direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×D4⋊2D7, C14.6C25, C28.41C24, D14.2C24, C24.60D14, Dic14⋊8C23, Dic7.14C24, (C2×D4)⋊48D14, (C7×D4)⋊7C23, D4⋊6(C22×D7), (C4×D7)⋊5C23, C7⋊D4⋊2C23, C2.7(D7×C24), (C22×D4)⋊14D7, (C2×C14).1C24, C4.41(C23×D7), (D4×C14)⋊50C22, C22.1(C23×D7), (C2×C28).563C23, (C23×Dic7)⋊11C2, (C2×Dic7)⋊11C23, (C22×C4).389D14, (C2×Dic14)⋊71C22, (C22×Dic14)⋊23C2, (C23×C14).81C22, C23.348(C22×D7), (C22×C28).299C22, (C22×C14).434C23, (C22×Dic7)⋊52C22, (C22×D7).246C23, (C23×D7).117C22, (D4×C2×C14)⋊10C2, C14⋊2(C2×C4○D4), (D7×C22×C4)⋊9C2, C7⋊2(C22×C4○D4), (C2×C4×D7)⋊59C22, (C2×C14)⋊16(C4○D4), (C22×C7⋊D4)⋊20C2, (C2×C7⋊D4)⋊51C22, (C2×C4).643(C22×D7), SmallGroup(448,1370)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×D4⋊2D7
G = < a,b,c,d,e,f | a2=b2=c4=d2=e7=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, fdf=c2d, fef=e-1 >
Subgroups: 3092 in 890 conjugacy classes, 463 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, Dic14, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C22×C4○D4, C2×Dic14, C2×C4×D7, D4⋊2D7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, C23×C14, C22×Dic14, D7×C22×C4, C2×D4⋊2D7, C23×Dic7, C22×C7⋊D4, D4×C2×C14, C22×D4⋊2D7
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, C25, C22×D7, C22×C4○D4, D4⋊2D7, C23×D7, C2×D4⋊2D7, D7×C24, C22×D4⋊2D7
►On 224 points
Generators in S224
(1 120)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 148)(30 149)(31 150)(32 151)(33 152)(34 153)(35 154)(36 141)(37 142)(38 143)(39 144)(40 145)(41 146)(42 147)(43 162)(44 163)(45 164)(46 165)(47 166)(48 167)(49 168)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 176)(58 177)(59 178)(60 179)(61 180)(62 181)(63 182)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 190)(72 191)(73 192)(74 193)(75 194)(76 195)(77 196)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 204)(86 205)(87 206)(88 207)(89 208)(90 209)(91 210)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 218)(100 219)(101 220)(102 221)(103 222)(104 223)(105 224)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 99)(51 100)(52 101)(53 102)(54 103)(55 104)(56 105)(113 176)(114 177)(115 178)(116 179)(117 180)(118 181)(119 182)(120 169)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(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)(141 204)(142 205)(143 206)(144 207)(145 208)(146 209)(147 210)(148 197)(149 198)(150 199)(151 200)(152 201)(153 202)(154 203)(155 218)(156 219)(157 220)(158 221)(159 222)(160 223)(161 224)(162 211)(163 212)(164 213)(165 214)(166 215)(167 216)(168 217)
(1 148 8 141)(2 149 9 142)(3 150 10 143)(4 151 11 144)(5 152 12 145)(6 153 13 146)(7 154 14 147)(15 162 22 155)(16 163 23 156)(17 164 24 157)(18 165 25 158)(19 166 26 159)(20 167 27 160)(21 168 28 161)(29 113 36 120)(30 114 37 121)(31 115 38 122)(32 116 39 123)(33 117 40 124)(34 118 41 125)(35 119 42 126)(43 127 50 134)(44 128 51 135)(45 129 52 136)(46 130 53 137)(47 131 54 138)(48 132 55 139)(49 133 56 140)(57 204 64 197)(58 205 65 198)(59 206 66 199)(60 207 67 200)(61 208 68 201)(62 209 69 202)(63 210 70 203)(71 218 78 211)(72 219 79 212)(73 220 80 213)(74 221 81 214)(75 222 82 215)(76 223 83 216)(77 224 84 217)(85 169 92 176)(86 170 93 177)(87 171 94 178)(88 172 95 179)(89 173 96 180)(90 174 97 181)(91 175 98 182)(99 183 106 190)(100 184 107 191)(101 185 108 192)(102 186 109 193)(103 187 110 194)(104 188 111 195)(105 189 112 196)
(1 155)(2 156)(3 157)(4 158)(5 159)(6 160)(7 161)(8 162)(9 163)(10 164)(11 165)(12 166)(13 167)(14 168)(15 141)(16 142)(17 143)(18 144)(19 145)(20 146)(21 147)(22 148)(23 149)(24 150)(25 151)(26 152)(27 153)(28 154)(29 127)(30 128)(31 129)(32 130)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 211)(58 212)(59 213)(60 214)(61 215)(62 216)(63 217)(64 218)(65 219)(66 220)(67 221)(68 222)(69 223)(70 224)(71 197)(72 198)(73 199)(74 200)(75 201)(76 202)(77 203)(78 204)(79 205)(80 206)(81 207)(82 208)(83 209)(84 210)(85 183)(86 184)(87 185)(88 186)(89 187)(90 188)(91 189)(92 190)(93 191)(94 192)(95 193)(96 194)(97 195)(98 196)(99 169)(100 170)(101 171)(102 172)(103 173)(104 174)(105 175)(106 176)(107 177)(108 178)(109 179)(110 180)(111 181)(112 182)
(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)(2 62)(3 61)(4 60)(5 59)(6 58)(7 57)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 91)(30 90)(31 89)(32 88)(33 87)(34 86)(35 85)(36 98)(37 97)(38 96)(39 95)(40 94)(41 93)(42 92)(43 112)(44 111)(45 110)(46 109)(47 108)(48 107)(49 106)(50 105)(51 104)(52 103)(53 102)(54 101)(55 100)(56 99)(113 175)(114 174)(115 173)(116 172)(117 171)(118 170)(119 169)(120 182)(121 181)(122 180)(123 179)(124 178)(125 177)(126 176)(127 196)(128 195)(129 194)(130 193)(131 192)(132 191)(133 190)(134 189)(135 188)(136 187)(137 186)(138 185)(139 184)(140 183)(141 203)(142 202)(143 201)(144 200)(145 199)(146 198)(147 197)(148 210)(149 209)(150 208)(151 207)(152 206)(153 205)(154 204)(155 224)(156 223)(157 222)(158 221)(159 220)(160 219)(161 218)(162 217)(163 216)(164 215)(165 214)(166 213)(167 212)(168 211)
G:=sub<Sym(224)| (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,141)(37,142)(38,143)(39,144)(40,145)(41,146)(42,147)(43,162)(44,163)(45,164)(46,165)(47,166)(48,167)(49,168)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,176)(58,177)(59,178)(60,179)(61,180)(62,181)(63,182)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,218)(100,219)(101,220)(102,221)(103,222)(104,223)(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(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)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,197)(149,198)(150,199)(151,200)(152,201)(153,202)(154,203)(155,218)(156,219)(157,220)(158,221)(159,222)(160,223)(161,224)(162,211)(163,212)(164,213)(165,214)(166,215)(167,216)(168,217), (1,148,8,141)(2,149,9,142)(3,150,10,143)(4,151,11,144)(5,152,12,145)(6,153,13,146)(7,154,14,147)(15,162,22,155)(16,163,23,156)(17,164,24,157)(18,165,25,158)(19,166,26,159)(20,167,27,160)(21,168,28,161)(29,113,36,120)(30,114,37,121)(31,115,38,122)(32,116,39,123)(33,117,40,124)(34,118,41,125)(35,119,42,126)(43,127,50,134)(44,128,51,135)(45,129,52,136)(46,130,53,137)(47,131,54,138)(48,132,55,139)(49,133,56,140)(57,204,64,197)(58,205,65,198)(59,206,66,199)(60,207,67,200)(61,208,68,201)(62,209,69,202)(63,210,70,203)(71,218,78,211)(72,219,79,212)(73,220,80,213)(74,221,81,214)(75,222,82,215)(76,223,83,216)(77,224,84,217)(85,169,92,176)(86,170,93,177)(87,171,94,178)(88,172,95,179)(89,173,96,180)(90,174,97,181)(91,175,98,182)(99,183,106,190)(100,184,107,191)(101,185,108,192)(102,186,109,193)(103,187,110,194)(104,188,111,195)(105,189,112,196), (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,161)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,127)(30,128)(31,129)(32,130)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,211)(58,212)(59,213)(60,214)(61,215)(62,216)(63,217)(64,218)(65,219)(66,220)(67,221)(68,222)(69,223)(70,224)(71,197)(72,198)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