178 112 171)
(1 67 20 77)(2 66 21 76)(3 65 22 75)(4 64 23 74)(5 63 24 73)(6 62 25 72)(7 61 26 71)(8 60 27 84)(9 59 28 83)(10 58 15 82)(11 57 16 81)(12 70 17 80)(13 69 18 79)(14 68 19 78)(29 105 48 87)(30 104 49 86)(31 103 50 85)(32 102 51 98)(33 101 52 97)(34 100 53 96)(35 99 54 95)(36 112 55 94)(37 111 56 93)(38 110 43 92)(39 109 44 91)(40 108 45 90)(41 107 46 89)(42 106 47 88)(113 194 136 182)(114 193 137 181)(115 192 138 180)(116 191 139 179)(117 190 140 178)(118 189 127 177)(119 188 128 176)(120 187 129 175)(121 186 130 174)(122 185 131 173)(123 184 132 172)(124 183 133 171)(125 196 134 170)(126 195 135 169)(141 209 162 222)(142 208 163 221)(143 207 164 220)(144 206 165 219)(145 205 166 218)(146 204 167 217)(147 203 168 216)(148 202 155 215)(149 201 156 214)(150 200 157 213)(151 199 158 212)(152 198 159 211)(153 197 160 224)(154 210 161 223)
(1 127 27 125)(2 128 28 126)(3 129 15 113)(4 130 16 114)(5 131 17 115)(6 132 18 116)(7 133 19 117)(8 134 20 118)(9 135 21 119)(10 136 22 120)(11 137 23 121)(12 138 24 122)(13 139 25 123)(14 140 26 124)(29 155 55 141)(30 156 56 142)(31 157 43 143)(32 158 44 144)(33 159 45 145)(34 160 46 146)(35 161 47 147)(36 162 48 148)(37 163 49 149)(38 164 50 150)(39 165 51 151)(40 166 52 152)(41 167 53 153)(42 168 54 154)(57 186 74 181)(58 187 75 182)(59 188 76 169)(60 189 77 170)(61 190 78 171)(62 191 79 172)(63 192 80 173)(64 193 81 174)(65 194 82 175)(66 195 83 176)(67 196 84 177)(68 183 71 178)(69 184 72 179)(70 185 73 180)(85 220 110 200)(86 221 111 201)(87 222 112 202)(88 223 99 203)(89 224 100 204)(90 211 101 205)(91 212 102 206)(92 213 103 207)(93 214 104 208)(94 215 105 209)(95 216 106 210)(96 217 107 197)(97 218 108 198)(98 219 109 199)
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), (1,168,27,154)(2,155,28,141)(3,156,15,142)(4,157,16,143)(5,158,17,144)(6,159,18,145)(7,160,19,146)(8,161,20,147)(9,162,21,148)(10,163,22,149)(11,164,23,150)(12,165,24,151)(13,166,25,152)(14,167,26,153)(29,128,55,126)(30,129,56,113)(31,130,43,114)(32,131,44,115)(33,132,45,116)(34,133,46,117)(35,134,47,118)(36,135,48,119)(37,136,49,120)(38,137,50,121)(39,138,51,122)(40,139,52,123)(41,140,53,124)(42,127,54,125)(57,220,74,200)(58,221,75,201)(59,222,76,202)(60,223,77,203)(61,224,78,204)(62,211,79,205)(63,212,80,206)(64,213,81,207)(65,214,82,208)(66,215,83,209)(67,216,84,210)(68,217,71,197)(69,218,72,198)(70,219,73,199)(85,186,110,181)(86,187,111,182)(87,188,112,169)(88,189,99,170)(89,190,100,171)(90,191,101,172)(91,192,102,173)(92,193,103,174)(93,194,104,175)(94,195,105,176)(95,196,106,177)(96,183,107,178)(97,184,108,179)(98,185,109,180), (1,154,8,147)(2,153,9,146)(3,152,10,145)(4,151,11,144)(5,150,12,143)(6,149,13,142)(7,148,14,141)(15,166,22,159)(16,165,23,158)(17,164,24,157)(18,163,25,156)(19,162,26,155)(20,161,27,168)(21,160,28,167)(29,133,36,140)(30,132,37,139)(31,131,38,138)(32,130,39,137)(33,129,40,136)(34,128,41,135)(35,127,42,134)(43,115,50,122)(44,114,51,121)(45,113,52,120)(46,126,53,119)(47,125,54,118)(48,124,55,117)(49,123,56,116)(57,199,64,206)(58,198,65,205)(59,197,66,204)(60,210,67,203)(61,209,68,202)(62,208,69,201)(63,207,70,200)(71,222,78,215)(72,221,79,214)(73,220,80,213)(74,219,81,212)(75,218,82,211)(76,217,83,224)(77,216,84,223)(85,192,92,185)(86,191,93,184)(87,190,94,183)(88,189,95,196)(89,188,96,195)(90,187,97,194)(91,186,98,193)(99,170,106,177)(100,169,107,176)(101,182,108,175)(102,181,109,174)(103,180,110,173)(104,179,111,172)(105,178,112,171), (1,67,20,77)(2,66,21,76)(3,65,22,75)(4,64,23,74)(5,