direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D4⋊2D7, C42.227D14, D4⋊8(C4×D7), (C4×D4)⋊28D7, (D4×C28)⋊5C2, (D7×C42)⋊2C2, C28⋊15(C4○D4), C4⋊C4.313D14, (D4×Dic7)⋊43C2, (C4×Dic14)⋊23C2, Dic14⋊12(C2×C4), (C2×D4).240D14, C28.31(C22×C4), (C2×C14).80C24, C14.20(C23×C4), D14.7(C22×C4), Dic7⋊11(C4○D4), Dic7⋊4D4⋊50C2, Dic7⋊3Q8⋊45C2, (C4×C28).143C22, (C2×C28).490C23, C22⋊C4.128D14, (C22×C4).319D14, C22.30(C23×D7), D14⋊C4.120C22, (D4×C14).247C22, C4⋊Dic7.361C22, Dic7.24(C22×C4), C23.161(C22×D7), C23.11D14⋊31C2, Dic7⋊C4.133C22, (C22×C14).150C23, (C22×C28).362C22, (C4×Dic7).291C22, (C2×Dic7).307C23, (C22×D7).167C23, C23.D7.100C22, (C2×Dic14).286C22, (C22×Dic7).218C22, C7⋊3(C4×C4○D4), C4.31(C2×C4×D7), (C4×D7)⋊6(C2×C4), C7⋊D4⋊1(C2×C4), C2.3(D7×C4○D4), (C7×D4)⋊10(C2×C4), (C4×C7⋊D4)⋊37C2, C22.1(C2×C4×D7), (C2×C4×Dic7)⋊34C2, C4⋊C4⋊7D7⋊46C2, C2.22(D7×C22×C4), C2.5(C2×D4⋊2D7), (C2×Dic7)⋊17(C2×C4), C14.135(C2×C4○D4), (C2×C14).1(C22×C4), (C2×C4×D7).291C22, (C2×D4⋊2D7).11C2, (C7×C4⋊C4).316C22, (C2×C4).577(C22×D7), (C2×C7⋊D4).107C22, (C7×C22⋊C4).140C22, SmallGroup(448,989)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D4⋊2D7
G = < a,b,c,d,e | a4=b4=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 1108 in 310 conjugacy classes, 157 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D7, C14, C14, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C2×C4○D4, Dic14, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×C14, C4×C4○D4, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C2×Dic14, C2×C4×D7, D4⋊2D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C4×Dic14, D7×C42, C23.11D14, Dic7⋊4D4, Dic7⋊3Q8, C4⋊C4⋊7D7, C2×C4×Dic7, C4×C7⋊D4, D4×Dic7, D4×C28, C2×D4⋊2D7, C4×D4⋊2D7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, C24, D14, C23×C4, C2×C4○D4, C4×D7, C22×D7, C4×C4○D4, C2×C4×D7, D4⋊2D7, C23×D7, D7×C22×C4, C2×D4⋊2D7, D7×C4○D4, C4×D4⋊2D7
►On 224 points
Generators in S224
(1 71 15 57)(2 72 16 58)(3 73 17 59)(4 74 18 60)(5 75 19 61)(6 76 20 62)(7 77 21 63)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 99 43 85)(30 100 44 86)(31 101 45 87)(32 102 46 88)(33 103 47 89)(34 104 48 90)(35 105 49 91)(36 106 50 92)(37 107 51 93)(38 108 52 94)(39 109 53 95)(40 110 54 96)(41 111 55 97)(42 112 56 98)(113 183 127 169)(114 184 128 170)(115 185 129 171)(116 186 130 172)(117 187 131 173)(118 188 132 174)(119 189 133 175)(120 190 134 176)(121 191 135 177)(122 192 136 178)(123 193 137 179)(124 194 138 180)(125 195 139 181)(126 196 140 182)(141 211 155 197)(142 212 156 198)(143 213 157 199)(144 214 158 200)(145 215 159 201)(146 216 160 202)(147 217 161 203)(148 218 162 204)(149 219 163 205)(150 220 164 206)(151 221 165 207)(152 222 166 208)(153 223 167 209)(154 224 168 210)
