direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C14×C22.D4, (C23×C4)⋊6C14, (C23×C28)⋊7C2, C23.49(C7×D4), C24.35(C2×C14), C22.61(D4×C14), (C2×C28).657C23, (C2×C14).344C24, (C22×C28)⋊59C22, (C22×D4).10C14, (C22×C14).171D4, C14.183(C22×D4), C23.5(C22×C14), (D4×C14).316C22, C22.18(C23×C14), (C23×C14).92C22, (C22×C14).259C23, C2.7(D4×C2×C14), (C2×C4⋊C4)⋊16C14, (C14×C4⋊C4)⋊43C2, C4⋊C4⋊11(C2×C14), (D4×C2×C14).23C2, C2.7(C14×C4○D4), (C7×C4⋊C4)⋊67C22, C22⋊C4⋊12(C2×C14), (C2×C22⋊C4)⋊10C14, (C14×C22⋊C4)⋊30C2, (C22×C4)⋊17(C2×C14), (C2×D4).61(C2×C14), C14.226(C2×C4○D4), (C2×C14).415(C2×D4), C22.31(C7×C4○D4), (C7×C22⋊C4)⋊66C22, (C2×C4).13(C22×C14), (C2×C14).231(C4○D4), SmallGroup(448,1307)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 530 in 342 conjugacy classes, 178 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×6], C4 [×10], C22, C22 [×10], C22 [×22], C7, C2×C4 [×10], C2×C4 [×18], D4 [×8], C23, C23 [×8], C23 [×10], C14, C14 [×6], C14 [×6], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×8], C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], C28 [×10], C2×C14, C2×C14 [×10], C2×C14 [×22], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, C2×C28 [×10], C2×C28 [×18], C7×D4 [×8], C22×C14, C22×C14 [×8], C22×C14 [×10], C2×C22.D4, C7×C22⋊C4 [×12], C7×C4⋊C4 [×8], C22×C28, C22×C28 [×8], C22×C28 [×4], D4×C14 [×4], D4×C14 [×4], C23×C14 [×2], C14×C22⋊C4, C14×C22⋊C4 [×2], C14×C4⋊C4 [×2], C7×C22.D4 [×8], C23×C28, D4×C2×C14, C14×C22.D4
Quotients:
C1, C2 [×15], C22 [×35], C7, D4 [×4], C23 [×15], C14 [×15], C2×D4 [×6], C4○D4 [×4], C24, C2×C14 [×35], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], C7×D4 [×4], C22×C14 [×15], C2×C22.D4, D4×C14 [×6], C7×C4○D4 [×4], C23×C14, C7×C22.D4 [×4], D4×C2×C14, C14×C4○D4 [×2], C14×C22.D4
Generators and relations
G = < a,b,c,d,e | a14=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
►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)
(29 60)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 57)(41 58)(42 59)(85 163)(86 164)(87 165)(88 166)(89 167)(90 168)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 161)(98 162)(99 202)(100 203)(101 204)(102 205)(103 206)(104 207)(105 208)(106 209)(107 210)(108 197)(109 198)(110 199)(111 200)(112 201)(141 177)(142 178)(143 179)(144 180)(145 181)(146 182)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)
(1 222)(2 223)(3 224)(4 211)(5 212)(6 213)(7 214)(8 215)(9 216)(10 217)(11 218)(12 219)(13 220)(14 221)(15 124)(16 125)(17 126)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 60)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 57)(41 58)(42 59)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(85 163)(86 164)(87 165)(88 166)(89 167)(90 168)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 161)(98 162)(99 202)(100 203)(101 204)(102 205)(103 206)(104 207)(105 208)(106 209)(107 210)(108 197)(109 198)(110 199)(111 200)(112 201)(127 190)(128 191)(129 192)(130 193)(131 194)(132 195)(133 196)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(141 177)(142 178)(143 179)(144 180)(145 181)(146 182)(147 169)(148 170)(149 171)(150 172)(151 173)(152 174)(153 175)(154 176)
(1 70 135 198)(2 57 136 199)(3 58 137 200)(4 59 138 201)(5 60 139 202)(6 61 140 203)(7 62 127 204)(8 63 128 205)(9 64 129 206)(10 65 130 207)(11 66 131 208)(12 67 132 209)(13 68 133 210)(14 69 134 197)(15 153 72 85)(16 154 73 86)(17 141 74 87)(18 142 75 88)(19 143 76 89)(20 144 77 90)(21 145 78 91)(22 146 79 92)(23 147 80 93)(24 148 81 94)(25 149 82 95)(26 150 83 96)(27 151 84 97)(28 152 71 98)(29 188 99 212)(30 189 100 213)(31 190 101 214)(32 191 102 215)(33 192 103 216)(34 193 104 217)(35 194 105 218)(36 195 106 219)(37 196 107 220)(38 183 108 221)(39 184 109 222)(40 185 110 223)(41 186 111 224)(42 187 112 211)(43 162 123 174)(44 163 124 175)(45 164 125 176)(46 165 126 177)(47 166 113 178)(48 167 114 179)(49 168 115 180)(50 155 116 181)(51 156 117 182)(52 157 118 169)(53 158 119 170)(54 159 120 171)(55 160 121 172)(56 161 122 173)