,183)(86,184)(87,185)(88,186)(89,187)(90,188)(91,189)(92,190)(93,191)(94,192)(95,193)(96,194)(97,195)(98,196)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(105,175)(106,176)(107,177)(108,178)(109,179)(110,180)(111,181)(112,182), (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)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,182)(121,181)(122,180)(123,179)(124,178)(125,177)(126,176)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,203)(142,202)(143,201)(144,200)(145,199)(146,198)(147,197)(148,210)(149,209)(150,208)(151,207)(152,206)(153,205)(154,204)(155,224)(156,223)(157,222)(158,221)(159,220)(160,219)(161,218)(162,217)(163,216)(164,215)(165,214)(166,213)(167,212)(168,211)>;
G:=Group( (1,120)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,148)(30,149)(31,150)(32,151)(33,152)(34,153)(35,154)(36,141)(37,142)(38,143)(39,144)(40,145)(41,146)(42,147)(43,162)(44,163)(45,164)(46,165)(47,166)(48,167)(49,168)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,176)(58,177)(59,178)(60,179)(61,180)(62,181)(63,182)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,218)(100,219)(101,220)(102,221)(103,222)(104,223)(105,224)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,57)(9,58)(10,59)(11,60)(12,61)(13,62)(14,63)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,85)(37,86)(38,87)(39,88)(40,89)(41,90)(42,91)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,99)(51,100)(52,101)(53,102)(54,103)(55,104)(56,105)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(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)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,197)(149,198)(150,199)(151,200)(152,201)(153,202)(154,203)(155,218)(156,219)(157,220)(158,221)(159,222)(160,223)(161,224)(162,211)(163,212)(164,213)(165,214)(166,215)(167,216)(168,217), (1,148,8,141)(2,149,9,142)(3,150,10,143)(4,151,11,144)(5,152,12,145)(6,153,13,146)(7,154,14,147)(15,162,22,155)(16,163,23,156)(17,164,24,157)(18,165,25,158)(19,166,26,159)(20,167,27,160)(21,168,28,161)(29,113,36,120)(30,114,37,121)(31,115,38,122)(32,116,39,123)(33,117,40,124)(34,118,41,125)(35,119,42,126)(43,127,50,134)(44,128,51,135)(45,129,52,136)(46,130,53,137)(47,131,54,138)(48,132,55,139)(49,133,56,140)(57,204,64,197)(58,205,65,198)(59,206,66,199)(60,207,67,200)(61,208,68,201)(62,209,69,202)(63,210,70,203)(71,218,78,211)(72,219,79,212)(73,220,80,213)(74,221,81,214)(75,222,82,215)(76,223,83,216)(77,224,84,217)(85,169,92,176)(86,170,93,177)(87,171,94,178)(88,172,95,179)(89,173,96,180)(90,174,97,181)(91,175,98,182)(99,183,106,190)(100,184,107,191)(101,185,108,192)(102,186,109,193)(103,187,110,194)(104,188,111,195)(105,189,112,196), (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,161)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,141)(16,142)(17,143)(18,144)(19,145)(20,146)(21,147)(22,148)(23,149)(24,150)(25,151)(26,152)(27,153)(28,154)(29,127)(30,128)(31,129)(32,130)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,211)(58,212)(59,213)(60,214)(61,215)(62,216)(63,217)(64,218)(65,219)(66,220)(67,221)(68,222)(69,223)(70,224)(71,197)(72,198)(73,199)(74,200)(75,201)(76,202)(77,203)(78,204)(79,205)(80,206)(81,207)(82,208)(83,209)(84,210)(85,183)(86,184)(87,185)(88,186)(89,187)(90,188)(91,189)(92,190)(93,191)(94,192)(95,193)(96,194)(97,195)(98,196)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(105,175)(106,176)(107,177)(108,178)(109,179)(110,180)(111,181)(112,182), (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