(1 162 22 141)(2 163 23 142)(3 164 24 143)(4 165 25 144)(5 166 26 145)(6 167 27 146)(7 168 28 147)(8 155 15 148)(9 156 16 149)(10 157 17 150)(11 158 18 151)(12 159 19 152)(13 160 20 153)(14 161 21 154)(29 134 50 113)(30 135 51 114)(31 136 52 115)(32 137 53 116)(33 138 54 117)(34 139 55 118)(35 140 56 119)(36 127 43 120)(37 128 44 121)(38 129 45 122)(39 130 46 123)(40 131 47 124)(41 132 48 125)(42 133 49 126)(57 218 78 197)(58 219 79 198)(59 220 80 199)(60 221 81 200)(61 222 82 201)(62 223 83 202)(63 224 84 203)(64 211 71 204)(65 212 72 205)(66 213 73 206)(67 214 74 207)(68 215 75 208)(69 216 76 209)(70 217 77 210)(85 190 106 169)(86 191 107 170)(87 192 108 171)(88 193 109 172)(89 194 110 173)(90 195 111 174)(91 196 112 175)(92 183 99 176)(93 184 100 177)(94 185 101 178)(95 186 102 179)(96 187 103 180)(97 188 104 181)(98 189 105 182)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 121)(10 122)(11 123)(12 124)(13 125)(14 126)(15 127)(16 128)(17 129)(18 130)(19 131)(20 132)(21 133)(22 134)(23 135)(24 136)(25 137)(26 138)(27 139)(28 140)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 161)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(57 77)(58 76)(59 75)(60 74)(61 73)(62 72)(63 71)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(113 126)(114 125)(115 124)(116 123)(117 122)(118 121)(119 120)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 161)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)(154 162)(169 182)(170 181)(171 180)(172 179)(173 178)(174 177)(175 176)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(197 217)(198 216)(199 215)(200 214)(201 213)(202 212)(203 211)(204 224)(205 223)(206 222)(207 221)(208 220)(209 219)(210 218)
G:=sub<Sym(224)| (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98)(113,183,127,169)(114,184,128,170)(115,185,129,171)(116,186,130,172)(117,187,131,173)(118,188,132,174)(119,189,133,175)(120,190,134,176)(121,191,135,177)(122,192,136,178)(123,193,137,179)(124,194,138,180)(125,195,139,181)(126,196,140,182)(141,211,155,197)(142,212,156,198)(143,213,157,199)(144,214,158,200)(145,215,159,201)(146,216,160,202)(147,217,161,203)(148,218,162,204)(149,219,163,205)(150,220,164,206)(151,221,165,207)(152,222,166,208)(153,223,167,209)(154,224,168,210), (1,162,22,141)(2,163,23,142)(3,164,24,143)(4,165,25,144)(5,166,26,145)(6,167,27,146)(7,168,28,147)(8,155,15,148)(9,156,16,149)(10,157,17,150)(11,158,18,151)(12,159,19,152)(13,160,20,153)(14,161,21,154)(29,134,50,113)(30,135,51,114)(31,136,52,115)(32,137,53,116)(33,138,54,117)(34,139,55,118)(35,140,56,119)(36,127,43,120)(37,128,44,121)(38,129,45,122)(39,130,46,123)(40,131,47,124)(41,132,48,125)(42,133,49,126)(57,218,78,197)(58,219,79,198)(59,220,80,199)(60,221,81,200)(61,222,82,201)(62,223,83,202)(63,224,84,203)(64,211,71,204)(65,212,72,205)(66,213,73,206)(67,214,74,207)(68,215,75,208)(69,216,76,209)(70,217,77,210)(85,190,106,169)(86,191,107,170)(87,192,108,171)(88,193,109,172)(89,194,110,173)(90,195,111,174)(91,196,112,175)(92,183,99,176)(93,184,100,177)(94,185,101,178)(95,186,102,179)(96,187,103,180)(97,188,104,181)(98,189,105,182), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(57,77)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(197,217)(198,216)(199,215)(200,214)(201,213)(202,212)(203,211)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218)>;