(1 151)(2 152)(3 153)(4 154)(5 141)(6 142)(7 143)(8 144)(9 145)(10 146)(11 147)(12 148)(13 149)(14 150)(15 41)(16 42)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(43 199)(44 200)(45 201)(46 202)(47 203)(48 204)(49 205)(50 206)(51 207)(52 208)(53 209)(54 210)(55 197)(56 198)(57 123)(58 124)(59 125)(60 126)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 110)(72 111)(73 112)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)(85 137)(86 138)(87 139)(88 140)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(155 192)(156 193)(157 194)(158 195)(159 196)(160 183)(161 184)(162 185)(163 186)(164 187)(165 188)(166 189)(167 190)(168 191)(169 218)(170 219)(171 220)(172 221)(173 222)(174 223)(175 224)(176 211)(177 212)(178 213)(179 214)(180 215)(181 216)(182 217)
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), (29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,57)(41,58)(42,59)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,161)(98,162)(99,202)(100,203)(101,204)(102,205)(103,206)(104,207)(105,208)(106,209)(107,210)(108,197)(109,198)(110,199)(111,200)(112,201)(141,177)(142,178)(143,179)(144,180)(145,181)(146,182)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176), (1,222)(2,223)(3,224)(4,211)(5,212)(6,213)(7,214)(8,215)(9,216)(10,217)(11,218)(12,219)(13,220)(14,221)(15,124)(16,125)(17,126)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,57)(41,58)(42,59)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,161)(98,162)(99,202)(100,203)(101,204)(102,205)(103,206)(104,207)(105,208)(106,209)(107,210)(108,197)(109,198)(110,199)(111,200)(112,201)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(141,177)(142,178)(143,179)(144,180)(145,181)(146,182)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176), (1,70,135,198)(2,57,136,199)(3,58,137,200)(4,59,138,201)(5,60,139,202)(6,61,140,203)(7,62,127,204)(8,63,128,205)(9,64,129,206)(10,65,130,207)(11,66,131,208)(12,67,132,209)(13,68,133,210)(14,69,134,197)(15,153,72,85)(16,154,73,86)(17,141,74,87)(18,142,75,88)(19,143,76,89)(20,144,77,90)(21,145,78,91)(22,146,79,92)(23,147,80,93)(24,148,81,94)(25,149,82,95)(26,150,83,96)(27,151,84,97)(28,152,71,98)(29,188,99,212)(30,189,100,213)(31,190,101,214)(32,191,102,215)(33,192,103,216)(34,193,104,217)(35,194,105,218)(36,195,106,219)(37,196,107,220)(38,183,108,221)(39,184,109,222)(40,185,110,223)(41,186,111,224)(42,187,112,211)(43,162,123,174)(44,163,124,175)(45,164,125,176)(46,165,126,177)(47,166,113,178)(48,167,114,179)(49,168,115,180)(50,155,116,181)(51,156,117,182)(52,157,118,169)(53,158,119,170)(54,159,120,171)(55,160,121,172)(56,161,122,173), (1,151)(2,152)(3,153)(4,154)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,41)(16,42)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(43,199)(44,200)(45,201)(46,202)(47,203)(48,204)(49,205)(50,206)(51,207)(52,208)(53,209)(54,210)(55,197)(56,198)(57,123)(58,124)(59,125)(60,126)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,110)(72,111)(73,112)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(85,137)(86,138)(87,139)(88,140)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(155,192)(156,193)(157,194)(158,195)(159,196)(160,183)(161,184)(162,185)(163,186)(164,187)(165,188)(166,189)(167,190)(168,191)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,211)(177,212)(178,213)(179,214)(180,215)(181,216)(182,217)>;