63,24,73)(6,62,25,72)(7,61,26,71)(8,60,27,84)(9,59,28,83)(10,58,15,82)(11,57,16,81)(12,70,17,80)(13,69,18,79)(14,68,19,78)(29,105,48,87)(30,104,49,86)(31,103,50,85)(32,102,51,98)(33,101,52,97)(34,100,53,96)(35,99,54,95)(36,112,55,94)(37,111,56,93)(38,110,43,92)(39,109,44,91)(40,108,45,90)(41,107,46,89)(42,106,47,88)(113,194,136,182)(114,193,137,181)(115,192,138,180)(116,191,139,179)(117,190,140,178)(118,189,127,177)(119,188,128,176)(120,187,129,175)(121,186,130,174)(122,185,131,173)(123,184,132,172)(124,183,133,171)(125,196,134,170)(126,195,135,169)(141,209,162,222)(142,208,163,221)(143,207,164,220)(144,206,165,219)(145,205,166,218)(146,204,167,217)(147,203,168,216)(148,202,155,215)(149,201,156,214)(150,200,157,213)(151,199,158,212)(152,198,159,211)(153,197,160,224)(154,210,161,223), (1,127,27,125)(2,128,28,126)(3,129,15,113)(4,130,16,114)(5,131,17,115)(6,132,18,116)(7,133,19,117)(8,134,20,118)(9,135,21,119)(10,136,22,120)(11,137,23,121)(12,138,24,122)(13,139,25,123)(14,140,26,124)(29,155,55,141)(30,156,56,142)(31,157,43,143)(32,158,44,144)(33,159,45,145)(34,160,46,146)(35,161,47,147)(36,162,48,148)(37,163,49,149)(38,164,50,150)(39,165,51,151)(40,166,52,152)(41,167,53,153)(42,168,54,154)(57,186,74,181)(58,187,75,182)(59,188,76,169)(60,189,77,170)(61,190,78,171)(62,191,79,172)(63,192,80,173)(64,193,81,174)(65,194,82,175)(66,195,83,176)(67,196,84,177)(68,183,71,178)(69,184,72,179)(70,185,73,180)(85,220,110,200)(86,221,111,201)(87,222,112,202)(88,223,99,203)(89,224,100,204)(90,211,101,205)(91,212,102,206)(92,213,103,207)(93,214,104,208)(94,215,105,209)(95,216,106,210)(96,217,107,197)(97,218,108,198)(98,219,109,199)>;
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), (1,168,27,154)(2,155,28,141)(3,156,15,142)(4,157,16,143)(5,158,17,144)(6,159,18,145)(7,160,19,146)(8,161,20,147)(9,162,21,148)(10,163,22,149)(11,164,23,150)(12,165,24,151)(13,166,25,152)(14,167,26,153)(29,128,55,126)(30,129,56,113)(31,130,43,114)(32,131,44,115)(33,132,45,116)(34,133,46,117)(35,134,47,118)(36,135,48,119)(37,136,49,120)(38,137,50,121)(39,138,51,122)(40,139,52,123)(41,140,53,124)(42,127,54,125)(57,220,74,200)(58,221,75,201)(59,222,76,202)(60,223,77,203)(61,224,78,204)(62,211,79,205)(63,212,80,206)(64,213,81,207)(65,214,82,208)(66,215,83,209)(67,216,84,210)(68,217,71,197)(69,218,72,198)(70,219,73,199)(85,186,110,181)(86,187,111,182)(87,188,112,169)(88,189,99,170)(89,190,100,171)(90,191,101,172)(91,192,102,173)(92,193,103,174)(93,194,104,175)(94,195,105,176)(95,196,106,177)(96,183,107,178)(97,184,108,179)(98,185,109,180), (1,154,8,147)(2,153,9,146)(3,152,10,145)(4,151,11,144)(5,150,12,143)(6,149,13,142)(7,148,14,141)(15,166,22,159)(16,165,23,158)(17,164,24,157)(18,163,25,156)(19,162,26,155)(20,161,27,168)(21,160,28,167)(29,133,36,140)(30,132,37,139)(31,131,38,138)(32,130,39,137)(33,129,40,136)(34,128,41,135)(35,127,42,134)(43,115,50,122)(44,114,51,121)(45,113,52,120)(46,126,53,119)(47,125,54,118)(48,124,55,117)(49,123,56,116)(57,199,64,206)(58,198,65,205)(59,197,66,204)(60,210,67,203)(61,209,68,202)(62,208,69,201)(63,207,70,200)(71,222,78,215)(72,221,79,214)(73,220,80,213)(74,219,81,212)(75,218,82,211)(76,217,83,224)(77,216,84,223)(85,192,92,185)(86,191,93,184)(87,190,94,183)(88,189,95,196)(89,188,96,195)(90,187,97,194)(91,186,98,193)(99,170,106,177)(100,169,107,176)(101,182,108,175)(102,181,109,174)(103,180,110,173)(104,179,111,172)(105,178,112,171), (1,67,20,77)(2,66,21,76)(3,65,22,75)(4,64,23,74)(5,63,24,73)(6,62,25,72)(7,61,26,71)(8,60,27,84)(9,59,28,83)(10