G:=Group( (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98)(113,183,127,169)(114,184,128,170)(115,185,129,171)(116,186,130,172)(117,187,131,173)(118,188,132,174)(119,189,133,175)(120,190,134,176)(121,191,135,177)(122,192,136,178)(123,193,137,179)(124,194,138,180)(125,195,139,181)(126,196,140,182)(141,211,155,197)(142,212,156,198)(143,213,157,199)(144,214,158,200)(145,215,159,201)(146,216,160,202)(147,217,161,203)(148,218,162,204)(149,219,163,205)(150,220,164,206)(151,221,165,207)(152,222,166,208)(153,223,167,209)(154,224,168,210), (1,162,22,141)(2,163,23,142)(3,164,24,143)(4,165,25,144)(5,166,26,145)(6,167,27,146)(7,168,28,147)(8,155,15,148)(9,156,16,149)(10,157,17,150)(11,158,18,151)(12,159,19,152)(13,160,20,153)(14,161,21,154)(29,134,50,113)(30,135,51,114)(31,136,52,115)(32,137,53,116)(33,138,54,117)(34,139,55,118)(35,140,56,119)(36,127,43,120)(37,128,44,121)(38,129,45,122)(39,130,46,123)(40,131,47,124)(41,132,48,125)(42,133,49,126)(57,218,78,197)(58,219,79,198)(59,220,80,199)(60,221,81,200)(61,222,82,201)(62,223,83,202)(63,224,84,203)(64,211,71,204)(65,212,72,205)(66,213,73,206)(67,214,74,207)(68,215,75,208)(69,216,76,209)(70,217,77,210)(85,190,106,169)(86,191,107,170)(87,192,108,171)(88,193,109,172)(89,194,110,173)(90,195,111,174)(91,196,112,175)(92,183,99,176)(93,184,100,177)(94,185,101,178)(95,186,102,179)(96,187,103,180)(97,188,104,181)(98,189,105,182), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,121)(10,122)(11,123)(12,124)(13,125)(14,126)(15,127)(16,128)(17,129)(18,130)(19,131)(20,132)(21,133)(22,134)(23,135)(24,136)(25,137)(26,138)(27,139)(28,140)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,155)(44,156)(45,157)(46,158)(47,159)(48,160)(49,161)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (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), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,28)(9,27)(10,26)(11,25)(12,24)(13,23)(14,22)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(57,77)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(113,126)(114,125)(115,124)(116,123)(117,122)(118,121)(119,120)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(197,217)(198,216)(199,215)(200,214)(201,213)(202,212)(203,211)(204,224)(205,223)(206,222)(207,221)(208,220)(209,219)(210,218) );
G=PermutationGroup([[(1,71,15,57),(2,72,16,58),(3,73,17,59),(4,74,18,60),(5,75,19,61),(6,76,20,62),(7,77,21,63),(8,78,22,64),(9,79,23,65),(10,80,24,66),(11,81,25,67),(12,82,26,68),(13,83,27,69),(14,84,28,70),(29,99,43,85),(30,100,44,86),(31,101,45,87),(32,102,46,88),(33,103,47,89),(34,104,48,90),(35,105,49,91),(36,106,50,92),(37,107,51,93),(38,108,52,94),(39,109,53,95),(40,110,54,96),(41,111,55,97),(42,112,56,98),(113,183,127,169),(114,184,128,170),(115,185,129,171),(116,186,130,172),(117,187,131,173),(118,188,132,174),(119,189,133,175),(120,190,134,176),(121,191,135,177),(122,192,136,178),(123,193,137,179),(124,194,138,180),(125,195,139,181),(126,196,140,182),(141,211,155,197),(142,212,156,198),(143,213,157,199),(144,214,158,200),(145,215,159,201),(146,216,160,202),(147,217,161,203),(148,218,162,204),(149,219,163,205),(150,220,164,206),(151,221,165,207),(152,222,166,208),(153,223,167,209),(154,224,168,210)], [(1,162,22,141),(2,163,23,142),(3,164,24,143),(4,165,25,144),(5,166,26,145),(6,167,27,146),(7,168,28,147),(8,155,15,148),(9,156,16,149),(10,157,17,150),(11,158,18,151),(12,159,19,152),(13,160,20,153),(14,161,21,154),(29,134,50,113),(30,135,51,114),(31,136,52,115),(32,137,53,116),(33,138,54,117),(34,139,55,118),(35,140,56,119),(36,127,43