direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.D14, Q16⋊11D14, D56⋊17C22, C28.12C24, C56.34C23, D28.7C23, C4.48(D4×D7), (C2×D56)⋊20C2, C14⋊4(C4○D8), (C2×Q16)⋊13D7, (C14×Q16)⋊8C2, (C4×D7).31D4, C28.87(C2×D4), C7⋊C8.23C23, D14.11(C2×D4), (C2×C8).247D14, (C8×D7)⋊15C22, Q8⋊D7⋊10C22, (C7×Q16)⋊9C22, C4.12(C23×D7), C8.40(C22×D7), (C7×Q8).6C23, Q8.6(C22×D7), (C2×C56).99C22, (C2×Q8).154D14, Dic7.70(C2×D4), Q8⋊2D7⋊7C22, (C22×D7).63D4, (C4×D7).28C23, C22.144(D4×D7), (C2×C28).529C23, (C2×Dic7).217D4, C14.113(C22×D4), (C2×D28).180C22, (Q8×C14).151C22, (D7×C2×C8)⋊6C2, C7⋊4(C2×C4○D8), C2.86(C2×D4×D7), (C2×Q8⋊D7)⋊28C2, (C2×Q8⋊2D7)⋊16C2, (C2×C14).402(C2×D4), (C2×C7⋊C8).286C22, (C2×C4×D7).260C22, (C2×C4).617(C22×D7), SmallGroup(448,1218)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1380 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×12], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×14], Q8 [×4], Q8 [×2], C23 [×3], D7 [×6], C14, C14 [×2], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×12], Dic7 [×2], C28 [×2], C28 [×4], D14 [×2], D14 [×10], C2×C14, C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C7⋊C8 [×2], C56 [×2], C4×D7 [×4], C4×D7 [×8], D28 [×4], D28 [×10], C2×Dic7, C2×C28, C2×C28 [×2], C7×Q8 [×4], C7×Q8 [×2], C22×D7, C22×D7 [×2], C2×C4○D8, C8×D7 [×4], D56 [×4], C2×C7⋊C8, Q8⋊D7 [×8], C2×C56, C7×Q16 [×4], C2×C4×D7, C2×C4×D7 [×2], C2×D28 [×2], C2×D28 [×2], Q8⋊2D7 [×8], Q8⋊2D7 [×4], Q8×C14 [×2], D7×C2×C8, C2×D56, Q8.D14 [×8], C2×Q8⋊D7 [×2], C14×Q16, C2×Q8⋊2D7 [×2], C2×Q8.D14
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C4○D8 [×2], C22×D4, C22×D7 [×7], C2×C4○D8, D4×D7 [×2], C23×D7, Q8.D14 [×2], C2×D4×D7, C2×Q8.D14
Generators and relations
G = < a,b,c,d,e | a2=b4=e2=1, c2=d14=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=b2d13 >
(1 200)(2 201)(3 202)(4 203)(5 204)(6 205)(7 206)(8 207)(9 208)(10 209)(11 210)(12 211)(13 212)(14 213)(15 214)(16 215)(17 216)(18 217)(19 218)(20 219)(21 220)(22 221)(23 222)(24 223)(25 224)(26 197)(27 198)(28 199)(29 143)(30 144)(31 145)(32 146)(33 147)(34 148)(35 149)(36 150)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 157)(44 158)(45 159)(46 160)(47 161)(48 162)(49 163)(50 164)(51 165)(52 166)(53 167)(54 168)(55 141)(56 142)(57 124)(58 125)(59 126)(60 127)(61 128)(62 129)(63 130)(64 131)(65 132)(66 133)(67 134)(68 135)(69 136)(70 137)(71 138)(72 139)(73 140)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 120)(82 121)(83 122)(84 123)(85 195)(86 196)(87 169)(88 170)(89 171)(90 172)(91 173)(92 174)(93 175)(94 176)(95 177)(96 178)(97 179)(98 180)(99 181)(100 182)(101 183)(102 184)(103 185)(104 186)(105 187)(106 188)(107 189)(108 190)(109 191)(110 192)(111 193)(112 194)
(1 190 15 176)(2 177 16 191)(3 192 17 178)(4 179 18 193)(5 194 19 180)(6 181 20 195)(7 196 21 182)(8 183 22 169)(9 170 23 184)(10 185 24 171)(11 172 25 186)(12 187 26 173)(13 174 27 188)(14 189 28 175)(29 64 43 78)(30 79 44 65)(31 66 45 80)(32 81 46 67)(33 68 47 82)(34 83 48 69)(35 70 49 84)(36 57 50 71)(37 72 51 58)(38 59 52 73)(39 74 53 60)(40 61 54 75)(41 76 55 62)(42 63 56 77)(85 205 99 219)(86 220 100 206)(87 207 101 221)(88 222 102 208)(89 209 103 223)(90 224 104 210)(91 211 105 197)(92 198 106 212)(93 213 107 199)(94 200 108 214)(95 215 109 201)(96 202 110 216)(97 217 111 203)(98 204 112 218)(113 167 127 153)(114 154 128 168)(115 141 129 155)(116 156 130 142)(117 143 131 157)(118 158 132 144)(119 145 133 159)(120 160 134 146)(121 147 135 161)(122 162 136 148)(123 149 137 163)(124 164 138 150)(125 151 139 165)(126 166 140 152)