)(2,62)(3,61)(4,60)(5,59)(6,58)(7,57)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,91)(30,90)(31,89)(32,88)(33,87)(34,86)(35,85)(36,98)(37,97)(38,96)(39,95)(40,94)(41,93)(42,92)(43,112)(44,111)(45,110)(46,109)(47,108)(48,107)(49,106)(50,105)(51,104)(52,103)(53,102)(54,101)(55,100)(56,99)(113,175)(114,174)(115,173)(116,172)(117,171)(118,170)(119,169)(120,182)(121,181)(122,180)(123,179)(124,178)(125,177)(126,176)(127,196)(128,195)(129,194)(130,193)(131,192)(132,191)(133,190)(134,189)(135,188)(136,187)(137,186)(138,185)(139,184)(140,183)(141,203)(142,202)(143,201)(144,200)(145,199)(146,198)(147,197)(148,210)(149,209)(150,208)(151,207)(152,206)(153,205)(154,204)(155,224)(156,223)(157,222)(158,221)(159,220)(160,219)(161,218)(162,217)(163,216)(164,215)(165,214)(166,213)(167,212)(168,211) );
G=PermutationGroup([[(1,120),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,140),(22,127),(23,128),(24,129),(25,130),(26,131),(27,132),(28,133),(29,148),(30,149),(31,150),(32,151),(33,152),(34,153),(35,154),(36,141),(37,142),(38,143),(39,144),(40,145),(41,146),(42,147),(43,162),(44,163),(45,164),(46,165),(47,166),(48,167),(49,168),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,161),(57,176),(58,177),(59,178),(60,179),(61,180),(62,181),(63,182),(64,169),(65,170),(66,171),(67,172),(68,173),(69,174),(70,175),(71,190),(72,191),(73,192),(74,193),(75,194),(76,195),(77,196),(78,183),(79,184),(80,185),(81,186),(82,187),(83,188),(84,189),(85,204),(86,205),(87,206),(88,207),(89,208),(90,209),(91,210),(92,197),(93,198),(94,199),(95,200),(96,201),(97,202),(98,203),(99,218),(100,219),(101,220),(102,221),(103,222),(104,223),(105,224),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,57),(9,58),(10,59),(11,60),(12,61),(13,62),(14,63),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,85),(37,86),(38,87),(39,88),(40,89),(41,90),(42,91),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,99),(51,100),(52,101),(53,102),(54,103),(55,104),(56,105),(113,176),(114,177),(115,178),(116,179),(117,180),(118,181),(119,182),(120,169),(121,170),(122,171),(123,172),(124,173),(125,174),(126,175),(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),(141,204),(142,205),(143,206),(144,207),(145,208),(146,209),(147,210),(148,197),(149,198),(150,199),(151,200),(152,201),(153,202),(154,203),(155,218),(156,219),(157,220),(158,221),(159,222),(160,223),(161,224),(162,211),(163,212),(164,213),(165,214),(166,215),(167,216),(168,217)], [(1,148,8,141),(2,149,9,142),(3,150,10,143),(4,151,11,144),(5,152,12,145),(6,153,13,146),(7,154,14,147),(15,162,22,155),(16,163,23,156),(17,164,24,157),(18,165,25,158),(19,166,26,159),(20,167,27,160),(21,168,28,161),(29,113,36,120),(30,114,37,121),(31,115,38,122),(32,116,39,123),(33,117,40,124),(34,118,41,125),(35,119,42,126),(43,127,50,134),(44,128,51,135),(45,129,52,136),(46,130,53,137),(47,131,54,138),(48,132,55,139),(49,133,56,140),(57,204,64,197),(58,205,65,198),(59,206,66,199),(60,207,67,200),(61,208,68,201),(62,209,69,202),(63,210,70,203),(71,218,78,211),(72,219,79,212),(73,220,80,213),(74,221,81,214),(75,222,82,215),(76,223,83,216),(77,224,84,217),(85,169,92,176),(86,170,93,177),(87,171,94,178),(88,172,95,179),(89,173,96,180),(90,174,97,181),(91,175,98,182),(99,183,106,190),(100,184,107,191),(101,185,108,192),(102,186,109,193),(103,187,110,194),(104,188,111,195),(105,189,112,196)], [(1,155),(2,156),(3,157),(4,158),(5,159),(6,160),(7,161),(8,162),(9,163),(10,164),(11,165),(12,166),(