,120),(37,128,44,121),(38,129,45,122),(39,130,46,123),(40,131,47,124),(41,132,48,125),(42,133,49,126),(57,218,78,197),(58,219,79,198),(59,220,80,199),(60,221,81,200),(61,222,82,201),(62,223,83,202),(63,224,84,203),(64,211,71,204),(65,212,72,205),(66,213,73,206),(67,214,74,207),(68,215,75,208),(69,216,76,209),(70,217,77,210),(85,190,106,169),(86,191,107,170),(87,192,108,171),(88,193,109,172),(89,194,110,173),(90,195,111,174),(91,196,112,175),(92,183,99,176),(93,184,100,177),(94,185,101,178),(95,186,102,179),(96,187,103,180),(97,188,104,181),(98,189,105,182)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,121),(10,122),(11,123),(12,124),(13,125),(14,126),(15,127),(16,128),(17,129),(18,130),(19,131),(20,132),(21,133),(22,134),(23,135),(24,136),(25,137),(26,138),(27,139),(28,140),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,155),(44,156),(45,157),(46,158),(47,159),(48,160),(49,161),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(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)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,28),(9,27),(10,26),(11,25),(12,24),(13,23),(14,22),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(57,77),(58,76),(59,75),(60,74),(61,73),(62,72),(63,71),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106),(113,126),(114,125),(115,124),(116,123),(117,122),(118,121),(119,120),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134),(141,161),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,168),(149,167),(150,166),(151,165),(152,164),(153,163),(154,162),(169,182),(170,181),(171,180),(172,179),(173,178),(174,177),(175,176),(183,196),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190),(197,217),(198,216),(199,215),(200,214),(201,213),(202,212),(203,211),(204,224),(205,223),(206,222),(207,221),(208,220),(209,219),(210,218)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AD | 7A | 7B | 7C | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | C4○D4 | C4○D4 | D14 | D14 | D14 | D14 | D14 | C4×D7 | D4⋊2D7 | D7×C4○D4 |
| kernel | C4×D4⋊2D7 | C4×Dic14 | D7×C42 | C23.11D14 | Dic7⋊4D4 | Dic7⋊3Q8 | C4⋊C4⋊7D7 | C2×C4×Dic7 | C4×C7⋊D4 | D4×Dic7 | D4×C28 | C2×D4⋊2D7 | D4⋊2D7 | C4×D4 | Dic7 | C28 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 16 | 3 | 4 | 4 | 3 | 6 | 3 | 6 | 3 | 24 | 6 | 6 |
Matrix representation of C4×D4⋊2D7 ►in GL4(𝔽29) generated by
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 2 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 28 |
| 0 | 0 | 27 | 17 |
| 0 | 28 | 0 | 0 |
| 1 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 1 | 0 | 0 |
| 25 | 11 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 5 | 1 |
G:=sub<GL(4,GF(29))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[28,0,0,0,0,28,0,0,0,0,17,2,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,27,0,0,28,17],[0,1,0,0,28,18,0,0,0,0,1,0,0,0,0,1],[18,25,0,0,1,11,0,0,0,0,28,5,0,0,0,1] >;
C4×D4⋊2D7 in GAP, Magma, Sage, TeX
C_4\times D_4\rtimes_2D_7
% in TeX
G:=Group("C4xD4:2D7"); // GroupNames label
G:=SmallGroup(448,989);
// by ID
G=gap.SmallGroup(448,989);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,100,794,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations