?
G = C14.452- (1+4) order 448 = 26·7
45th non-split extension by C14 of 2- (1+4) acting via 2- (1+4)/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C14.452- (1+4), (C7×Q8)⋊17D4, C7⋊6(Q8⋊5D4), Q8⋊8(C7⋊D4), (C22×Q8)⋊9D7, C28⋊7D4⋊38C2, (Q8×Dic7)⋊27C2, C28.262(C2×D4), D14⋊3Q8⋊42C2, (C2×Q8).189D14, C28.23D4⋊30C2, (C2×C28).649C23, (C2×C14).309C24, D14⋊C4.78C22, C22⋊3(Q8⋊2D7), (C22×C4).279D14, C14.157(C22×D4), (C2×D28).183C22, C4⋊Dic7.258C22, (Q8×C14).236C22, C22.320(C23×D7), C23.240(C22×D7), Dic7⋊C4.171C22, (C22×C14).427C23, (C22×C28).287C22, (C2×Dic7).290C23, (C4×Dic7).172C22, (C22×D7).135C23, C23.D7.146C22, C2.45(Q8.10D14), (Q8×C2×C14)⋊8C2, (C4×C7⋊D4)⋊27C2, C4.70(C2×C7⋊D4), (C2×C14)⋊18(C4○D4), (C2×Q8⋊2D7)⋊18C2, C14.129(C2×C4○D4), C2.36(C2×Q8⋊2D7), (C2×C4×D7).166C22, C2.30(C22×C7⋊D4), (C2×C4).244(C22×D7), (C2×C7⋊D4).139C22, SmallGroup(448,1270)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1236 in 290 conjugacy classes, 115 normal (22 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×2], C22 [×11], C7, C2×C4 [×6], C2×C4 [×17], D4 [×12], Q8 [×4], Q8 [×6], C23, C23 [×3], D7 [×3], C14 [×3], C14 [×2], C42 [×3], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C2×D4 [×6], C2×Q8, C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×4], Dic7 [×5], C28 [×6], C28 [×3], D14 [×9], C2×C14, C2×C14 [×2], C2×C14 [×2], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, C4×D7 [×6], D28 [×6], C2×Dic7 [×2], C2×Dic7 [×3], C7⋊D4 [×6], C2×C28 [×6], C2×C28 [×6], C7×Q8 [×4], C7×Q8 [×6], C22×D7 [×3], C22×C14, Q8⋊5D4, C4×Dic7 [×3], Dic7⋊C4 [×3], C4⋊Dic7 [×3], D14⋊C4 [×9], C23.D7, C2×C4×D7 [×3], C2×D28 [×3], Q8⋊2D7 [×4], C2×C7⋊D4 [×3], C22×C28 [×3], Q8×C14, Q8×C14 [×3], Q8×C14 [×4], C4×C7⋊D4 [×3], C28⋊7D4 [×3], Q8×Dic7, D14⋊3Q8 [×3], C28.23D4 [×3], C2×Q8⋊2D7, Q8×C2×C14, C14.452- (1+4)
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C4○D4 [×2], C24, D14 [×7], C22×D4, C2×C4○D4, 2- (1+4), C7⋊D4 [×4], C22×D7 [×7], Q8⋊5D4, Q8⋊2D7 [×2], C2×C7⋊D4 [×6], C23×D7, C2×Q8⋊2D7, Q8.10D14, C22×C7⋊D4, C14.452- (1+4)
Generators and relations
G = < a,b,c,d,e | a14=b4=c2=1, d2=e2=a7b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-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)
(1 150 100 203)(2 149 101 202)(3 148 102 201)(4 147 103 200)(5 146 104 199)(6 145 105 198)(7 144 106 197)(8 143 107 210)(9 142 108 209)(10 141 109 208)(11 154 110 207)(12 153 111 206)(13 152 112 205)(14 151 99 204)(15 43 183 42)(16 56 184 41)(17 55 185 40)(18 54 186 39)(19 53 187 38)(20 52 188 37)(21 51 189 36)(22 50 190 35)(23 49 191 34)(24 48 192 33)(25 47 193 32)(26 46 194 31)(27 45 195 30)(28 44 196 29)(57 118 162 178)(58 117 163 177)(59 116 164 176)(60 115 165 175)(61 114 166 174)(62 113 167 173)(63 126 168 172)(64 125 155 171)(65 124 156 170)(66 123 157 169)(67 122 158 182)(68 121 159 181)(69 120 160 180)(70 119 161 179)(71 212 132 94)(72 211 133 93)(73 224 134 92)(74 223 135 91)(75 222 136 90)(76 221 137 89)(77 220 138 88)(78 219 139 87)(79 218 140 86)(80 217 127 85)(81 216 128 98)(82 215 129 97)(83 214 130 96)(84 213 131 95)
(15 183)(16 184)(17 185)(18 186)(19 187)(20 188)(21 189)(22 190)(23 191)(24 192)(25 193)(26 194)(27 195)(28 196)(85 217)(86 218)(87 219)(88 220)(89 221)(90 222)(91 223)(92 224)(93 211)(94 212)(95 213)(96 214)(97 215)(98 216)(113 173)(114 174)(115 175)(116 176)(117 177)(118 178)(119 179)(120 180)(121 181)(122 182)(123 169)(124 170)(125 171)(126 172)(141 208)(142 209)(143 210)(144 197)(145 198)(146 199)(147 200)(148 201)(149 202)(150 203)(151 204)(152 205)(153 206)(154 207)
(1 52 107 30)(2 53 108 31)(3 54 109 32)(4 55 110 33)(5 56 111 34)(6 43 112 35)(7 44 99 36)(8 45 100 37)(9 46 101 38)(10 47 102 39)(11 48 103 40)(12 49 104 41)(13 50 105 42)(14 51 106 29)(15 145 190 205)(16 146 191 206)(17 147 192 207)(18 148 193 208)(19 149 194 209)(20 150 195 210)(21 151 196 197)(22 152 183 198)(23 153 184 199)(24 154 185 200)(25 141 186 201)(26 142 187 202)(27 143 188 203)(28 144 189 204)(57 139 155 71)(58 140 156 72)(59 127 157 73)(60 128 158 74)(61 129 159 75)(62 130 160 76)(63 131 161 77)(64 132 162 78)(65 133 163 79)(66 134 164 80)(67 135 165 81)(68 136 166 82)(69 137 167 83)(70 138 168 84)(85 176 224 123)(86 177 211 124)(87 178 212 125)(88 179 213 126)(89 180 214 113)(90 181 215 114)(91 182 216 115)(92 169 217 116)(93 170 218 117)(94 171 219 118)(95 172 220 119)(96 173 221 120)(97 174 222 121)(98 175 223 122)
(1 57 107 155)(2 58 108 156)(3 59 109 157)(4 60 110 158)(5 61 111 159)(6 62 112 160)(7 63 99 161)(8 64 100 162)(9 65 101 163)(10 66 102 164)(11 67 103 165)(12 68 104 166)(13 69 105 167)(14 70 106 168)(15 89 190 214)(16 90 191 215)(17 91 192 216)(18 92 193 217)(19 93 194 218)(20 94 195 219)(21 95 196 220)(22 96 183 221)(23 97 184 222)(24 98 185 223)(25 85 186 224)(26 86 187 211)(27 87 188 212)(28 88 189 213)(29 138 51 84)(30 139 52 71)(31 140 53 72)(32 127 54 73)(33 128 55 74)(34 129 56 75)(35 130 43 76)(36 131 44 77)(37 132 45 78)(38 133 46 79)(39 134 47 80)(40 135 48 81)(41 136 49 82)(42 137 50 83)(113 205 180 145)(114 206 181 146)(115 207 182 147)(116 208 169 148)(117 209 170 149)(118 210 171 150)(119 197 172 151)(120 198 173 152)(121 199 174 153)(122 200 175 154)(123 201 176 141)(124 202 177 142)(125 203 178 143)(126 204 179 144)
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), (1,150,100,203)(2,149,101,202)(3,148,102,201)(4,147,103,200)(5,146,104,199)(6,145,105,198)(7,144,106,197)(8,143,107,210)(9,142,108,209)(10,141,109,208)(11,154,110,207)(12,153,111,206)(13,152,112,205)(14,151,99,204)(15,43,183,42)(16,56,184,41)(17,55,185,40)(18,54,186,39)(19,53,187,38)(20,52,188,37)(21,51,189,36)(22,50,190,35)(23,49,191,34)(24,48,192,33)(25,47,193,32)(26,46,194,31)(27,45,195,30)(28,44,196,29)(57,118,162,178)(58,117,163,177)(59,116,164,176)(60,115,165,175)(61,114,166,174)(62,113,167,173)(63,126,168,172)(64,125,155,171)(65,124,156,170)(66,123,157,169)(67,122,158,182)(68,121,159,181)(69,120,160,180)(70,119,161,179)(71,212,132,94)(72,211,133,93)(73,224,134,92)(74,223,135,91)(75,222,136,90)(76,221,137,89)(77,220,138,88)(78,219,139,87)(79,218,140,86)(80,217,127,85)(81,216,128,98)(82,215,129,97)(83,214,130,96)(84,213,131,95), (15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,193)(26,194)(27,195)(28,196)(85,217)(86,218)(87,219)(88,220)(89,221)(90,222)(91,223)(92,224)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(113,173)(114,174)(115,175)(116,176)(117,177)(118,178)(119,179)(120,180)(121,181)(122,182)(123,169)(124,170)(125,171)(126,172)(141,208)(142,209)(143,210)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,204)(152,205)(153,206)(154,207), (1,52,107,30)(2,53,108,31)(3,54,109,32)(4,55,110,33)(5,56,111,34)(6,43,112,35)(7,44,99,36)(8,45,100,37)(9,46,101,38)(10,47,102,39)(11,48,103,40)(12,49,104,41)(13,50,105,42)(14,51,106,29)(15,145,190,205)(16,146,191,206)(17,147,192,207)(18,148,193,208)(19,149,194,209)(20,150,195,210)(21,151,196,197)(22,152,183,198)(23,153,184,199)(24,154,185,200)(25,141,186,201)(26,142,187,202)(27,143,188,203)(28,144,189,204)(57,139,155,71)(58,140,156,72)(59,127,157,73)(60,128,158,74)(61,129,159,75)(62,130,160,76)(63,131,161,77)(64,132,162,78)(65,133,163,79)(66,134,164,80)(67,135,165,81)(68,136,166,82)(69,137,167,83)(70,138,168,84)(85,176,224,123)(86,177,211,124)(87,178,212,125)(88,179,213,126)(89,180,214,113)(90,181,215,114)(91,182,216,115)(92,169,217,116)(93,170,218,117)(94,171,219,118)(95,172,220,119)(96,173,221,120)(97,174,222,121)(98,175,223,122), (1,57,107,155)(2,58,108,156)(3,59,109,157)(4,60,110,158)(5,61,111,159)(6,62,112,160)(7,63,99,161)(8,64,100,162)(9,65,101,163)(10,66,102,164)(11,67,103,165)(12,68,104,166)(13,69,105,167)(14,70,106,168)(15,89,190,214)(16,90,191,215)(17,91,192,216)(18,92,193,217)(19,93,194,218)(20,94,195,219)(21,95,196,220)(22,96,183,221)(23,97,184,222)(24,98,185,223)(25,85,186,224)(26,86,187,211)(27,87,188,212)(28,88,189,213)(29,138,51,84)(30,139,52,71)(31,140,53,72)(32,127,54,73)(33,128,55,74)(34,129,56,75)(35,130,43,76)(36,131,44,77)(37,132,45,78)(38,133,46,79)(39,134,47,80)(40,135,48,81)(41,136,49,82)(42,137,50,83)(113,205,180,145)(114,206,181,146)(115,207,182,147)(116,208,169,148)(117,209,170,149)(118,210,171,150)(119,197,172,151)(120,198,173,152)(121,199,174,153)(122,200,175,154)(123,201,176,141)(124,202,177,142)(125,203,178,143)(126,204,179,144)>;
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), (1,150,100,203)(2,149,101,202)(3,148,102,201)(4,147,103,200)(5,146,104,199)(6,145,105,198)(7,144,106,197)(8,143,107,210)(9,142,108,209)(10,141,109,208)(11,154,110,207)(12,153,111,206)(13,152,112,205)(14,151,99,204)(15,43,183,42)(16,56,184,41)(17,55,185,40)(18,54,186,39)(19,53,187,38)(20,52,188,37)(21,51,189,36)(22,50,190,35)(23,49,191,34)(24,48,192,33)(25,47,193,32)(26,46,194,31)(27,45,195,30)(28,44,196,29)(57,118,162,178)(58,117,163,177)(59,116,164,176)(60,115,165,175)(61,114,166,174)(62,113,167,173)(63,126,168,172)(64,125,155,171)(65,124,156,170)(66,123,157,169)(67,122,158,182)(68,121,159,181)(69,120,160,180)(70,119,161,179)(71,212,132,94)(72,211,133,93)(73,224,134,92)(74,223,135,91)(75,222,136,90)(76,221,137,89)(77,220,138,88)(78,219,139,87)(79,218,140,86)(80,217,127,85)(81,216,128,98)(82,215,129,97)(83,214,130,96)(84,213,131,95), (15,183)(16,184)(17,185)(18,186)(19,187)(20,188)(21,189)(22,190)(23,191)(24,192)(25,193)(26,194)(27,195)(28,196)(85,217)(86,218)(87,219)(88,220)(89,221)(90,222)(91,223)(92,224)(93,211)(94,212)(95,213)(96,214)(97,215)(98,216)(113,173)(114,174)(115,175)(116,176)(117,177)(118,178)(119,179)(120,180)(121,181)(122,182)(123,169)(124,170)(125,171)(126,172)(141,208)(142,209)(143,210)(144,197)(145,198)(146,199)(147,200)(148,201)(149,202)(150,203)(151,204)(152,205)(153,206)(154,207), (1,52,107,30)(2,53,108,31)(3,54,109,32)(4,55,110,33)(5,56,111,34)(6,43,112,35)(7,44,99,36)(8,45,100,37)(9,46,101,38)(10,47,102,39)(11,48,103,40)(12,49,104,41)(13,50,105,42)(14,51,106,29)(15,145,190,205)(16,146,191,206)(17,147,192,207)(18,148,193,208)(19,149,194,209)(20,150,195,210)(21,151,196,197)(22,152,183,198)(23,153,184,199)(24,154,185,200)(25,141,186,201)(26,142,187,202)(27,143,188,203)(28,144,189,204)(57,139,155,71)(58,140,156,72)(59,127,157,73)(60,128,158,74)(61,129,159,75)(62,130,160,76)(63,131,161,77)(64,132,162,78)(65,133,163,79)(66,134,164,80)(67,135,165,81)(68,136,166,82)(69,137,167,83)(70,138,168,84)(85,176,224,123)(86,177,211,124)(87,178,212,125)(88,179,213,126)(89,180,214,113)(90,181,215,114)(91,182,216,115)(92,169,217,116)(93,170,218,117)(94,171,219,118)(95,172,220,119)(96,173,221,120)(97,174,222,121)(98,175,223,122), (1,57,107,155)(2,58,108,156)(3,59,109,157)(4,60,110,158)(5,61,111,159)(6,62,112,160)(7,63,99,161)(8,64,100,162)(9,65,101,163)(10,66,102,164)(11,67,103,165)(12,68,104,166)(13,69,105,167)(14,70,106,168)(15,89,190,214)(16,90,191,215)(17,91,192,216)(18,92,193,217)(19,93,194,218)(20,94,195,219)(21,95,196,220)(22,96,183,221)(23,97,184,222)(24,98,185,223)(25,85,186,224)(26,86,187,211)(27,87,188,212)(28,88,189,213)(29,138,51,84)(30,139,52,71)(31,140,53,72)(32,127,54,73)(33,128,55,74)(34,129,56,75)(35,130,43,76)(36,131,44,77)(37,132,45,78)(38,133,46,79)(39,134,47,80)(40,135,48,81)(41,136,49,82)(42,137,50,83)(113,205,180,145)(114,206,181,146)(115,207,182,147)(116,208,169,148)(117,209,170,149)(118,210,171,150)(119,197,172,151)(120,198,173,152)(121,199,174,153)(122,200,175,154)(123,201,176,141)(124,202,177,142)(125,203,178,143)(126,204,179,144) );
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)], [(1,150,100,203),(2,149,101,202),(3,148,102,201),(4,147,103,200),(5,146,104,199),(6,145,105,198),(7,144,106,197),(8,143,107,210),(9,142,108,209),(10,141,109,208),(11,154,110,207),(12,153,111,206),(13,152,112,205),(14,151,99,204),(15,43,183,42),(16,56,184,41),(17,55,185,40),(18,54,186,39),(19,53,187,38),(20,52,188,37),(21,51,189,36),(22,50,190,35),(23,49,191,34),(24,48,192,33),(25,47,193,32),(26,46,194,31),(27,45,195,30),(28,44,196,29),(57,118,162,178),(58,117,163,177),(59,116,164,176),(60,115,165,175),(61,114,166,174),(62,113,167,173),(63,126,168,172),(64,125,155,171),(65,124,156,170),(66,123,157,169),(67,122,158,182),(68,121,159,181),(69,120,160,180),(70,119,161,179),(71,212,132,94),(72,211,133,93),(73,224,134,92),(74,223,135,91),(75,222,136,90),(76,221,137,89),(77,220,138,88),(78,219,139,87),(79,218,140,86),(80,217,127,85),(81,216,128,98),(82,215,129,97),(83,214,130,96),(84,213,131,95)], [(15,183),(16,184),(17,185),(18,186),(19,187),(20,188),(21,189),(22,190),(23,191),(24,192),(25,193),(26,194),(27,195),(28,196),(85,217),(86,218),(87,219),(88,220),(89,221),(90,222),(91,223),(92,224),(93,211),(94,212),(95,213),(96,214),(97,215),(98,216),(113,173),(114,174),(115,175),(116,176),(117,177),(118,178),(119,179),(120,180),(121,181),(122,182),(123,169),(124,170),(125,171),(126,172),(141,208),(142,209),(143,210),(144,197),(145,198),(146,199),(147,200),(148,201),(149,202),(150,203),(151,204),(152,205),(153,206),(154,207)], [(1,52,107,30),(2,53,108,31),(3,54,109,32),(4,55,110,33),(5,56,111,34),(6,43,112,35),(7,44,99,36),(8,45,100,37),(9,46,101,38),(10,47,102,39),(11,48,103,40),(12,49,104,41),(13,50,105,42),(14,51,106,29),(15,145,190,205),(16,146,191,206),(17,147,192,207),(18,148,193,208),(19,149,194,209),(20,150,195,210),(21,151,196,197),(22,152,183,198),(23,153,184,199),(24,154,185,200),(25,141,186,201),(26,142,187,202),(27,143,188,203),(28,144,189,204),(57,139,155,71),(58,140,156,72),(59,127,157,73),(60,128,158,74),(61,129,159,75),(62,130,160,76),(63,131,161,77),(64,132,162,78),(65,133,163,79),(66,134,164,80),(67,135,165,81),(68,136,166,82),(69,137,167,83),(70,138,168,84),(85,176,224,123),(86,177,211,124),(87,178,212,125),(88,179,213,126),(89,180,214,113),(90,181,215,114),(91,182,216,115),(92,169,217,116),(93,170,218,117),(94,171,219,118),(95,172,220,119),(96,173,221,120),(97,174,222,121),(98,175,223,122)], [(1,57,107,155),(2,58,108,156),(3,59,109,157),(4,60,110,158),(5,61,111,159),(6,62,112,160),(7,63,99,161),(8,64,100,162),(9,65,101,163),(10,66,102,164),(11,67,103,165),(12,68,104,166),(13,69,105,167),(14,70,106,168),(15,89,190,214),(16,90,191,215),(17,91,192,216),(18,92,193,217),(19,93,194,218),(20,94,195,219),(21,95,196,220),(22,96,183,221),(23,97,184,222),(24,98,185,223),(25,85,186,224),(26,86,187,211),(27,87,188,212),(28,88,189,213),(29,138,51,84),(30,139,52,71),(31,140,53,72),(32,127,54,73),(33,128,55,74),(34,129,56,75),(35,130,43,76),(36,131,44,77),(37,132,45,78),(38,133,46,79),(39,134,47,80),(40,135,48,81),(41,136,49,82),(42,137,50,83),(113,205,180,145),(114,206,181,146),(115,207,182,147),(116,208,169,148),(117,209,170,149),(118,210,171,150),(119,197,172,151),(120,198,173,152),(121,199,174,153),(122,200,175,154),(123,201,176,141),(124,202,177,142),(125,203,178,143),(126,204,179,144)])
Matrix representation ►G ⊆ GL4(𝔽29) generated by
| 4 | 0 | 0 | 0 |
| 2 | 22 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 12 | 21 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 0 | 0 | 17 | 24 |
| 0 | 0 | 17 | 12 |
| 1 | 0 | 0 | 0 |
| 3 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 3 | 28 | 0 | 0 |
| 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 17 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 2 |
| 0 | 0 | 28 | 1 |
G:=sub<GL(4,GF(29))| [4,2,0,0,0,22,0,0,0,0,28,0,0,0,0,28],[12,0,0,0,21,17,0,0,0,0,17,17,0,0,24,12],[1,3,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,3,0,0,0,28,0,0,0,0,12,0,0,0,5,17],[28,0,0,0,0,28,0,0,0,0,28,28,0,0,2,1] >;
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 7A | 7B | 7C | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | C7⋊D4 | 2- (1+4) | Q8⋊2D7 | Q8.10D14 |
| kernel | C14.452- (1+4) | C4×C7⋊D4 | C28⋊7D4 | Q8×Dic7 | D14⋊3Q8 | C28.23D4 | C2×Q8⋊2D7 | Q8×C2×C14 | C7×Q8 | C22×Q8 | C2×C14 | C22×C4 | C2×Q8 | Q8 | C14 | C22 | C2 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 4 | 3 | 4 | 9 | 12 | 24 | 1 | 6 | 6 |
In GAP, Magma, Sage, TeX
C_{14}._{45}2_-^{(1+4)} % in TeX
G:=Group("C14.45ES-(2,2)"); // GroupNames label
G:=SmallGroup(448,1270);
// by ID
G=gap.SmallGroup(448,1270);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,758,387,184,675,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^4=c^2=1,d^2=e^2=a^7*b^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=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
×
⋊
ℤ
𝔽
○
≀