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), (29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,57)(41,58)(42,59)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,161)(98,162)(99,202)(100,203)(101,204)(102,205)(103,206)(104,207)(105,208)(106,209)(107,210)(108,197)(109,198)(110,199)(111,200)(112,201)(141,177)(142,178)(143,179)(144,180)(145,181)(146,182)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176), (1,222)(2,223)(3,224)(4,211)(5,212)(6,213)(7,214)(8,215)(9,216)(10,217)(11,218)(12,219)(13,220)(14,221)(15,124)(16,125)(17,126)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,57)(41,58)(42,59)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,161)(98,162)(99,202)(100,203)(101,204)(102,205)(103,206)(104,207)(105,208)(106,209)(107,210)(108,197)(109,198)(110,199)(111,200)(112,201)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)(133,196)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(141,177)(142,178)(143,179)(144,180)(145,181)(146,182)(147,169)(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176), (1,70,135,198)(2,57,136,199)(3,58,137,200)(4,59,138,201)(5,60,139,202)(6,61,140,203)(7,62,127,204)(8,63,128,205)(9,64,129,206)(10,65,130,207)(11,66,131,208)(12,67,132,209)(13,68,133,210)(14,69,134,197)(15,153,72,85)(16,154,73,86)(17,141,74,87)(18,142,75,88)(19,143,76,89)(20,144,77,90)(21,145,78,91)(22,146,79,92)(23,147,80,93)(24,148,81,94)(25,149,82,95)(26,150,83,96)(27,151,84,97)(28,152,71,98)(29,188,99,212)(30,189,100,213)(31,190,101,214)(32,191,102,215)(33,192,103,216)(34,193,104,217)(35,194,105,218)(36,195,106,219)(37,196,107,220)(38,183,108,221)(39,184,109,222)(40,185,110,223)(41,186,111,224)(42,187,112,211)(43,162,123,174)(44,163,124,175)(45,164,125,176)(46,165,126,177)(47,166,113,178)(48,167,114,179)(49,168,115,180)(50,155,116,181)(51,156,117,182)(52,157,118,169)(53,158,119,170)(54,159,120,171)(55,160,121,172)(56,161,122,173), (1,151)(2,152)(3,153)(4,154)(5,141)(6,142)(7,143)(8,144)(9,145)(10,146)(11,147)(12,148)(13,149)(14,150)(15,41)(16,42)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(43,199)(44,200)(45,201)(46,202)(47,203)(48,204)(49,205)(50,206)(51,207)(52,208)(53,209)(54,210)(55,197)(56,198)(57,123)(58,124)(59,125)(60,126)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,121)(70,122)(71,110)(72,111)(73,112)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(85,137)(86,138)(87,139)(88,140)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(155,192)(156,193)(157,194)(158,195)(159,196)(160,183)(161,184)(162,185)(163,186)(164,187)(165,188)(166,189)(167,190)(168,191)(169,218)(170,219)(171,220)(172,221)(173,222)(174,223)(175,224)(176,211)(177,212)(178,213)(179,214)(180,215)(181,216)(182,217) );
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)], [(29,60),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,57),(41,58),(42,59),(85,163),(86,164),(87,165),(88,166),(89,167),(90,168),(91,155),(92,156),(93,157),(94,158),(95,159),(96,160),(97,161),(98,162),(99,202),(100,203),(101,204),(102,205),(103,206),(104,207),(105,208),(106,209),(107,210),(108,197),(109,198),(110,199),(111,200),(112,201),(141,177),(142,178),(143,179),(144,180),(145,181),(146,182),(147,169),(148,170),(149,171),(150,172),(151,173),(152,174),(153,175),(154,176)], [(1,222),(2,223),(3,224),(4,211),(5,212),(6,213),(7,214),(8,215),(9,216),(10,217),(11,218),(12,219),(13,220),(14,221),(15,124),(16,125),(17,126),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,121),(27,122),(28,123),(29,60),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,57),(41,58),(42,59),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(85,163),(86,164),(87,165),(88,166),(89,167),(90,168),(91,155),(92,156),(93,157),(94,158),(95,159),(96,160),(97,161),(98,162),(99,202),(100,203),(101,