direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.9D14, C28.36C24, Dic14.31C23, C7⋊C8.15C23, C4○D4.41D14, (C2×C28).219D4, C28.428(C2×D4), C4.36(C23×D7), (C2×D4).233D14, C14⋊5(C8.C22), D4.D7⋊18C22, (C2×Q8).191D14, (C7×D4).24C23, C7⋊Q16⋊17C22, D4.24(C22×D7), Q8.24(C22×D7), (C7×Q8).24C23, (C2×C28).558C23, (C22×C14).125D4, C14.161(C22×D4), (C22×C4).283D14, C23.69(C7⋊D4), C4.Dic7⋊38C22, (C2×Dic14)⋊70C22, (C22×Dic14)⋊21C2, (D4×C14).273C22, (Q8×C14).238C22, (C22×C28).293C22, C7⋊6(C2×C8.C22), C4.31(C2×C7⋊D4), (C2×D4.D7)⋊31C2, (C2×C4○D4).10D7, (C2×C7⋊Q16)⋊31C2, (C2×C14).77(C2×D4), (C14×C4○D4).11C2, (C2×C4).96(C7⋊D4), (C2×C7⋊C8).184C22, (C2×C4.Dic7)⋊32C2, C2.34(C22×C7⋊D4), (C7×C4○D4).50C22, (C2×C4).247(C22×D7), C22.120(C2×C7⋊D4), SmallGroup(448,1276)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 916 in 258 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×6], C7, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8 [×11], C23, C23, C14, C14 [×2], C14 [×4], C2×C8 [×2], M4(2) [×4], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×9], C4○D4 [×4], C4○D4 [×2], Dic7 [×4], C28 [×2], C28 [×2], C28 [×2], C2×C14, C2×C14 [×2], C2×C14 [×6], C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C22×Q8, C2×C4○D4, C7⋊C8 [×4], Dic14 [×4], Dic14 [×6], C2×Dic7 [×6], C2×C28 [×2], C2×C28 [×4], C2×C28 [×5], C7×D4 [×2], C7×D4 [×5], C7×Q8 [×2], C7×Q8, C22×C14, C22×C14, C2×C8.C22, C2×C7⋊C8 [×2], C4.Dic7 [×4], D4.D7 [×8], C7⋊Q16 [×8], C2×Dic14 [×6], C2×Dic14 [×3], C22×Dic7, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4 [×4], C7×C4○D4 [×2], C2×C4.Dic7, C2×D4.D7 [×2], C2×C7⋊Q16 [×2], D4.9D14 [×8], C22×Dic14, C14×C4○D4, C2×D4.9D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C8.C22 [×2], C22×D4, C7⋊D4 [×4], C22×D7 [×7], C2×C8.C22, C2×C7⋊D4 [×6], C23×D7, D4.9D14 [×2], C22×C7⋊D4, C2×D4.9D14
Generators and relations
G = < a,b,c,d,e | a2=b4=c2=d14=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >
►On 224 points
Generators in S224
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 133)(16 134)(17 135)(18 136)(19 137)(20 138)(21 139)(22 140)(23 127)(24 128)(25 129)(26 130)(27 131)(28 132)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 165)(44 166)(45 167)(46 168)(47 155)(48 156)(49 157)(50 158)(51 159)(52 160)(53 161)(54 162)(55 163)(56 164)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 151)(72 152)(73 153)(74 154)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 224)(184 211)(185 212)(186 213)(187 214)(188 215)(189 216)(190 217)(191 218)(192 219)(193 220)(194 221)(195 222)(196 223)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)
(1 199 119 41)(2 200 120 42)(3 201 121 29)(4 202 122 30)(5 203 123 31)(6 204 124 32)(7 205 125 33)(8 206 126 34)(9 207 113 35)(10 208 114 36)(11 209 115 37)(12 210 116 38)(13 197 117 39)(14 198 118 40)(15 83 140 142)(16 84 127 143)(17 71 128 144)(18 72 129 145)(19 73 130 146)(20 74 131 147)(21 75 132 148)(22 76 133 149)(23 77 134 150)(24 78 135 151)(25 79 136 152)(26 80 137 153)(27 81 138 154)(28 82 139 141)(43 219 158 185)(44 220 159 186)(45 221 160 187)(46 222 161 188)(47 223 162 189)(48 224 163 190)(49 211 164 191)(50 212 165 192)(51 213 166 193)(52 214 167 194)(53 215 168 195)(54 216 155 196)(55 217 156 183)(56 218 157 184)(57 89 110 172)(58 90 111 173)(59 91 112 174)(60 92 99 175)(61 93 100 176)(62 94 101 177)(63 95 102 178)(64 96 103 179)(65 97 104 180)(66 98 105 181)(67 85 106 182)(68 86 107 169)(69 87 108 170)(70 88 109 171)
