metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.55D14, (C2×C28).70D4, C4⋊D4.3D7, (C2×D4).35D14, C28.55D4⋊8C2, D4⋊Dic7⋊13C2, C4.Dic14⋊33C2, C14.54(C2×SD16), (C2×C14).15SD16, (C22×C14).80D4, C28.181(C4○D4), C4.91(D4⋊2D7), (C2×C28).353C23, (D4×C14).51C22, (C22×C4).117D14, C23.57(C7⋊D4), C7⋊5(C23.46D4), C22.3(D4.D7), C2.11(D4⋊D14), C14.113(C8⋊C22), C4⋊Dic7.335C22, (C22×C28).157C22, C14.78(C22.D4), C2.12(C23.18D14), C2.8(C2×D4.D7), (C2×C4⋊Dic7)⋊32C2, (C7×C4⋊D4).2C2, (C2×C14).484(C2×D4), (C2×C4).48(C7⋊D4), (C2×C7⋊C8).106C22, (C7×C4⋊C4).102C22, (C2×C4).453(C22×D7), C22.159(C2×C7⋊D4), SmallGroup(448,568)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊D4.D7
G = < a,b,c,d,e | a4=b4=c2=d7=1, e2=a2b2, bab-1=cac=eae-1=a-1, ad=da, cbc=b-1, bd=db, ebe-1=a-1b, cd=dc, ece-1=a-1c, ede-1=d-1 >
Subgroups: 460 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C23.46D4, C2×C7⋊C8, C4⋊Dic7, C4⋊Dic7, C7×C22⋊C4, C7×C4⋊C4, C22×Dic7, C22×C28, D4×C14, D4×C14, C4.Dic14, C28.55D4, D4⋊Dic7, C2×C4⋊Dic7, C7×C4⋊D4, C4⋊D4.D7
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, C4○D4, D14, C22.D4, C2×SD16, C8⋊C22, C7⋊D4, C22×D7, C23.46D4, D4.D7, D4⋊2D7, C2×C7⋊D4, C2×D4.D7, C23.18D14, D4⋊D14, C4⋊D4.D7
►On 224 points
Generators in S224
(1 41 13 34)(2 42 14 35)(3 36 8 29)(4 37 9 30)(5 38 10 31)(6 39 11 32)(7 40 12 33)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 85 64 92)(58 86 65 93)(59 87 66 94)(60 88 67 95)(61 89 68 96)(62 90 69 97)(63 91 70 98)(71 99 78 106)(72 100 79 107)(73 101 80 108)(74 102 81 109)(75 103 82 110)(76 104 83 111)(77 105 84 112)(113 141 120 148)(114 142 121 149)(115 143 122 150)(116 144 123 151)(117 145 124 152)(118 146 125 153)(119 147 126 154)(127 155 134 162)(128 156 135 163)(129 157 136 164)(130 158 137 165)(131 159 138 166)(132 160 139 167)(133 161 140 168)(169 204 176 197)(170 205 177 198)(171 206 178 199)(172 207 179 200)(173 208 180 201)(174 209 181 202)(175 210 182 203)(183 218 190 211)(184 219 191 212)(185 220 192 213)(186 221 193 214)(187 222 194 215)(188 223 195 216)(189 224 196 217)
(1 83 20 69)(2 84 21 70)(3 78 15 64)(4 79 16 65)(5 80 17 66)(6 81 18 67)(7 82 19 68)(8 71 22 57)(9 72 23 58)(10 73 24 59)(11 74 25 60)(12 75 26 61)(13 76 27 62)(14 77 28 63)(29 106 43 92)(30 107 44 93)(31 108 45 94)(32 109 46 95)(33 110 47 96)(34 111 48 97)(35 112 49 98)(36 99 50 85)(37 100 51 86)(38 101 52 87)(39 102 53 88)(40 103 54 89)(41 104 55 90)(42 105 56 91)(113 218 127 204)(114 219 128 205)(115 220 129 206)(116 221 130 207)(117 222 131 208)(118 223 132 209)(119 224 133 210)(120 211 134 197)(121 212 135 198)(122 213 136 199)(123 214 137 200)(124 215 138 201)(125 216 139 202)(126 217 140 203)(141 183 155 169)(142 184 156 170)(143 185 157 171)(144 186 158 172)(145 187 159 173)(146 188 160 174)(147 189 161 175)(148 190 162 176)(149 191 163 177)(150 192 164 178)(151 193 165 179)(152 194 166 180)(153 195 167 181)(154 196 168 182)
(1 62)(2 63)(3 57)(4 58)(5 59)(6 60)(7 61)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(113 197)(114 198)(115 199)(116 200)(117 201)(118 202)(119 203)(120 204)(121 205)(122 206)(123 207)(124 208)(125 209)(126 210)(127 211)(128 212)(129 213)(130 214)(131 215)(132 216)(133 217)(134 218)(135 219)(136 220)(137 221)(138 222)(139 223)(140 224)(141 176)(142 177)(143 178)(144 179)(145 180)(146 181)(147 182)(148 169)(149 170)(150 171)(151 172)(152 173)(153 174)(154 175)(155 190)(156 191)(157 192)(158 193)(159 194)(160 195)(161 196)(162 183)(163 184)(164 185)(165 186)(166 187)(167 188)(168 189)
