direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×D4⋊2Q8, C28.45SD16, C4⋊Q8⋊4C14, C4⋊C8⋊10C14, D4⋊2(C7×Q8), (C7×D4)⋊9Q8, C4.Q8⋊7C14, (C4×D4).8C14, C4.14(Q8×C14), (D4×C28).23C2, (C2×C28).332D4, C28.120(C2×Q8), C4.10(C7×SD16), D4⋊C4.5C14, C42.22(C2×C14), C2.11(C14×SD16), C14.91(C2×SD16), C22.97(D4×C14), C28.313(C4○D4), (C2×C56).303C22, (C2×C28).932C23, (C4×C28).264C22, C14.95(C22⋊Q8), C14.139(C8⋊C22), (D4×C14).299C22, (C7×C4⋊C8)⋊29C2, (C7×C4⋊Q8)⋊25C2, (C7×C4.Q8)⋊22C2, C4.25(C7×C4○D4), C4⋊C4.13(C2×C14), (C2×C8).40(C2×C14), (C2×C4).133(C7×D4), C2.14(C7×C8⋊C22), C2.14(C7×C22⋊Q8), (C2×D4).59(C2×C14), (C2×C14).653(C2×D4), (C7×D4⋊C4).14C2, (C7×C4⋊C4).235C22, (C2×C4).107(C22×C14), SmallGroup(448,884)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4⋊2Q8
G = < a,b,c,d,e | a7=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b2c, ece-1=bc, ede-1=d-1 >
Subgroups: 202 in 108 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C28, C28, C28, C2×C14, C2×C14, D4⋊C4, C4⋊C8, C4.Q8, C4×D4, C4⋊Q8, C56, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4⋊2Q8, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C22×C28, D4×C14, Q8×C14, C7×D4⋊C4, C7×C4⋊C8, C7×C4.Q8, D4×C28, C7×C4⋊Q8, C7×D4⋊2Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, SD16, C2×D4, C2×Q8, C4○D4, C2×C14, C22⋊Q8, C2×SD16, C8⋊C22, C7×D4, C7×Q8, C22×C14, D4⋊2Q8, C7×SD16, D4×C14, Q8×C14, C7×C4○D4, C7×C22⋊Q8, C14×SD16, C7×C8⋊C22, C7×D4⋊2Q8
►On 224 points
Generators in S224
(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 67 78 76)(2 68 79 77)(3 69 80 71)(4 70 81 72)(5 64 82 73)(6 65 83 74)(7 66 84 75)(8 213 222 45)(9 214 223 46)(10 215 224 47)(11 216 218 48)(12 217 219 49)(13 211 220 43)(14 212 221 44)(15 42 24 33)(16 36 25 34)(17 37 26 35)(18 38 27 29)(19 39 28 30)(20 40 22 31)(21 41 23 32)(50 87 59 95)(51 88 60 96)(52 89 61 97)(53 90 62 98)(54 91 63 92)(55 85 57 93)(56 86 58 94)(99 117 153 108)(100 118 154 109)(101 119 148 110)(102 113 149 111)(103 114 150 112)(104 115 151 106)(105 116 152 107)(120 138 129 147)(121 139 130 141)(122 140 131 142)(123 134 132 143)(124 135 133 144)(125 136 127 145)(126 137 128 146)(155 164 209 173)(156 165 210 174)(157 166 204 175)(158 167 205 169)(159 168 206 170)(160 162 207 171)(161 163 208 172)(176 203 185 194)(177 197 186 195)(178 198 187 196)(179 199 188 190)(180 200 189 191)(181 201 183 192)(182 202 184 193)
(1 95)(2 96)(3 97)(4 98)(5 92)(6 93)(7 94)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 36)(15 211)(16 212)(17 213)(18 214)(19 215)(20 216)(21 217)(22 48)(23 49)(24 43)(25 44)(26 45)(27 46)(28 47)(29 223)(30 224)(31 218)(32 219)(33 220)(34 221)(35 222)(50 76)(51 77)(52 71)(53 72)(54 73)(55 74)(56 75)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(99 136)(100 137)(101 138)(102 139)(103 140)(104 134)(105 135)(106 132)(107 133)(108 127)(109 128)(110 129)(111 130)(112 131)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 120)(141 149)(142 150)(143 151)(144 152)(145 153)(146 154)(147 148)(155 181)(156 182)(157 176)(158 177)(159 178)(160 179)(161 180)(162 190)(163 191)(164 192)(165 193)(166 194)(167 195)(168 196)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(183 209)(184 210)(185 204)(186 205)(187 206)(188 207)(189 208)
(1 151 50 132)(2 152 51 133)(3 153 52 127)(4 154 53 128)(5 148 54 129)(6 149 55 130)(7 150 56 131)(8 185 35 204)(9 186 29 205)(10 187 30 206)(11 188 31 207)(12 189 32 208)(13 183 33 209)(14 184 34 210)(15 173 211 192)(16 174 212 193)(17 175 213 194)(18 169 214 195)(19 170 215 196)(20 171 216 190)(21 172 217 191)(22 162 48 199)(23 163 49 200)(24 164 43 201)(25 165 44 202)(26 166 45 203)(27 167 46 197)(28 168 47 198)(36 156 221 182)(37 157 222 176)(38 158 223 177)(39 159 224 178)(40 160 218 179)(41 161 219 180)(42 155 220 181)(57 121 83 102)(58 122 84 103)(59 123 78 104)(60 124 79 105)(61 125 80 99)(62 126 81 100)(63 120 82 101)(64 110 91 147)(65 111 85 141)(66 112 86 142)(67 106 87 143)(68 107 88 144)(69 108 89 145)(70 109 90 146)(71 117 97 136)(72 118 98 137)(73 119 92 138)(74 113 93 139)(75 114 94 140)(76 115 95 134)(77 116 96 135)
(1 162 50 199)(2 163 51 200)(3 164 52 201)(4 165 53 202)(5 166 54 203)(6 167 55 197)(7 168 56 198)(8 119 35 138)(9 113 29 139)(10 114 30 140)(11 115 31 134)(12 116 32 135)(13 117 33 136)(14 118 34 137)(15 125 211 99)(16 126 212 100)(17 120 213 101)(18 121 214 102)(19 122 215 103)(20 123 216 104)(21 124 217 105)(22 132 48 151)(23 133 49 152)(24 127 43 153)(25 128 44 154)(26 129 45 148)(27 130 46 149)(28 131 47 150)(36 146 221 109)(37 147 222 110)(38 141 223 111)(39 142 224 112)(40 143 218 106)(41 144 219 107)(42 145 220 108)(57 195 83 169)(58 196 84 170)(59 190 78 171)(60 191 79 172)(61 192 80 173)(62 193 81 174)(63 194 82 175)(64 157 91 176)(65 158 85 177)(66 159 86 178)(67 160 87 179)(68 161 88 180)(69 155 89 181)(70 156 90 182)(71 209 97 183)(72 210 98 184)(73 204 92 185)(74 205 93 186)(75 206 94 187)(76 207 95 188)(77 208 96 189)
G:=sub<Sym(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), (1,67,78,76)(2,68,79,77)(3,69,80,71)(4,70,81,72)(5,64,82,73)(6,65,83,74)(7,66,84,75)(8,213,222,45)(9,214,223,46)(10,215,224,47)(11,216,218,48)(12,217,219,49)(13,211,220,43)(14,212,221,44)(15,42,24,33)(16,36,25,34)(17,37,26,35)(18,38,27,29)(19,39,28,30)(20,40,22,31)(21,41,23,32)(50,87,59,95)(51,88,60,96)(52,89,61,97)(53,90,62,98)(54,91,63,92)(55,85,57,93)(56,86,58,94)(99,117,153,108)(100,118,154,109)(101,119,148,110)(102,113,149,111)(103,114,150,112)(104,115,151,106)(105,116,152,107)(120,138,129,147)(121,139,130,141)(122,140,131,142)(123,134,132,143)(124,135,133,144)(125,136,127,145)(126,137,128,146)(155,164,209,173)(156,165,210,174)(157,166,204,175)(158,167,205,169)(159,168,206,170)(160,162,207,171)(161,163,208,172)(176,203,185,194)(177,197,186,195)(178,198,187,196)(179,199,188,190)(180,200,189,191)(181,201,183,192)(182,202,184,193), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,36)(15,211)(16,212)(17,213)(18,214)(19,215)(20,216)(21,217)(22,48)(23,49)(24,43)(25,44)(26,45)(27,46)(28,47)(29,223)(30,224)(31,218)(32,219)(33,220)(34,221)(35,222)(50,76)(51,77)(52,71)(53,72)(54,73)(55,74)(56,75)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(99,136)(100,137)(101,138)(102,139)(103,140)(104,134)(105,135)(106,132)(107,133)(108,127)(109,128)(110,129)(111,130)(112,131)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(155,181)(156,182)(157,176)(158,177)(159,178)(160,179)(161,180)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(183,209)(184,210)(185,204)(186,205)(187,206)(188,207)(189,208), (1,151,50,132)(2,152,51,133)(3,153,52,127)(4,154,53,128)(5,148,54,129)(6,149,55,130)(7,150,56,131)(8,185,35,204)(9,186,29,205)(10,187,30,206)(11,188,31,207)(12,189,32,208)(13,183,33,209)(14,184,34,210)(15,173,211,192)(16,174,212,193)(17,175,213,194)(18,169,214,195)(19,170,215,196)(20,171,216,190)(21,172,217,191)(22,162,48,199)(23,163,49,200)(24,164,43,201)(25,165,44,202)(26,166,45,203)(27,167,46,197)(28,168,47,198)(36,156,221,182)(37,157,222,176)(38,158,223,177)(39,159,224,178)(40,160,218,179)(41,161,219,180)(42,155,220,181)(57,121,83,102)(58,122,84,103)(59,123,78,104)(60,124,79,105)(61,125,80,99)(62,126,81,100)(63,120,82,101)(64,110,91,147)(65,111,85,141)(66,112,86,142)(67,106,87,143)(68,107,88,144)(69,108,89,145)(70,109,90,146)(71,117,97,136)(72,118,98,137)(73,119,92,138)(74,113,93,139)(75,114,94,140)(76,115,95,134)(77,116,96,135), (1,162,50,199)(2,163,51,200)(3,164,52,201)(4,165,53,202)(5,166,54,203)(6,167,55,197)(7,168,56,198)(8,119,35,138)(9,113,29,139)(10,114,30,140)(11,115,31,134)(12,116,32,135)(13,117,33,136)(14,118,34,137)(15,125,211,99)(16,126,212,100)(17,120,213,101)(18,121,214,102)(19,122,215,103)(20,123,216,104)(21,124,217,105)(22,132,48,151)(23,133,49,152)(24,127,43,153)(25,128,44,154)(26,129,45,148)(27,130,46,149)(28,131,47,150)(36,146,221,109)(37,147,222,110)(38,141,223,111)(39,142,224,112)(40,143,218,106)(41,144,219,107)(42,145,220,108)(57,195,83,169)(58,196,84,170)(59,190,78,171)(60,191,79,172)(61,192,80,173)(62,193,81,174)(63,194,82,175)(64,157,91,176)(65,158,85,177)(66,159,86,178)(67,160,87,179)(68,161,88,180)(69,155,89,181)(70,156,90,182)(71,209,97,183)(72,210,98,184)(73,204,92,185)(74,205,93,186)(75,206,94,187)(76,207,95,188)(77,208,96,189)>;
G:=Group( (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), (1,67,78,76)(2,68,79,77)(3,69,80,71)(4,70,81,72)(5,64,82,73)(6,65,83,74)(7,66,84,75)(8,213,222,45)(9,214,223,46)(10,215,224,47)(11,216,218,48)(12,217,219,49)(13,211,220,43)(14,212,221,44)(15,42,24,33)(16,36,25,34)(17,37,26,35)(18,38,27,29)(19,39,28,30)(20,40,22,31)(21,41,23,32)(50,87,59,95)(51,88,60,96)(52,89,61,97)(53,90,62,98)(54,91,63,92)(55,85,57,93)(56,86,58,94)(99,117,153,108)(100,118,154,109)(101,119,148,110)(102,113,149,111)(103,114,150,112)(104,115,151,106)(105,116,152,107)(120,138,129,147)(121,139,130,141)(122,140,131,142)(123,134,132,143)(124,135,133,144)(125,136,127,145)(126,137,128,146)(155,164,209,173)(156,165,210,174)(157,166,204,175)(158,167,205,169)(159,168,206,170)(160,162,207,171)(161,163,208,172)(176,203,185,194)(177,197,186,195)(178,198,187,196)(179,199,188,190)(180,200,189,191)(181,201,183,192)(182,202,184,193), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,36)(15,211)(16,212)(17,213)(18,214)(19,215)(20,216)(21,217)(22,48)(23,49)(24,43)(25,44)(26,45)(27,46)(28,47)(29,223)(30,224)(31,218)(32,219)(33,220)(34,221)(35,222)(50,76)(51,77)(52,71)(53,72)(54,73)(55,74)(56,75)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(99,136)(100,137)(101,138)(102,139)(103,140)(104,134)(105,135)(106,132)(107,133)(108,127)(109,128)(110,129)(111,130)(112,131)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(155,181)(156,182)(157,176)(158,177)(159,178)(160,179)(161,180)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(183,209)(184,210)(185,204)(186,205)(187,206)(188,207)(189,208), (1,151,50,132)(2,152,51,133)(3,153,52,127)(4,154,53,128)(5,148,54,129)(6,149,55,130)(7,150,56,131)(8,185,35,204)(9,186,29,205)(10,187,30,206)(11,188,31,207)(12,189,32,208)(13,183,33,209)(14,184,34,210)(15,173,211,192)(16,174,212,193)(17,175,213,194)(18,169,214,195)(19,170,215,196)(20,171,216,190)(21,172,217,191)(22,162,48,199)(23,163,49,200)(24,164,43,201)(25,165,44,202)(26,166,45,203)(27,167,46,197)(28,168,47,198)(36,156,221,182)(37,157,222,176)(38,158,223,177)(39,159,224,178)(40,160,218,179)(41,161,219,180)(42,155,220,181)(57,121,83,102)(58,122,84,103)(59,123,78,104)(60,124,79,105)(61,125,80,99)(62,126,81,100)(63,120,82,101)(64,110,91,147)(65,111,85,141)(66,112,86,142)(67,106,87,143)(68,107,88,144)(69,108,89,145)(70,109,90,146)(71,117,97,136)(72,118,98,137)(73,119,92,138)(74,113,93,139)(75,114,94,140)(76,115,95,134)(77,116,96,135), (1,162,50,199)(2,163,51,200)(3,164,52,201)(4,165,53,202)(5,166,54,203)(6,167,55,197)(7,168,56,198)(8,119,35,138)(9,113,29,139)(10,114,30,140)(11,115,31,134)(12,116,32,135)(13,117,33,136)(14,118,34,137)(15,125,211,99)(16,126,212,100)(17,120,213,101)(18,121,214,102)(19,122,215,103)(20,123,216,104)(21,124,217,105)(22,132,48,151)(23,133,49,152)(24,127,43,153)(25,128,44,154)(26,129,45,148)(27,130,46,149)(28,131,47,150)(36,146,221,109)(37,147,222,110)(38,141,223,111)(39,142,224,112)(40,143,218,106)(41,144,219,107)(42,145,220,108)(57,195,83,169)(58,196,84,170)(59,190,78,171)(60,191,79,172)(61,192,80,173)(62,193,81,174)(63,194,82,175)(64,157,91,176)(65,158,85,177)(66,159,86,178)(67,160,87,179)(68,161,88,180)(69,155,89,181)(70,156,90,182)(71,209,97,183)(72,210,98,184)(73,204,92,185)(74,205,93,186)(75,206,94,187)(76,207,95,188)(77,208,96,189) );
G=PermutationGroup([[(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)], [(1,67,78,76),(2,68,79,77),(3,69,80,71),(4,70,81,72),(5,64,82,73),(6,65,83,74),(7,66,84,75),(8,213,222,45),(9,214,223,46),(10,215,224,47),(11,216,218,48),(12,217,219,49),(13,211,220,43),(14,212,221,44),(15,42,24,33),(16,36,25,34),(17,37,26,35),(18,38,27,29),(19,39,28,30),(20,40,22,31),(21,41,23,32),(50,87,59,95),(51,88,60,96),(52,89,61,97),(53,90,62,98),(54,91,63,92),(55,85,57,93),(56,86,58,94),(99,117,153,108),(100,118,154,109),(101,119,148,110),(102,113,149,111),(103,114,150,112),(104,115,151,106),(105,116,152,107),(120,138,129,147),(121,139,130,141),(122,140,131,142),(123,134,132,143),(124,135,133,144),(125,136,127,145),(126,137,128,146),(155,164,209,173),(156,165,210,174),(157,166,204,175),(158,167,205,169),(159,168,206,170),(160,162,207,171),(161,163,208,172),(176,203,185,194),(177,197,186,195),(178,198,187,196),(179,199,188,190),(180,200,189,191),(181,201,183,192),(182,202,184,193)], [(1,95),(2,96),(3,97),(4,98),(5,92),(6,93),(7,94),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,36),(15,211),(16,212),(17,213),(18,214),(19,215),(20,216),(21,217),(22,48),(23,49),(24,43),(25,44),(26,45),(27,46),(28,47),(29,223),(30,224),(31,218),(32,219),(33,220),(34,221),(35,222),(50,76),(51,77),(52,71),(53,72),(54,73),(55,74),(56,75),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(99,136),(100,137),(101,138),(102,139),(103,140),(104,134),(105,135),(106,132),(107,133),(108,127),(109,128),(110,129),(111,130),(112,131),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,120),(141,149),(142,150),(143,151),(144,152),(145,153),(146,154),(147,148),(155,181),(156,182),(157,176),(158,177),(159,178),(160,179),(161,180),(162,190),(163,191),(164,192),(165,193),(166,194),(167,195),(168,196),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(183,209),(184,210),(185,204),(186,205),(187,206),(188,207),(189,208)], [(1,151,50,132),(2,152,51,133),(3,153,52,127),(4,154,53,128),(5,148,54,129),(6,149,55,130),(7,150,56,131),(8,185,35,204),(9,186,29,205),(10,187,30,206),(11,188,31,207),(12,189,32,208),(13,183,33,209),(14,184,34,210),(15,173,211,192),(16,174,212,193),(17,175,213,194),(18,169,214,195),(19,170,215,196),(20,171,216,190),(21,172,217,191),(22,162,48,199),(23,163,49,200),(24,164,43,201),(25,165,44,202),(26,166,45,203),(27,167,46,197),(28,168,47,198),(36,156,221,182),(37,157,222,176),(38,158,223,177),(39,159,224,178),(40,160,218,179),(41,161,219,180),(42,155,220,181),(57,121,83,102),(58,122,84,103),(59,123,78,104),(60,124,79,105),(61,125,80,99),(62,126,81,100),(63,120,82,101),(64,110,91,147),(65,111,85,141),(66,112,86,142),(67,106,87,143),(68,107,88,144),(69,108,89,145),(70,109,90,146),(71,117,97,136),(72,118,98,137),(73,119,92,138),(74,113,93,139),(75,114,94,140),(76,115,95,134),(77,116,96,135)], [(1,162,50,199),(2,163,51,200),(3,164,52,201),(4,165,53,202),(5,166,54,203),(6,167,55,197),(7,168,56,198),(8,119,35,138),(9,113,29,139),(10,114,30,140),(11,115,31,134),(12,116,32,135),(13,117,33,136),(14,118,34,137),(15,125,211,99),(16,126,212,100),(17,120,213,101),(18,121,214,102),(19,122,215,103),(20,123,216,104),(21,124,217,105),(22,132,48,151),(23,133,49,152),(24,127,43,153),(25,128,44,154),(26,129,45,148),(27,130,46,149),(28,131,47,150),(36,146,221,109),(37,147,222,110),(38,141,223,111),(39,142,224,112),(40,143,218,106),(41,144,219,107),(42,145,220,108),(57,195,83,169),(58,196,84,170),(59,190,78,171),(60,191,79,172),(61,192,80,173),(62,193,81,174),(63,194,82,175),(64,157,91,176),(65,158,85,177),(66,159,86,178),(67,160,87,179),(68,161,88,180),(69,155,89,181),(70,156,90,182),(71,209,97,183),(72,210,98,184),(73,204,92,185),(74,205,93,186),(75,206,94,187),(76,207,95,188),(77,208,96,189)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 28A | ··· | 28X | 28Y | ··· | 28AP | 28AQ | ··· | 28BB | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | D4 | Q8 | SD16 | C4○D4 | C7×D4 | C7×Q8 | C7×SD16 | C7×C4○D4 | C8⋊C22 | C7×C8⋊C22 |
| kernel | C7×D4⋊2Q8 | C7×D4⋊C4 | C7×C4⋊C8 | C7×C4.Q8 | D4×C28 | C7×C4⋊Q8 | D4⋊2Q8 | D4⋊C4 | C4⋊C8 | C4.Q8 | C4×D4 | C4⋊Q8 | C2×C28 | C7×D4 | C28 | C28 | C2×C4 | D4 | C4 | C4 | C14 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 6 | 12 | 6 | 12 | 6 | 6 | 2 | 2 | 4 | 2 | 12 | 12 | 24 | 12 | 1 | 6 |
Matrix representation of C7×D4⋊2Q8 ►in GL4(𝔽113) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 112 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 15 | 109 |
| 0 | 0 | 0 | 98 |
| 100 | 13 | 0 | 0 |
| 13 | 13 | 0 | 0 |
| 0 | 0 | 97 | 69 |
| 0 | 0 | 106 | 16 |
G:=sub<GL(4,GF(113))| [28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[0,112,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,112,0,0,112,0,0,0,0,0,112,0,0,0,0,112],[0,1,0,0,112,0,0,0,0,0,15,0,0,0,109,98],[100,13,0,0,13,13,0,0,0,0,97,106,0,0,69,16] >;
C7×D4⋊2Q8 in GAP, Magma, Sage, TeX
C_7\times D_4\rtimes_2Q_8
% in TeX
G:=Group("C7xD4:2Q8"); // GroupNames label
G:=SmallGroup(448,884);
// by ID
G=gap.SmallGroup(448,884);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1960,813,400,2438,14117,3547,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e^-1=b*c,e*d*e^-1=d^-1>;
// generators/relations