(1 156)(2 56)(3 158)(4 44)(5 160)(6 46)(7 162)(8 48)(9 164)(10 50)(11 166)(12 52)(13 168)(14 54)(15 68)(16 108)(17 70)(18 110)(19 58)(20 112)(21 60)(22 100)(23 62)(24 102)(25 64)(26 104)(27 66)(28 106)(29 185)(30 220)(31 187)(32 222)(33 189)(34 224)(35 191)(36 212)(37 193)(38 214)(39 195)(40 216)(41 183)(42 218)(43 121)(45 123)(47 125)(49 113)(51 115)(53 117)(55 119)(57 129)(59 131)(61 133)(63 135)(65 137)(67 139)(69 127)(71 171)(72 89)(73 173)(74 91)(75 175)(76 93)(77 177)(78 95)(79 179)(80 97)(81 181)(82 85)(83 169)(84 87)(86 142)(88 144)(90 146)(92 148)(94 150)(96 152)(98 154)(99 132)(101 134)(103 136)(105 138)(107 140)(109 128)(111 130)(114 165)(116 167)(118 155)(120 157)(122 159)(124 161)(126 163)(141 182)(143 170)(145 172)(147 174)(149 176)(151 178)(153 180)(184 200)(186 202)(188 204)(190 206)(192 208)(194 210)(196 198)(197 215)(199 217)(201 219)(203 221)(205 223)(207 211)(209 213)
(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 63 119 102)(2 62 120 101)(3 61 121 100)(4 60 122 99)(5 59 123 112)(6 58 124 111)(7 57 125 110)(8 70 126 109)(9 69 113 108)(10 68 114 107)(11 67 115 106)(12 66 116 105)(13 65 117 104)(14 64 118 103)(15 192 140 212)(16 191 127 211)(17 190 128 224)(18 189 129 223)(19 188 130 222)(20 187 131 221)(21 186 132 220)(22 185 133 219)(23 184 134 218)(24 183 135 217)(25 196 136 216)(26 195 137 215)(27 194 138 214)(28 193 139 213)(29 93 201 176)(30 92 202 175)(31 91 203 174)(32 90 204 173)(33 89 205 172)(34 88 206 171)(35 87 207 170)(36 86 208 169)(37 85 209 182)(38 98 210 181)(39 97 197 180)(40 96 198 179)(41 95 199 178)(42 94 200 177)(43 76 158 149)(44 75 159 148)(45 74 160 147)(46 73 161 146)(47 72 162 145)(48 71 163 144)(49 84 164 143)(50 83 165 142)(51 82 166 141)(52 81 167 154)(53 80 168 153)(54 79 155 152)(55 78 156 151)(56 77 157 150)
G:=sub<Sym(224)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,165)(44,166)(45,167)(46,168)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,161)(54,162)(55,163)(56,164)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,151)(72,152)(73,153)(74,154)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,224)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (1,199,119,41)(2,200,120,42)(3,201,121,29)(4,202,122,30)(5,203,123,31)(6,204,124,32)(7,205,125,33)(8,206,126,34)(9,207,113,35)(10,208,114,36)(11,209,115,37)(12,210,116,38)(13,197,117,39)(14,198,118,40)(15,83,140,142)(16,84,127,143)(17,71,128,144)(18,72,129,145)(19,73,130,146)(20,74,131,147)(21,75,132,148)(22,76,133,149)(23,77,134,150)(24,78,135,151)(25,79,136,152)(26,80,137,153)(27,81,138,154)(28,82,139,141)(43,219,158,185)(44,220,159,186)(45,221,160,187)(46,222,161,188)(47,223,162,189)(48,224,163,190)(49,211,164,191)(50,212,165,192)(51,213,166,193)(52,214,167,194)(53,215,168,195)(54,216,155,196)(55,217,156,183)(56,218,157,184)(57,89,110,172)(58,90,111,173)(59,91,112,174)(60,92,99,175)(61,93,100,176)(62,94,101,177)(63,95,102,178)(64,96,103,179)(65,97,104,180)(66,98,105,181)(67,85,106,182)(68,86,107,169)(69,87,108,170)(70,88,109,171), (1,156)(2,56)(3,158)(4,44)(5,160)(6,46)(7,162)(8,48)(9,164)(10,50)(11,166)(12,52)(13,168)(14,54)(15,68)(16,108)(17,70)(18,110)(19,58)(20,112)(21,60)(22,100)(23,62)(24,102)(25,64)(26,104)(27,66)(28,106)(29,185)(30,220)(31,187)(32,222)(33,189)(34,224)(35,191)(36,212)(37,193)(38,214)(39,195)(40,216)(41,183)(42,218)(43,121)(45