(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)
(1 140 27 119)(2 139 28 118)(3 138 22 117)(4 137 23 116)(5 136 24 115)(6 135 25 114)(7 134 26 113)(8 131 15 124)(9 130 16 123)(10 129 17 122)(11 128 18 121)(12 127 19 120)(13 133 20 126)(14 132 21 125)(29 166 50 145)(30 165 51 144)(31 164 52 143)(32 163 53 142)(33 162 54 141)(34 168 55 147)(35 167 56 146)(36 159 43 152)(37 158 44 151)(38 157 45 150)(39 156 46 149)(40 155 47 148)(41 161 48 154)(42 160 49 153)(57 194 78 173)(58 193 79 172)(59 192 80 171)(60 191 81 170)(61 190 82 169)(62 196 83 175)(63 195 84 174)(64 187 71 180)(65 186 72 179)(66 185 73 178)(67 184 74 177)(68 183 75 176)(69 189 76 182)(70 188 77 181)(85 222 106 201)(86 221 107 200)(87 220 108 199)(88 219 109 198)(89 218 110 197)(90 224 111 203)(91 223 112 202)(92 215 99 208)(93 214 100 207)(94 213 101 206)(95 212 102 205)(96 211 103 204)(97 217 104 210)(98 216 105 209)
G:=sub<Sym(224)| (1,41,13,34)(2,42,14,35)(3,36,8,29)(4,37,9,30)(5,38,10,31)(6,39,11,32)(7,40,12,33)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,85,64,92)(58,86,65,93)(59,87,66,94)(60,88,67,95)(61,89,68,96)(62,90,69,97)(63,91,70,98)(71,99,78,106)(72,100,79,107)(73,101,80,108)(74,102,81,109)(75,103,82,110)(76,104,83,111)(77,105,84,112)(113,141,120,148)(114,142,121,149)(115,143,122,150)(116,144,123,151)(117,145,124,152)(118,146,125,153)(119,147,126,154)(127,155,134,162)(128,156,135,163)(129,157,136,164)(130,158,137,165)(131,159,138,166)(132,160,139,167)(133,161,140,168)(169,204,176,197)(170,205,177,198)(171,206,178,199)(172,207,179,200)(173,208,180,201)(174,209,181,202)(175,210,182,203)(183,218,190,211)(184,219,191,212)(185,220,192,213)(186,221,193,214)(187,222,194,215)(188,223,195,216)(189,224,196,217), (1,83,20,69)(2,84,21,70)(3,78,15,64)(4,79,16,65)(5,80,17,66)(6,81,18,67)(7,82,19,68)(8,71,22,57)(9,72,23,58)(10,73,24,59)(11,74,25,60)(12,75,26,61)(13,76,27,62)(14,77,28,63)(29,106,43,92)(30,107,44,93)(31,108,45,94)(32,109,46,95)(33,110,47,96)(34,111,48,97)(35,112,49,98)(36,99,50,85)(37,100,51,86)(38,101,52,87)(39,102,53,88)(40,103,54,89)(41,104,55,90)(42,105,56,91)(113,218,127,204)(114,219,128,205)(115,220,129,206)(116,221,130,207)(117,222,131,208)(118,223,132,209)(119,224,133,210)(120,211,134,197)(121,212,135,198)(122,213,136,199)(123,214,137,200)(124,215,138,201)(125,216,139,202)(126,217,140,203)(141,183,155,169)(142,184,156,170)(143,185,157,171)(144,186,158,172)(145,187,159,173)(146,188,160,174)(147,189,161,175)(148,190,162,176)(149,191,163,177)(150,192,164,178)(151,193,165,179)(152,194,166,180)(153,195,167,181)(154,196,168,182), (1,62)(2,63)(3,57)(4,58)(5,59)(6,60)(7,61)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(113,197)(114,198)(115,199)(116,200)(117,201)(118,202)(119,203)(120,204)(121,205)(122,206)(123,207)(124,208)(125,209)(126,210)(127,211)(128,212)(129,213)(130,214)(131,215)(132,216)(133,217)(134,218)(135,219)(136,220)(137,221)(138,222)(139,223)(140,224)(141,176)(142,177)(143,