13,167),(14,168),(15,141),(16,142),(17,143),(18,144),(19,145),(20,146),(21,147),(22,148),(23,149),(24,150),(25,151),(26,152),(27,153),(28,154),(29,127),(30,128),(31,129),(32,130),(33,131),(34,132),(35,133),(36,134),(37,135),(38,136),(39,137),(40,138),(41,139),(42,140),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,211),(58,212),(59,213),(60,214),(61,215),(62,216),(63,217),(64,218),(65,219),(66,220),(67,221),(68,222),(69,223),(70,224),(71,197),(72,198),(73,199),(74,200),(75,201),(76,202),(77,203),(78,204),(79,205),(80,206),(81,207),(82,208),(83,209),(84,210),(85,183),(86,184),(87,185),(88,186),(89,187),(90,188),(91,189),(92,190),(93,191),(94,192),(95,193),(96,194),(97,195),(98,196),(99,169),(100,170),(101,171),(102,172),(103,173),(104,174),(105,175),(106,176),(107,177),(108,178),(109,179),(110,180),(111,181),(112,182)], [(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),(2,62),(3,61),(4,60),(5,59),(6,58),(7,57),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,84),(16,83),(17,82),(18,81),(19,80),(20,79),(21,78),(22,77),(23,76),(24,75),(25,74),(26,73),(27,72),(28,71),(29,91),(30,90),(31,89),(32,88),(33,87),(34,86),(35,85),(36,98),(37,97),(38,96),(39,95),(40,94),(41,93),(42,92),(43,112),(44,111),(45,110),(46,109),(47,108),(48,107),(49,106),(50,105),(51,104),(52,103),(53,102),(54,101),(55,100),(56,99),(113,175),(114,174),(115,173),(116,172),(117,171),(118,170),(119,169),(120,182),(121,181),(122,180),(123,179),(124,178),(125,177),(126,176),(127,196),(128,195),(129,194),(130,193),(131,192),(132,191),(133,190),(134,189),(135,188),(136,187),(137,186),(138,185),(139,184),(140,183),(141,203),(142,202),(143,201),(144,200),(145,199),(146,198),(147,197),(148,210),(149,209),(150,208),(151,207),(152,206),(153,205),(154,204),(155,224),(156,223),(157,222),(158,221),(159,220),(160,219),(161,218),(162,217),(163,216),(164,215),(165,214),(166,213),(167,212),(168,211)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 2P | 2Q | 2R | 2S | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | D14 | D4⋊2D7 |
| kernel | C22×D4⋊2D7 | C22×Dic14 | D7×C22×C4 | C2×D4⋊2D7 | C23×Dic7 | C22×C7⋊D4 | D4×C2×C14 | C22×D4 | C2×C14 | C22×C4 | C2×D4 | C24 | C22 |
| # reps | 1 | 1 | 1 | 24 | 2 | 2 | 1 | 3 | 8 | 3 | 36 | 6 | 12 |
Matrix representation of C22×D4⋊2D7 ►in GL6(𝔽29)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 11 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 18 |
| 0 | 0 | 0 | 0 | 13 | 17 |
| 28 | 28 | 0 | 0 | 0 | 0 |
| 5 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 28 | 0 | 0 |
| 0 | 0 | 2 | 22 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 18 | 26 | 0 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 25 | 0 | 0 |
| 0 | 0 | 6 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 13 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,11,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,12,13,0,0,0,0,18,17],[28,5,0,0,0,0,28,4,0,0,0,0,0,0,25,2,0,0,0,0,28,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[18,11,0,0,0,0,26,11,0,0,0,0,0,0,24,6,0,0,0,0,25,5,0,0,0,0,0,0,28,0,0,0,0,0,13,1] >;
C22×D4⋊2D7 in GAP, Magma, Sage, TeX
C_2^2\times D_4\rtimes_2D_7
% in TeX
G:=Group("C2^2xD4:2D7"); // GroupNames label
G:=SmallGroup(448,1370);
// by ID
G=gap.SmallGroup(448,1370);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,136,1684,235,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=d^2=e^7=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=c^2*d,f*e*f=e^-1>;
// generators/relations