,123)(47,125)(49,113)(51,115)(53,117)(55,119)(57,129)(59,131)(61,133)(63,135)(65,137)(67,139)(69,127)(71,171)(72,89)(73,173)(74,91)(75,175)(76,93)(77,177)(78,95)(79,179)(80,97)(81,181)(82,85)(83,169)(84,87)(86,142)(88,144)(90,146)(92,148)(94,150)(96,152)(98,154)(99,132)(101,134)(103,136)(105,138)(107,140)(109,128)(111,130)(114,165)(116,167)(118,155)(120,157)(122,159)(124,161)(126,163)(141,182)(143,170)(145,172)(147,174)(149,176)(151,178)(153,180)(184,200)(186,202)(188,204)(190,206)(192,208)(194,210)(196,198)(197,215)(199,217)(201,219)(203,221)(205,223)(207,211)(209,213), (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), (1,63,119,102)(2,62,120,101)(3,61,121,100)(4,60,122,99)(5,59,123,112)(6,58,124,111)(7,57,125,110)(8,70,126,109)(9,69,113,108)(10,68,114,107)(11,67,115,106)(12,66,116,105)(13,65,117,104)(14,64,118,103)(15,192,140,212)(16,191,127,211)(17,190,128,224)(18,189,129,223)(19,188,130,222)(20,187,131,221)(21,186,132,220)(22,185,133,219)(23,184,134,218)(24,183,135,217)(25,196,136,216)(26,195,137,215)(27,194,138,214)(28,193,139,213)(29,93,201,176)(30,92,202,175)(31,91,203,174)(32,90,204,173)(33,89,205,172)(34,88,206,171)(35,87,207,170)(36,86,208,169)(37,85,209,182)(38,98,210,181)(39,97,197,180)(40,96,198,179)(41,95,199,178)(42,94,200,177)(43,76,158,149)(44,75,159,148)(45,74,160,147)(46,73,161,146)(47,72,162,145)(48,71,163,144)(49,84,164,143)(50,83,165,142)(51,82,166,141)(52,81,167,154)(53,80,168,153)(54,79,155,152)(55,78,156,151)(56,77,157,150)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,133)(16,134)(17,135)(18,136)(19,137)(20,138)(21,139)(22,140)(23,127)(24,128)(25,129)(26,130)(27,131)(28,132)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,165)(44,166)(45,167)(46,168)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,161)(54,162)(55,163)(56,164)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,151)(72,152)(73,153)(74,154)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,224)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (1,199,119,41)(2,200,120,42)(3,201,121,29)(4,202,122,30)(5,203,123,31)(6,204,124,32)(7,205,125,33)(8,206,126,34)(9,207,113,35)(10,208,114,36)(11,209,115,37)(12,210,116,38)(13,197,117,39)(14,198,118,40)(15,83,140,142)(16,84,127,143)(17,71,128,144)(18,72,129,145)(19,73,130,146)(20,74,131,147)(21,75,132,148)(22,76,133,149)(23,77,134,150)(24,78,135,151)(25,79,136,152)(26,80,137,153)(27,81,138,154)(28,82,139,141)(43,219,158,185)(44,220,159,186)(45,221,160,187)(46,222,161,188)(47,223,162,189)(48,224,163,190)(49,211,164,191)(50,212,165,192)(51,213,166,193)(52,214,167,194)(53,215,168,195)(54,216,155,196)(55,217,156,183)(56,218,157,184)(57,89,110,172)(58,90,111,173)(59,91,112,174)(60,92,99,175)(61,93,100,176)(62,94,101,177)(63,95,102,178)(64,96,103,179)(65,97,104,180)(66,98,105,181)(67,85,106,182)(68,86,107,169)(69,87,108,170)(70,88,109,171), (1,156)(2,56)(3,158)(4,44)(5,160)(6,46)(7,162)(8,48)(9,164)(10,50)(11,166)(12,52)(13,168)(14,54)(15,68)(16,108)(17,70)(18,110)(19,58)(20,112)(21,60)(22,100)(23,62)(24,102)(25,64)(26,104)(27,66)(28,106)(29,185)(30,220)(31,187)(32,222)(33,189)(34,224)(35,191)(36,212)(37,193)(38,214)(39,195)(40,216)(41,183)(42,218)(43,121)(45,123)(47,125)(49,113)(51,115)(53,117)(55,119)(57,129)(59,131)(61,133)(63,135)(65,137)(67,139)(69,127)(71,171)(72,89)(73,173)(74,91)(75,175)(76,93)(77,177)(78,95)(79,179)(80,97)(81,181)(82,85)(83,169)(84,87)(86,142)(88,144)(90