(1 137 15 123)(2 164 16 150)(3 139 17 125)(4 166 18 152)(5 113 19 127)(6 168 20 154)(7 115 21 129)(8 142 22 156)(9 117 23 131)(10 144 24 158)(11 119 25 133)(12 146 26 160)(13 121 27 135)(14 148 28 162)(29 88 43 102)(30 223 44 209)(31 90 45 104)(32 197 46 211)(33 92 47 106)(34 199 48 213)(35 94 49 108)(36 201 50 215)(37 96 51 110)(38 203 52 217)(39 98 53 112)(40 205 54 219)(41 100 55 86)(42 207 56 221)(57 109 71 95)(58 202 72 216)(59 111 73 97)(60 204 74 218)(61 85 75 99)(62 206 76 220)(63 87 77 101)(64 208 78 222)(65 89 79 103)(66 210 80 224)(67 91 81 105)(68 212 82 198)(69 93 83 107)(70 214 84 200)(114 181 128 195)(116 183 130 169)(118 185 132 171)(120 187 134 173)(122 189 136 175)(124 191 138 177)(126 193 140 179)(141 196 155 182)(143 170 157 184)(145 172 159 186)(147 174 161 188)(149 176 163 190)(151 178 165 192)(153 180 167 194)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 82)(58 81)(59 80)(60 79)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(83 84)(85 88)(86 87)(89 112)(90 111)(91 110)(92 109)(93 108)(94 107)(95 106)(96 105)(97 104)(98 103)(99 102)(100 101)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(121 124)(122 123)(133 140)(134 139)(135 138)(136 137)(141 156)(142 155)(143 154)(144 153)(145 152)(146 151)(147 150)(148 149)(157 168)(158 167)(159 166)(160 165)(161 164)(162 163)(169 196)(170 195)(171 194)(172 193)(173 192)(174 191)(175 190)(176 189)(177 188)(178 187)(179 186)(180 185)(181 184)(182 183)(197 216)(198 215)(199 214)(200 213)(201 212)(202 211)(203 210)(204 209)(205 208)(206 207)(217 224)(218 223)(219 222)(220 221)
G:=sub<Sym(224)| (1,200)(2,201)(3,202)(4,203)(5,204)(6,205)(7,206)(8,207)(9,208)(10,209)(11,210)(12,211)(13,212)(14,213)(15,214)(16,215)(17,216)(18,217)(19,218)(20,219)(21,220)(22,221)(23,222)(24,223)(25,224)(26,197)(27,198)(28,199)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,141)(56,142)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,121)(83,122)(84,123)(85,195)(86,196)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(97,179)(98,180)(99,181)(100,182)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,193)(112,194), (1,190,15,176)(2,177,16,191)(3,192,17,178)(4,179,18,193)(5,194,19,180)(6,181,20,195)(7,196,21,182)(8,183,22,169)(9,170,23,184)(10,185,24,171)(11,172,25,186)(12,187,26,173)(13,174,27,188)(14,189,28,175)(29,64,43,78)(30,79,44,65)(31,66,45,80)(32,81,46,67)(33,68,47,82)(34,83,48,69)(35,70,49,84)(36,57,50,71)(37,72,51,58)(38,59,52,73)(39,74,53,60)(40,61,54,75)(41,76,55,62)(42,63,56,77)(85,205,99,219)(86,220,100,206)(87,207,101,221)(88,222,102,208)(89,209,103,223)(90,224,104,210)(91,211,105,197)(92,198,106,212)(93,213,107,199)(94,200,108,214)(95,215,109,201)(96,202,110,216)(97,217,111,203)(98,204,112,218)(113,167,127,153)(114,154,128,168)(115,141,129,155)(116,156,130,142)(117,143,131,157)(118,158,132,144)(119,145,133,159)(120,160,134,146)(121,147,135,161)(122,162,136,148)(123,149,137,163)(124,164,138,150)(125,151,139,165)(126,166,140,152), (1,137,15,123)(2,164,16,150)(3,139,17,125)(4,166,18,152)