,58,15,82)(11,57,16,81)(12,70,17,80)(13,69,18,79)(14,68,19,78)(29,105,48,87)(30,104,49,86)(31,103,50,85)(32,102,51,98)(33,101,52,97)(34,100,53,96)(35,99,54,95)(36,112,55,94)(37,111,56,93)(38,110,43,92)(39,109,44,91)(40,108,45,90)(41,107,46,89)(42,106,47,88)(113,194,136,182)(114,193,137,181)(115,192,138,180)(116,191,139,179)(117,190,140,178)(118,189,127,177)(119,188,128,176)(120,187,129,175)(121,186,130,174)(122,185,131,173)(123,184,132,172)(124,183,133,171)(125,196,134,170)(126,195,135,169)(141,209,162,222)(142,208,163,221)(143,207,164,220)(144,206,165,219)(145,205,166,218)(146,204,167,217)(147,203,168,216)(148,202,155,215)(149,201,156,214)(150,200,157,213)(151,199,158,212)(152,198,159,211)(153,197,160,224)(154,210,161,223), (1,127,27,125)(2,128,28,126)(3,129,15,113)(4,130,16,114)(5,131,17,115)(6,132,18,116)(7,133,19,117)(8,134,20,118)(9,135,21,119)(10,136,22,120)(11,137,23,121)(12,138,24,122)(13,139,25,123)(14,140,26,124)(29,155,55,141)(30,156,56,142)(31,157,43,143)(32,158,44,144)(33,159,45,145)(34,160,46,146)(35,161,47,147)(36,162,48,148)(37,163,49,149)(38,164,50,150)(39,165,51,151)(40,166,52,152)(41,167,53,153)(42,168,54,154)(57,186,74,181)(58,187,75,182)(59,188,76,169)(60,189,77,170)(61,190,78,171)(62,191,79,172)(63,192,80,173)(64,193,81,174)(65,194,82,175)(66,195,83,176)(67,196,84,177)(68,183,71,178)(69,184,72,179)(70,185,73,180)(85,220,110,200)(86,221,111,201)(87,222,112,202)(88,223,99,203)(89,224,100,204)(90,211,101,205)(91,212,102,206)(92,213,103,207)(93,214,104,208)(94,215,105,209)(95,216,106,210)(96,217,107,197)(97,218,108,198)(98,219,109,199) );
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)], [(1,168,27,154),(2,155,28,141),(3,156,15,142),(4,157,16,143),(5,158,17,144),(6,159,18,145),(7,160,19,146),(8,161,20,147),(9,162,21,148),(10,163,22,149),(11,164,23,150),(12,165,24,151),(13,166,25,152),(14,167,26,153),(29,128,55,126),(30,129,56,113),(31,130,43,114),(32,131,44,115),(33,132,45,116),(34,133,46,117),(35,134,47,118),(36,135,48,119),(37,136,49,120),(38,137,50,121),(39,138,51,122),(40,139,52,123),(41,140,53,124),(42,127,54,125),(57,220,74,200),(58,221,75,201),(59,222,76,202),(60,223,77,203),(61,224,78,204),(62,211,79,205),(63,212,80,206),(64,213,81,207),(65,214,82,208),(66,215,83,209),(67,216,84,210),(68,217,71,197),(69,218,72,198),(70,219,73,199),(85,186,110,181),(86,187,111,182),(87,188,112,169),(88,189,99,170),(89,190,100,171),(90,191,101,172),(91,192,102,173),(92,193,103,174),(93,194,104,175),(94,195,105,176),(95,196,106,177),(96,183,107,178),(97,184,108,179),(98,185,109,180)], [(1,154,8,147),(2,153,9,146),(3,152,10,145),(4,151,11,144),(5,150,12,143),(6,149,13,142),(7,148,14,141),(15,166,22,159),(16,165,23,158),(17,164,24,157),(18,163,25,156),(19,162,26,155),(20,161,27,168),(21,160,28,167),(29,133,36,140),(30,132,37,139),(31,131,38,138),(32,130,39,137),(33,129,40,136),(34,128,41,135),(35,127,42,134),(43,115,50,122),(44,114,51,121),(45,113,52,120),(46,126,53,119),(47,125,54,118),(48,124,55,117),(49,123,56,116),(57,199,64,206),(58,198,65,205),(59,197,66,204),(60,210,67,203),(61,209,68,202),(62,208,69,201),(63,207,70,200),(71,222,78,215),(72,221,79,214),(73,220,80,213),(74,219,81,212),(75,218,82,211),(76,217,83,224),(77,216,84,223),(85,192,92,185),(86,191,93,184),(87,190,94,183),(88,189,95,196),(89,188,96,195),(90,187,97,194),(91,186,98,193),(99,170,106,177),(100,169,107,176),(101,182,108,175),(102,181,109,174),(103,180,110,173),(104,179,111,172),(105,178,112,171)], [(1,67,20,77),(2,66,21,76),(3,65,22,75),(4,64,23,74),(5