,146)(92,148)(94,150)(96,152)(98,154)(99,132)(101,134)(103,136)(105,138)(107,140)(109,128)(111,130)(114,165)(116,167)(118,155)(120,157)(122,159)(124,161)(126,163)(141,182)(143,170)(145,172)(147,174)(149,176)(151,178)(153,180)(184,200)(186,202)(188,204)(190,206)(192,208)(194,210)(196,198)(197,215)(199,217)(201,219)(203,221)(205,223)(207,211)(209,213), (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), (1,63,119,102)(2,62,120,101)(3,61,121,100)(4,60,122,99)(5,59,123,112)(6,58,124,111)(7,57,125,110)(8,70,126,109)(9,69,113,108)(10,68,114,107)(11,67,115,106)(12,66,116,105)(13,65,117,104)(14,64,118,103)(15,192,140,212)(16,191,127,211)(17,190,128,224)(18,189,129,223)(19,188,130,222)(20,187,131,221)(21,186,132,220)(22,185,133,219)(23,184,134,218)(24,183,135,217)(25,196,136,216)(26,195,137,215)(27,194,138,214)(28,193,139,213)(29,93,201,176)(30,92,202,175)(31,91,203,174)(32,90,204,173)(33,89,205,172)(34,88,206,171)(35,87,207,170)(36,86,208,169)(37,85,209,182)(38,98,210,181)(39,97,197,180)(40,96,198,179)(41,95,199,178)(42,94,200,177)(43,76,158,149)(44,75,159,148)(45,74,160,147)(46,73,161,146)(47,72,162,145)(48,71,163,144)(49,84,164,143)(50,83,165,142)(51,82,166,141)(52,81,167,154)(53,80,168,153)(54,79,155,152)(55,78,156,151)(56,77,157,150) );
G=PermutationGroup([(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,133),(16,134),(17,135),(18,136),(19,137),(20,138),(21,139),(22,140),(23,127),(24,128),(25,129),(26,130),(27,131),(28,132),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,165),(44,166),(45,167),(46,168),(47,155),(48,156),(49,157),(50,158),(51,159),(52,160),(53,161),(54,162),(55,163),(56,164),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,151),(72,152),(73,153),(74,154),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146),(81,147),(82,148),(83,149),(84,150),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(183,224),(184,211),(185,212),(186,213),(187,214),(188,215),(189,216),(190,217),(191,218),(192,219),(193,220),(194,221),(195,222),(196,223),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210)], [(1,199,119,41),(2,200,120,42),(3,201,121,29),(4,202,122,30),(5,203,123,31),(6,204,124,32),(7,205,125,33),(8,206,126,34),(9,207,113,35),(10,208,114,36),(11,209,115,37),(12,210,116,38),(13,197,117,39),(14,198,118,40),(15,83,140,142),(16,84,127,143),(17,71,128,144),(18,72,129,145),(19,73,130,146),(20,74,131,147),(21,75,132,148),(22,76,133,149),(23,77,134,150),(24,78,135,151),(25,79,136,152),(26,80,137,153),(27,81,138,154),(28,82,139,141),(43,219,158,185),(44,220,159,186),(45,221,160,187),(46,222,161,188),(47,223,162,189),(48,224,163,190),(49,211,164,191),(50,212,165,192),(51,213,166,193),(52,214,167,194),(53,215,168,195),(54,216,155,196),(55,217,156,183),(56,218,157,184),(57,89,110,172),(58,90,111,173),(59,91,112,174),(60,92,99,175),(61,93,100,176),(62,94,101,177),(63,95,102,178),(64,96,103,179),(65,97,104,180),(66,98,105,181),(67,85,106,182),(68,86,107,169),(69,87,108,170),(70,88,109,171)], [(1,156),(2,56),(3,158),(4,44),(5,160),(6,46),(7,162),(8,48),(9,164),(10,50),(11,166),(12,52),(13,168),(14,54),(15,68),(16,108),(17,70),(18,110),(19,58),(20,112),(21,60),(22,100),(23,62),(24,102),(25,64),(26,104),(27,66),(28,106),(29,185),(30,220),(31,187),(32,222),(33,189),(34,224),(35,191),(36,212),(37,193),(38,214),(39,195),(40,216),(41,183),(42,218),(43,121),(45,123),(47,125),(49,113),(51,115),(53,117),(55,119),(57,129