(5,113,19,127)(6,168,20,154)(7,115,21,129)(8,142,22,156)(9,117,23,131)(10,144,24,158)(11,119,25,133)(12,146,26,160)(13,121,27,135)(14,148,28,162)(29,88,43,102)(30,223,44,209)(31,90,45,104)(32,197,46,211)(33,92,47,106)(34,199,48,213)(35,94,49,108)(36,201,50,215)(37,96,51,110)(38,203,52,217)(39,98,53,112)(40,205,54,219)(41,100,55,86)(42,207,56,221)(57,109,71,95)(58,202,72,216)(59,111,73,97)(60,204,74,218)(61,85,75,99)(62,206,76,220)(63,87,77,101)(64,208,78,222)(65,89,79,103)(66,210,80,224)(67,91,81,105)(68,212,82,198)(69,93,83,107)(70,214,84,200)(114,181,128,195)(116,183,130,169)(118,185,132,171)(120,187,134,173)(122,189,136,175)(124,191,138,177)(126,193,140,179)(141,196,155,182)(143,170,157,184)(145,172,159,186)(147,174,161,188)(149,176,163,190)(151,178,165,192)(153,180,167,194), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(83,84)(85,88)(86,87)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)(133,140)(134,139)(135,138)(136,137)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163)(169,196)(170,195)(171,194)(172,193)(173,192)(174,191)(175,190)(176,189)(177,188)(178,187)(179,186)(180,185)(181,184)(182,183)(197,216)(198,215)(199,214)(200,213)(201,212)(202,211)(203,210)(204,209)(205,208)(206,207)(217,224)(218,223)(219,222)(220,221)>;
G:=Group( (1,200)(2,201)(3,202)(4,203)(5,204)(6,205)(7,206)(8,207)(9,208)(10,209)(11,210)(12,211)(13,212)(14,213)(15,214)(16,215)(17,216)(18,217)(19,218)(20,219)(21,220)(22,221)(23,222)(24,223)(25,224)(26,197)(27,198)(28,199)(29,143)(30,144)(31,145)(32,146)(33,147)(34,148)(35,149)(36,150)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,157)(44,158)(45,159)(46,160)(47,161)(48,162)(49,163)(50,164)(51,165)(52,166)(53,167)(54,168)(55,141)(56,142)(57,124)(58,125)(59,126)(60,127)(61,128)(62,129)(63,130)(64,131)(65,132)(66,133)(67,134)(68,135)(69,136)(70,137)(71,138)(72,139)(73,140)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,121)(83,122)(84,123)(85,195)(86,196)(87,169)(88,170)(89,171)(90,172)(91,173)(92,174)(93,175)(94,176)(95,177)(96,178)(97,179)(98,180)(99,181)(100,182)(101,183)(102,184)(103,185)(104,186)(105,187)(106,188)(107,189)(108,190)(109,191)(110,192)(111,193)(112,194), (1,190,15,176)(2,177,16,191)(3,192,17,178)(4,179,18,193)(5,194,19,180)(6,181,20,195)(7,196,21,182)(8,183,22,169)(9,170,23,184)(10,185,24,171)(11,172,25,186)(12,187,26,173)(13,174,27,188)(14,189,28,175)(29,64,43,78)(30,79,44,65)(31,66,45,80)(32,81,46,67)(33,68,47,82)(34,83,48,69)(35,70,49,84)(36,57,50,71)(37,72,51,58)(38,59,52,73)(39,74,53,60)(40,61,54,75)(41,76,55,62)(42,63,56,77)(85,205,99,219)(86,220,100,206)(87,207,101,221)(88,222,102,208)(89,209,103,223)(90,224,104,210)(91,211,105,197)(92,198,106,212)(93,213,107,199)(94,200,108,214)(95,215,109,201)(96,202,110,216)(97,217,111,203)(98,204,112,218)(113,167,127,153)(114,154,128,168)(115,141,129,155)(116,156,130,142)(117,143,131,157)(118,158,132,144)(119,145,133,159)(120,160,134,146)(121,147,135,161)(122,162,136,148)(123,149,137,163)(124,164,138,150)(125,151,139,165)(126,166,140,152), (1,137,15,123)(2,164,16,150)(3,139,17,125)(4,166,18,152)(5,113,19,127)(6,168,20,154)(7,115,21,129)(8,142,22,156)(9,117,23,131)(10,144,24,158)(11,119,25,133)(12,146,26,160)(13,121,27,135)(14,148,28,162)(29,88,43,102)(30,223,44,209)(31,90,45,104)(32,197,46,211)(33,92,47,106