,63,24,73),(6,62,25,72),(7,61,26,71),(8,60,27,84),(9,59,28,83),(10,58,15,82),(11,57,16,81),(12,70,17,80),(13,69,18,79),(14,68,19,78),(29,105,48,87),(30,104,49,86),(31,103,50,85),(32,102,51,98),(33,101,52,97),(34,100,53,96),(35,99,54,95),(36,112,55,94),(37,111,56,93),(38,110,43,92),(39,109,44,91),(40,108,45,90),(41,107,46,89),(42,106,47,88),(113,194,136,182),(114,193,137,181),(115,192,138,180),(116,191,139,179),(117,190,140,178),(118,189,127,177),(119,188,128,176),(120,187,129,175),(121,186,130,174),(122,185,131,173),(123,184,132,172),(124,183,133,171),(125,196,134,170),(126,195,135,169),(141,209,162,222),(142,208,163,221),(143,207,164,220),(144,206,165,219),(145,205,166,218),(146,204,167,217),(147,203,168,216),(148,202,155,215),(149,201,156,214),(150,200,157,213),(151,199,158,212),(152,198,159,211),(153,197,160,224),(154,210,161,223)], [(1,127,27,125),(2,128,28,126),(3,129,15,113),(4,130,16,114),(5,131,17,115),(6,132,18,116),(7,133,19,117),(8,134,20,118),(9,135,21,119),(10,136,22,120),(11,137,23,121),(12,138,24,122),(13,139,25,123),(14,140,26,124),(29,155,55,141),(30,156,56,142),(31,157,43,143),(32,158,44,144),(33,159,45,145),(34,160,46,146),(35,161,47,147),(36,162,48,148),(37,163,49,149),(38,164,50,150),(39,165,51,151),(40,166,52,152),(41,167,53,153),(42,168,54,154),(57,186,74,181),(58,187,75,182),(59,188,76,169),(60,189,77,170),(61,190,78,171),(62,191,79,172),(63,192,80,173),(64,193,81,174),(65,194,82,175),(66,195,83,176),(67,196,84,177),(68,183,71,178),(69,184,72,179),(70,185,73,180),(85,220,110,200),(86,221,111,201),(87,222,112,202),(88,223,99,203),(89,224,100,204),(90,211,101,205),(91,212,102,206),(92,213,103,207),(93,214,104,208),(94,215,105,209),(95,216,106,210),(96,217,107,197),(97,218,108,198),(98,219,109,199)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 25 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 7 | 25 | 0 | 0 | 0 | 0 |
| 12 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 7 | 20 | 0 | 0 | 0 | 0 |
| 12 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 3 | 19 | 0 | 0 | 0 | 0 |
| 1 | 26 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 18 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,25,4,0,0,0,0,25,11,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[7,12,0,0,0,0,25,22,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[7,12,0,0,0,0,20,22,0,0,0,0,0,0,28,11,0,0,0,0,0,1,0,0,0,0,0,0,17,0,0,0,0,0,0,12],[3,1,0,0,0,0,19,26,0,0,0,0,0,0,1,18,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,28,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,12] >;
67 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 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 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | 2- (1+4) | D4⋊2D7 | Q8.10D14 | D7×C4○D4 |
| kernel | C14.232- (1+4) | C23.D14 | Dic7.D4 | Dic7⋊3Q8 | C28⋊Q8 | C4⋊C4⋊7D7 | C4⋊C4⋊D7 | C23.21D14 | C4×C7⋊D4 | Q8×Dic7 | D14⋊3Q8 | C7×C22⋊Q8 | C22⋊Q8 | Dic7 | C28 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C14 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 6 | 9 | 3 | 3 | 1 | 6 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{23}2_-^{(1+4)} % in TeX
G:=Group("C14.23ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1094);
// by ID
G=gap.SmallGroup(448,1094);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,758,219,100,1571,297,18822]);
// 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,b*d=d*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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