178)(144,179)(145,180)(146,181)(147,182)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)(161,196)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189), (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), (1,140,27,119)(2,139,28,118)(3,138,22,117)(4,137,23,116)(5,136,24,115)(6,135,25,114)(7,134,26,113)(8,131,15,124)(9,130,16,123)(10,129,17,122)(11,128,18,121)(12,127,19,120)(13,133,20,126)(14,132,21,125)(29,166,50,145)(30,165,51,144)(31,164,52,143)(32,163,53,142)(33,162,54,141)(34,168,55,147)(35,167,56,146)(36,159,43,152)(37,158,44,151)(38,157,45,150)(39,156,46,149)(40,155,47,148)(41,161,48,154)(42,160,49,153)(57,194,78,173)(58,193,79,172)(59,192,80,171)(60,191,81,170)(61,190,82,169)(62,196,83,175)(63,195,84,174)(64,187,71,180)(65,186,72,179)(66,185,73,178)(67,184,74,177)(68,183,75,176)(69,189,76,182)(70,188,77,181)(85,222,106,201)(86,221,107,200)(87,220,108,199)(88,219,109,198)(89,218,110,197)(90,224,111,203)(91,223,112,202)(92,215,99,208)(93,214,100,207)(94,213,101,206)(95,212,102,205)(96,211,103,204)(97,217,104,210)(98,216,105,209)>;
G:=Group( (1,41,13,34)(2,42,14,35)(3,36,8,29)(4,37,9,30)(5,38,10,31)(6,39,11,32)(7,40,12,33)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,85,64,92)(58,86,65,93)(59,87,66,94)(60,88,67,95)(61,89,68,96)(62,90,69,97)(63,91,70,98)(71,99,78,106)(72,100,79,107)(73,101,80,108)(74,102,81,109)(75,103,82,110)(76,104,83,111)(77,105,84,112)(113,141,120,148)(114,142,121,149)(115,143,122,150)(116,144,123,151)(117,145,124,152)(118,146,125,153)(119,147,126,154)(127,155,134,162)(128,156,135,163)(129,157,136,164)(130,158,137,165)(131,159,138,166)(132,160,139,167)(133,161,140,168)(169,204,176,197)(170,205,177,198)(171,206,178,199)(172,207,179,200)(173,208,180,201)(174,209,181,202)(175,210,182,203)(183,218,190,211)(184,219,191,212)(185,220,192,213)(186,221,193,214)(187,222,194,215)(188,223,195,216)(189,224,196,217), (1,83,20,69)(2,84,21,70)(3,78,15,64)(4,79,16,65)(5,80,17,66)(6,81,18,67)(7,82,19,68)(8,71,22,57)(9,72,23,58)(10,73,24,59)(11,74,25,60)(12,75,26,61)(13,76,27,62)(14,77,28,63)(29,106,43,92)(30,107,44,93)(31,108,45,94)(32,109,46,95)(33,110,47,96)(34,111,48,97)(35,112,49,98)(36,99,50,85)(37,100,51,86)(38,101,52,87)(39,102,53,88)(40,103,54,89)(41,104,55,90)(42,105,56,91)(113,218,127,204)(114,219,128,205)(115,220,129,206)(116,221,130,207)(117,222,131,208)(118,223,132,209)(119,224,133,210)(120,211,134,197)(121,212,135,198)(122,213,136,199)(123,214,137,200)(124,215,138,201)(125,216,139,202)(126,217,140,203)(141,183,155,169)(142,184,156,170)(143,185,157,171)(144,186,158,172)(145,187,159,173)(146,188,160,174)(147,189,161,175)(148,190,162,176)(149,191,163,177)(150,192,164,178)(151,193,165,179)(152,194,166,180)(153,195,167,181)(154,196,168,182), (1,62)(2,63)(3,57)(4,58)(5,59)(6,60)(7,61)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(113,197)(114,198)(115,199)(116,200)(117,201)(118,202)(119,203)(120,204)(121,205)(122,206)(123,207)(124,208)(125,209)(126,210)(127,211)(128,212)(129,213)(130,214)(131,215)(132,216)(133,217)(134,218)(135,219)(136,220)(137,221)(138,222)(139,223)(140,224)(141,176)(142,177)(143,178)(144,179)(145,180)(146,181)(147,182)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)(161,196)