)(34,199,48,213)(35,94,49,108)(36,201,50,215)(37,96,51,110)(38,203,52,217)(39,98,53,112)(40,205,54,219)(41,100,55,86)(42,207,56,221)(57,109,71,95)(58,202,72,216)(59,111,73,97)(60,204,74,218)(61,85,75,99)(62,206,76,220)(63,87,77,101)(64,208,78,222)(65,89,79,103)(66,210,80,224)(67,91,81,105)(68,212,82,198)(69,93,83,107)(70,214,84,200)(114,181,128,195)(116,183,130,169)(118,185,132,171)(120,187,134,173)(122,189,136,175)(124,191,138,177)(126,193,140,179)(141,196,155,182)(143,170,157,184)(145,172,159,186)(147,174,161,188)(149,176,163,190)(151,178,165,192)(153,180,167,194), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(83,84)(85,88)(86,87)(89,112)(90,111)(91,110)(92,109)(93,108)(94,107)(95,106)(96,105)(97,104)(98,103)(99,102)(100,101)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)(121,124)(122,123)(133,140)(134,139)(135,138)(136,137)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)(157,168)(158,167)(159,166)(160,165)(161,164)(162,163)(169,196)(170,195)(171,194)(172,193)(173,192)(174,191)(175,190)(176,189)(177,188)(178,187)(179,186)(180,185)(181,184)(182,183)(197,216)(198,215)(199,214)(200,213)(201,212)(202,211)(203,210)(204,209)(205,208)(206,207)(217,224)(218,223)(219,222)(220,221) );
G=PermutationGroup([(1,200),(2,201),(3,202),(4,203),(5,204),(6,205),(7,206),(8,207),(9,208),(10,209),(11,210),(12,211),(13,212),(14,213),(15,214),(16,215),(17,216),(18,217),(19,218),(20,219),(21,220),(22,221),(23,222),(24,223),(25,224),(26,197),(27,198),(28,199),(29,143),(30,144),(31,145),(32,146),(33,147),(34,148),(35,149),(36,150),(37,151),(38,152),(39,153),(40,154),(41,155),(42,156),(43,157),(44,158),(45,159),(46,160),(47,161),(48,162),(49,163),(50,164),(51,165),(52,166),(53,167),(54,168),(55,141),(56,142),(57,124),(58,125),(59,126),(60,127),(61,128),(62,129),(63,130),(64,131),(65,132),(66,133),(67,134),(68,135),(69,136),(70,137),(71,138),(72,139),(73,140),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(81,120),(82,121),(83,122),(84,123),(85,195),(86,196),(87,169),(88,170),(89,171),(90,172),(91,173),(92,174),(93,175),(94,176),(95,177),(96,178),(97,179),(98,180),(99,181),(100,182),(101,183),(102,184),(103,185),(104,186),(105,187),(106,188),(107,189),(108,190),(109,191),(110,192),(111,193),(112,194)], [(1,190,15,176),(2,177,16,191),(3,192,17,178),(4,179,18,193),(5,194,19,180),(6,181,20,195),(7,196,21,182),(8,183,22,169),(9,170,23,184),(10,185,24,171),(11,172,25,186),(12,187,26,173),(13,174,27,188),(14,189,28,175),(29,64,43,78),(30,79,44,65),(31,66,45,80),(32,81,46,67),(33,68,47,82),(34,83,48,69),(35,70,49,84),(36,57,50,71),(37,72,51,58),(38,59,52,73),(39,74,53,60),(40,61,54,75),(41,76,55,62),(42,63,56,77),(85,205,99,219),(86,220,100,206),(87,207,101,221),(88,222,102,208),(89,209,103,223),(90,224,104,210),(91,211,105,197),(92,198,106,212),(93,213,107,199),(94,200,108,214),(95,215,109,201),(96,202,110,216),(97,217,111,203),(98,204,112,218),(113,167,127,153),(114,154,128,168),(115,141,129,155),(116,156,130,142),(117,143,131,157),(118,158,132,144),(119,145,133,159),(120,160,134,146),(121,147,135,161),(122,162,136,148),(123,149,137,163),(124,164,138,150),(125,151,139,165),(126,166,140,152)], [(1,137,15,123),(2,164,16,150),(3,139,17,125),(4,166,18,152),(5,113,19,127),(6,168,20,154),(7,115,21,129),(8,142,22,156),(9,117,23,131),(10,