204),(102,205),(103,206),(104,207),(105,208),(106,209),(107,210),(108,197),(109,198),(110,199),(111,200),(112,201),(127,190),(128,191),(129,192),(130,193),(131,194),(132,195),(133,196),(134,183),(135,184),(136,185),(137,186),(138,187),(139,188),(140,189),(141,177),(142,178),(143,179),(144,180),(145,181),(146,182),(147,169),(148,170),(149,171),(150,172),(151,173),(152,174),(153,175),(154,176)], [(1,70,135,198),(2,57,136,199),(3,58,137,200),(4,59,138,201),(5,60,139,202),(6,61,140,203),(7,62,127,204),(8,63,128,205),(9,64,129,206),(10,65,130,207),(11,66,131,208),(12,67,132,209),(13,68,133,210),(14,69,134,197),(15,153,72,85),(16,154,73,86),(17,141,74,87),(18,142,75,88),(19,143,76,89),(20,144,77,90),(21,145,78,91),(22,146,79,92),(23,147,80,93),(24,148,81,94),(25,149,82,95),(26,150,83,96),(27,151,84,97),(28,152,71,98),(29,188,99,212),(30,189,100,213),(31,190,101,214),(32,191,102,215),(33,192,103,216),(34,193,104,217),(35,194,105,218),(36,195,106,219),(37,196,107,220),(38,183,108,221),(39,184,109,222),(40,185,110,223),(41,186,111,224),(42,187,112,211),(43,162,123,174),(44,163,124,175),(45,164,125,176),(46,165,126,177),(47,166,113,178),(48,167,114,179),(49,168,115,180),(50,155,116,181),(51,156,117,182),(52,157,118,169),(53,158,119,170),(54,159,120,171),(55,160,121,172),(56,161,122,173)], [(1,151),(2,152),(3,153),(4,154),(5,141),(6,142),(7,143),(8,144),(9,145),(10,146),(11,147),(12,148),(13,149),(14,150),(15,41),(16,42),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(43,199),(44,200),(45,201),(46,202),(47,203),(48,204),(49,205),(50,206),(51,207),(52,208),(53,209),(54,210),(55,197),(56,198),(57,123),(58,124),(59,125),(60,126),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,121),(70,122),(71,110),(72,111),(73,112),(74,99),(75,100),(76,101),(77,102),(78,103),(79,104),(80,105),(81,106),(82,107),(83,108),(84,109),(85,137),(86,138),(87,139),(88,140),(89,127),(90,128),(91,129),(92,130),(93,131),(94,132),(95,133),(96,134),(97,135),(98,136),(155,192),(156,193),(157,194),(158,195),(159,196),(160,183),(161,184),(162,185),(163,186),(164,187),(165,188),(166,189),(167,190),(168,191),(169,218),(170,219),(171,220),(172,221),(173,222),(174,223),(175,224),(176,211),(177,212),(178,213),(179,214),(180,215),(181,216),(182,217)])
Matrix representation ►G ⊆ GL6(𝔽29)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 10 | 28 |
| 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 17 | 0 | 0 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 12 |
| 0 | 0 | 0 | 0 | 7 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 28 |
| 0 | 0 | 0 | 0 | 24 | 24 |
G:=sub<GL(6,GF(29))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,10,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,17,0,0,0,0,17,0,0,0,0,0,0,0,0,17,0,0,0,0,17,0,0,0,0,0,0,0,27,7,0,0,0,0,12,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,5,24,0,0,0,0,28,24] >;
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4H | 4I | ··· | 4N | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14BN | 14BO | ··· | 14BZ | 28A | ··· | 28AV | 28AW | ··· | 28CF |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C7×D4 | C7×C4○D4 |
| kernel | C14×C22.D4 | C14×C22⋊C4 | C14×C4⋊C4 | C7×C22.D4 | C23×C28 | D4×C2×C14 | C2×C22.D4 | C2×C22⋊C4 | C2×C4⋊C4 | C22.D4 | C23×C4 | C22×D4 | C22×C14 | C2×C14 | C23 | C22 |
| # reps | 1 | 3 | 2 | 8 | 1 | 1 | 6 | 18 | 12 | 48 | 6 | 6 | 4 | 8 | 24 | 48 |
In GAP, Magma, Sage, TeX
C_{14}\times C_2^2.D_4 % in TeX
G:=Group("C14xC2^2.D4"); // GroupNames label
G:=SmallGroup(448,1307);
// by ID
G=gap.SmallGroup(448,1307);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,4790,604]);
// Polycyclic
G:=Group<a,b,c,d,e|a^14=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