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189), (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), (1,140,27,119)(2,139,28,118)(3,138,22,117)(4,137,23,116)(5,136,24,115)(6,135,25,114)(7,134,26,113)(8,131,15,124)(9,130,16,123)(10,129,17,122)(11,128,18,121)(12,127,19,120)(13,133,20,126)(14,132,21,125)(29,166,50,145)(30,165,51,144)(31,164,52,143)(32,163,53,142)(33,162,54,141)(34,168,55,147)(35,167,56,146)(36,159,43,152)(37,158,44,151)(38,157,45,150)(39,156,46,149)(40,155,47,148)(41,161,48,154)(42,160,49,153)(57,194,78,173)(58,193,79,172)(59,192,80,171)(60,191,81,170)(61,190,82,169)(62,196,83,175)(63,195,84,174)(64,187,71,180)(65,186,72,179)(66,185,73,178)(67,184,74,177)(68,183,75,176)(69,189,76,182)(70,188,77,181)(85,222,106,201)(86,221,107,200)(87,220,108,199)(88,219,109,198)(89,218,110,197)(90,224,111,203)(91,223,112,202)(92,215,99,208)(93,214,100,207)(94,213,101,206)(95,212,102,205)(96,211,103,204)(97,217,104,210)(98,216,105,209) );
G=PermutationGroup([[(1,41,13,34),(2,42,14,35),(3,36,8,29),(4,37,9,30),(5,38,10,31),(6,39,11,32),(7,40,12,33),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49),(57,85,64,92),(58,86,65,93),(59,87,66,94),(60,88,67,95),(61,89,68,96),(62,90,69,97),(63,91,70,98),(71,99,78,106),(72,100,79,107),(73,101,80,108),(74,102,81,109),(75,103,82,110),(76,104,83,111),(77,105,84,112),(113,141,120,148),(114,142,121,149),(115,143,122,150),(116,144,123,151),(117,145,124,152),(118,146,125,153),(119,147,126,154),(127,155,134,162),(128,156,135,163),(129,157,136,164),(130,158,137,165),(131,159,138,166),(132,160,139,167),(133,161,140,168),(169,204,176,197),(170,205,177,198),(171,206,178,199),(172,207,179,200),(173,208,180,201),(174,209,181,202),(175,210,182,203),(183,218,190,211),(184,219,191,212),(185,220,192,213),(186,221,193,214),(187,222,194,215),(188,223,195,216),(189,224,196,217)], [(1,83,20,69),(2,84,21,70),(3,78,15,64),(4,79,16,65),(5,80,17,66),(6,81,18,67),(7,82,19,68),(8,71,22,57),(9,72,23,58),(10,73,24,59),(11,74,25,60),(12,75,26,61),(13,76,27,62),(14,77,28,63),(29,106,43,92),(30,107,44,93),(31,108,45,94),(32,109,46,95),(33,110,47,96),(34,111,48,97),(35,112,49,98),(36,99,50,85),(37,100,51,86),(38,101,52,87),(39,102,53,88),(40,103,54,89),(41,104,55,90),(42,105,56,91),(113,218,127,204),(114,219,128,205),(115,220,129,206),(116,221,130,207),(117,222,131,208),(118,223,132,209),(119,224,133,210),(120,211,134,197),(121,212,135,198),(122,213,136,199),(123,214,137,200),(124,215,138,201),(125,216,139,202),(126,217,140,203),(141,183,155,169),(142,184,156,170),(143,185,157,171),(144,186,158,172),(145,187,159,173),(146,188,160,174),(147,189,161,175),(148,190,162,176),(149,191,163,177),(150,192,164,178),(151,193,165,179),(152,194,166,180),(153,195,167,181),(154,196,168,182)], [(1,62),(2,63),(3,57),(4,58),(5,59),(6,60),(7,61),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(113,197),(114,198),(115,199),(116,200),(117,201),(118,202),(119,203),(120,204),(121,205),(122,206),(123,207),(124,208),(125,209),(126,210),(127,211),(128,212),(129,213),(130,214),(131,215),(132,216),(133,217),(134,218),(135,219),(136,220),(137,221),(138,222),(139,223),(140,224),(141,176),(142,177),(143,178),(144,179),(145,180),(146,181),(147,182),(148,169),(149,170),(150,171),(151,172),(152,173),(153,174),(154,175),(155,190),(156,191),(157,192),(158,193),(159,194),(160,195),(161,196),(162,183),(163,184),(164,185),(165,186),(166,187),(167,188),(168,189)], [(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)], [(1,140,27,119),(2,139,28,118),(3,138,22,117),(4,137,23,116),(5,136,24,115),(6,135,25,114),(7,134,26,113),(8,131,15,124),(9,130,16,123),(10,129,17,122),(11,128,18,121),(12,127,19,120),(13,133,20,126),(14,132,21,125),(29,166,50,145),(30,165,51,144),(31,164,52,143),(32,163,53,142),(33,162,54,141),(34,168,55,147),(35,167,56,146),(36,159,43,152),(37,158,44,151),(38,157,45,150),(39,156,46,149),(40,155,47,148),(41,161,48,154),(42,160,49,153),(57,194,78,173),(58,193,79,172),(59,192,80,171),(60,191,81,170),(61,190,82,169),(62,196,83,175),(63,195,84,174),(64,187,71,180),(65,186,72,179),(66,185,73,178),(67,184,74,177),(68,183,75,176),(69,189,76,182),(70,188,77,181),(85,222,106,201),(86,221,107,200),(87,220,108,199),(88,219,109,198),(89,218,110,197),(90,224,111,203),(91,223,112,202),(92,215,99,208),(93,214,100,207),(94,213,101,206),(95,212,102,205),(96,211,103,204),(97,217,104,210),(98,216,105,209)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28L | 28M | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 8 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | SD16 | D14 | D14 | D14 | C7⋊D4 | C7⋊D4 | C8⋊C22 | D4⋊2D7 | D4.D7 | D4⋊D14 |
| kernel | C4⋊D4.D7 | C4.Dic14 | C28.55D4 | D4⋊Dic7 | C2×C4⋊Dic7 | C7×C4⋊D4 | C2×C28 | C22×C14 | C4⋊D4 | C28 | C2×C14 | C4⋊C4 | C22×C4 | C2×D4 | C2×C4 | C23 | C14 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 4 | 4 | 3 | 3 | 3 | 6 | 6 | 1 | 6 | 6 | 6 |
Matrix representation of C4⋊D4.D7 ►in GL6(𝔽113)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 112 | 0 | 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 |
| 27 | 17 | 0 | 0 | 0 | 0 |
| 17 | 86 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 2 | 0 | 0 |
| 0 | 0 | 105 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 86 | 96 | 0 | 0 | 0 | 0 |
| 96 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 2 | 0 | 0 |
| 0 | 0 | 106 | 44 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 88 | 1 |
| 0 | 0 | 0 | 0 | 53 | 34 |
| 7 | 96 | 0 | 0 | 0 | 0 |
| 96 | 106 | 0 | 0 | 0 | 0 |
| 0 | 0 | 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 80 |
| 0 | 0 | 0 | 0 | 0 | 112 |
G:=sub<GL(6,GF(113))| [0,112,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,17,0,0,0,0,17,86,0,0,0,0,0,0,69,105,0,0,0,0,2,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[86,96,0,0,0,0,96,27,0,0,0,0,0,0,69,106,0,0,0,0,2,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,88,53,0,0,0,0,1,34],[7,96,0,0,0,0,96,106,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,80,112] >;
C4⋊D4.D7 in GAP, Magma, Sage, TeX
C_4\rtimes D_4.D_7
% in TeX
G:=Group("C4:D4.D7"); // GroupNames label
G:=SmallGroup(448,568);
// by ID
G=gap.SmallGroup(448,568);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,336,254,219,1123,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^7=1,e^2=a^2*b^2,b*a*b^-1=c*a*c=e*a*e^-1=a^-1,a*d=d*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^-1*b,c*d=d*c,e*c*e^-1=a^-1*c,e*d*e^-1=d^-1>;
// generators/relations