144,24,158),(11,119,25,133),(12,146,26,160),(13,121,27,135),(14,148,28,162),(29,88,43,102),(30,223,44,209),(31,90,45,104),(32,197,46,211),(33,92,47,106),(34,199,48,213),(35,94,49,108),(36,201,50,215),(37,96,51,110),(38,203,52,217),(39,98,53,112),(40,205,54,219),(41,100,55,86),(42,207,56,221),(57,109,71,95),(58,202,72,216),(59,111,73,97),(60,204,74,218),(61,85,75,99),(62,206,76,220),(63,87,77,101),(64,208,78,222),(65,89,79,103),(66,210,80,224),(67,91,81,105),(68,212,82,198),(69,93,83,107),(70,214,84,200),(114,181,128,195),(116,183,130,169),(118,185,132,171),(120,187,134,173),(122,189,136,175),(124,191,138,177),(126,193,140,179),(141,196,155,182),(143,170,157,184),(145,172,159,186),(147,174,161,188),(149,176,163,190),(151,178,165,192),(153,180,167,194)], [(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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(29,40),(30,39),(31,38),(32,37),(33,36),(34,35),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(57,82),(58,81),(59,80),(60,79),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(83,84),(85,88),(86,87),(89,112),(90,111),(91,110),(92,109),(93,108),(94,107),(95,106),(96,105),(97,104),(98,103),(99,102),(100,101),(113,132),(114,131),(115,130),(116,129),(117,128),(118,127),(119,126),(120,125),(121,124),(122,123),(133,140),(134,139),(135,138),(136,137),(141,156),(142,155),(143,154),(144,153),(145,152),(146,151),(147,150),(148,149),(157,168),(158,167),(159,166),(160,165),(161,164),(162,163),(169,196),(170,195),(171,194),(172,193),(173,192),(174,191),(175,190),(176,189),(177,188),(178,187),(179,186),(180,185),(181,184),(182,183),(197,216),(198,215),(199,214),(200,213),(201,212),(202,211),(203,210),(204,209),(205,208),(206,207),(217,224),(218,223),(219,222),(220,221)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
112 | 0 | 0 | 0 |
0 | 112 | 0 | 0 |
0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 98 | 0 |
0 | 0 | 53 | 15 |
1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 |
0 | 0 | 88 | 69 |
0 | 0 | 81 | 25 |
104 | 80 | 0 | 0 |
33 | 33 | 0 | 0 |
0 | 0 | 111 | 1 |
0 | 0 | 108 | 2 |
104 | 80 | 0 | 0 |
112 | 9 | 0 | 0 |
0 | 0 | 2 | 112 |
0 | 0 | 3 | 111 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,53,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,88,81,0,0,69,25],[104,33,0,0,80,33,0,0,0,0,111,108,0,0,1,2],[104,112,0,0,80,9,0,0,0,0,2,3,0,0,112,111] >;
70 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
size | 1 | 1 | 1 | 1 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 7 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
70 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | D4×D7 | D4×D7 | Q8.D14 |
kernel | C2×Q8.D14 | D7×C2×C8 | C2×D56 | Q8.D14 | C2×Q8⋊D7 | C14×Q16 | C2×Q8⋊2D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C14 | C4 | C22 | C2 |
# reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 12 | 6 | 8 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_2\times Q_8.D_{14}
% in TeX
G:=Group("C2xQ8.D14");
// GroupNames label
G:=SmallGroup(448,1218);
// by ID
G=gap.SmallGroup(448,1218);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,184,1123,185,136,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=e^2=1,c^2=d^14=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=b^2*d^13>;
// generators/relations