),(59,131),(61,133),(63,135),(65,137),(67,139),(69,127),(71,171),(72,89),(73,173),(74,91),(75,175),(76,93),(77,177),(78,95),(79,179),(80,97),(81,181),(82,85),(83,169),(84,87),(86,142),(88,144),(90,146),(92,148),(94,150),(96,152),(98,154),(99,132),(101,134),(103,136),(105,138),(107,140),(109,128),(111,130),(114,165),(116,167),(118,155),(120,157),(122,159),(124,161),(126,163),(141,182),(143,170),(145,172),(147,174),(149,176),(151,178),(153,180),(184,200),(186,202),(188,204),(190,206),(192,208),(194,210),(196,198),(197,215),(199,217),(201,219),(203,221),(205,223),(207,211),(209,213)], [(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)], [(1,63,119,102),(2,62,120,101),(3,61,121,100),(4,60,122,99),(5,59,123,112),(6,58,124,111),(7,57,125,110),(8,70,126,109),(9,69,113,108),(10,68,114,107),(11,67,115,106),(12,66,116,105),(13,65,117,104),(14,64,118,103),(15,192,140,212),(16,191,127,211),(17,190,128,224),(18,189,129,223),(19,188,130,222),(20,187,131,221),(21,186,132,220),(22,185,133,219),(23,184,134,218),(24,183,135,217),(25,196,136,216),(26,195,137,215),(27,194,138,214),(28,193,139,213),(29,93,201,176),(30,92,202,175),(31,91,203,174),(32,90,204,173),(33,89,205,172),(34,88,206,171),(35,87,207,170),(36,86,208,169),(37,85,209,182),(38,98,210,181),(39,97,197,180),(40,96,198,179),(41,95,199,178),(42,94,200,177),(43,76,158,149),(44,75,159,148),(45,74,160,147),(46,73,161,146),(47,72,162,145),(48,71,163,144),(49,84,164,143),(50,83,165,142),(51,82,166,141),(52,81,167,154),(53,80,168,153),(54,79,155,152),(55,78,156,151),(56,77,157,150)])
Matrix representation ►G ⊆ GL6(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 8 | 0 | 0 |
| 0 | 0 | 105 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 105 |
| 0 | 0 | 0 | 0 | 8 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 105 |
| 0 | 0 | 0 | 0 | 8 | 17 |
| 0 | 0 | 17 | 8 | 0 | 0 |
| 0 | 0 | 105 | 96 | 0 | 0 |
| 106 | 0 | 0 | 0 | 0 | 0 |
| 59 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 103 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 89 | 10 |
| 0 | 0 | 0 | 0 | 103 | 103 |
| 86 | 68 | 0 | 0 | 0 | 0 |
| 89 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 18 | 0 | 0 |
| 0 | 0 | 56 | 102 | 0 | 0 |
| 0 | 0 | 0 | 0 | 70 | 105 |
| 0 | 0 | 0 | 0 | 90 | 43 |
G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,105,0,0,0,0,8,96,0,0,0,0,0,0,96,8,0,0,0,0,105,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,17,105,0,0,0,0,8,96,0,0,96,8,0,0,0,0,105,17,0,0],[106,59,0,0,0,0,0,16,0,0,0,0,0,0,24,10,0,0,0,0,103,10,0,0,0,0,0,0,89,103,0,0,0,0,10,103],[86,89,0,0,0,0,68,27,0,0,0,0,0,0,11,56,0,0,0,0,18,102,0,0,0,0,0,0,70,90,0,0,0,0,105,43] >;
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | D14 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8.C22 | D4.9D14 |
| kernel | C2×D4.9D14 | C2×C4.Dic7 | C2×D4.D7 | C2×C7⋊Q16 | D4.9D14 | C22×Dic14 | C14×C4○D4 | C2×C28 | C22×C14 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 3 | 1 | 3 | 3 | 3 | 3 | 12 | 18 | 6 | 2 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_4._9D_{14} % in TeX
G:=Group("C2xD4.9D14"); // GroupNames label
G:=SmallGroup(448,1276);
// by ID
G=gap.SmallGroup(448,1276);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,675,297,1684,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^14=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations
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