direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×Q8⋊D4, Q8⋊2(C7×D4), (C7×Q8)⋊20D4, C22⋊C8⋊8C14, C4.22(D4×C14), Q8⋊C4⋊9C14, (C2×SD16)⋊7C14, (C2×C14)⋊13SD16, C28.383(C2×D4), (C2×C28).318D4, C4⋊D4.2C14, C2.5(C14×SD16), (C22×Q8)⋊2C14, C22⋊3(C7×SD16), C23.42(C7×D4), C14.95C22≀C2, (C14×SD16)⋊24C2, C14.85(C2×SD16), C22.78(D4×C14), (C2×C28).913C23, (C2×C56).297C22, (C22×C14).164D4, (D4×C14).180C22, (Q8×C14).258C22, C14.131(C8.C22), (C22×C28).420C22, (Q8×C2×C14)⋊14C2, C4⋊C4.1(C2×C14), (C2×C4).27(C7×D4), (C7×C22⋊C8)⋊25C2, (C2×C8).34(C2×C14), C2.9(C7×C22≀C2), (C2×D4).5(C2×C14), C2.6(C7×C8.C22), (C7×Q8⋊C4)⋊31C2, (C2×C14).634(C2×D4), (C7×C4⋊D4).12C2, (C2×Q8).43(C2×C14), (C7×C4⋊C4).223C22, (C22×C4).38(C2×C14), (C2×C4).88(C22×C14), SmallGroup(448,856)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×Q8⋊D4
G = < a,b,c,d,e | a7=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=ece=b-1c, ede=d-1 >
Subgroups: 290 in 158 conjugacy classes, 62 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C56, C2×C28, C2×C28, C7×D4, C7×Q8, C7×Q8, C22×C14, C22×C14, Q8⋊D4, C7×C22⋊C4, C7×C4⋊C4, C2×C56, C7×SD16, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, Q8×C14, C7×C22⋊C8, C7×Q8⋊C4, C7×C4⋊D4, C14×SD16, Q8×C2×C14, C7×Q8⋊D4
Quotients: C1, C2, C22, C7, D4, C23, C14, SD16, C2×D4, C2×C14, C22≀C2, C2×SD16, C8.C22, C7×D4, C22×C14, Q8⋊D4, C7×SD16, D4×C14, C7×C22≀C2, C14×SD16, C7×C8.C22, C7×Q8⋊D4
►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 79 77 95)(2 80 71 96)(3 81 72 97)(4 82 73 98)(5 83 74 92)(6 84 75 93)(7 78 76 94)(8 18 222 33)(9 19 223 34)(10 20 224 35)(11 21 218 29)(12 15 219 30)(13 16 220 31)(14 17 221 32)(22 46 37 214)(23 47 38 215)(24 48 39 216)(25 49 40 217)(26 43 41 211)(27 44 42 212)(28 45 36 213)(50 67 59 85)(51 68 60 86)(52 69 61 87)(53 70 62 88)(54 64 63 89)(55 65 57 90)(56 66 58 91)(99 128 137 153)(100 129 138 154)(101 130 139 148)(102 131 140 149)(103 132 134 150)(104 133 135 151)(105 127 136 152)(106 141 115 123)(107 142 116 124)(108 143 117 125)(109 144 118 126)(110 145 119 120)(111 146 113 121)(112 147 114 122)(155 184 193 209)(156 185 194 210)(157 186 195 204)(158 187 196 205)(159 188 190 206)(160 189 191 207)(161 183 192 208)(162 197 171 179)(163 198 172 180)(164 199 173 181)(165 200 174 182)(166 201 175 176)(167 202 169 177)(168 203 170 178)
(1 123 77 141)(2 124 71 142)(3 125 72 143)(4 126 73 144)(5 120 74 145)(6 121 75 146)(7 122 76 147)(8 204 222 186)(9 205 223 187)(10 206 224 188)(11 207 218 189)(12 208 219 183)(13 209 220 184)(14 210 221 185)(15 192 30 161)(16 193 31 155)(17 194 32 156)(18 195 33 157)(19 196 34 158)(20 190 35 159)(21 191 29 160)(22 167 37 169)(23 168 38 170)(24 162 39 171)(25 163 40 172)(26 164 41 173)(27 165 42 174)(28 166 36 175)(43 181 211 199)(44 182 212 200)(45 176 213 201)(46 177 214 202)(47 178 215 203)(48 179 216 197)(49 180 217 198)(50 104 59 135)(51 105 60 136)(52 99 61 137)(53 100 62 138)(54 101 63 139)(55 102 57 140)(56 103 58 134)(64 148 89 130)(65 149 90 131)(66 150 91 132)(67 151 85 133)(68 152 86 127)(69 153 87 128)(70 154 88 129)(78 114 94 112)(79 115 95 106)(80 116 96 107)(81 117 97 108)(82 118 98 109)(83 119 92 110)(84 113 93 111)
(1 197 85 207)(2 198 86 208)(3 199 87 209)(4 200 88 210)(5 201 89 204)(6 202 90 205)(7 203 91 206)(8 119 213 101)(9 113 214 102)(10 114 215 103)(11 115 216 104)(12 116 217 105)(13 117 211 99)(14 118 212 100)(15 142 25 152)(16 143 26 153)(17 144 27 154)(18 145 28 148)(19 146 22 149)(20 147 23 150)(21 141 24 151)(29 123 39 133)(30 124 40 127)(31 125 41 128)(32 126 42 129)(33 120 36 130)(34 121 37 131)(35 122 38 132)(43 137 220 108)(44 138 221 109)(45 139 222 110)(46 140 223 111)(47 134 224 112)(48 135 218 106)(49 136 219 107)(50 191 79 162)(51 192 80 163)(52 193 81 164)(53 194 82 165)(54 195 83 166)(55 196 84 167)(56 190 78 168)(57 158 93 169)(58 159 94 170)(59 160 95 171)(60 161 96 172)(61 155 97 173)(62 156 98 174)(63 157 92 175)(64 186 74 176)(65 187 75 177)(66 188 76 178)(67 189 77 179)(68 183 71 180)(69 184 72 181)(70 185 73 182)
(1 207)(2 208)(3 209)(4 210)(5 204)(6 205)(7 206)(8 119)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 142)(16 143)(17 144)(18 145)(19 146)(20 147)(21 141)(22 149)(23 150)(24 151)(25 152)(26 153)(27 154)(28 148)(29 123)(30 124)(31 125)(32 126)(33 120)(34 121)(35 122)(36 130)(37 131)(38 132)(39 133)(40 127)(41 128)(42 129)(43 137)(44 138)(45 139)(46 140)(47 134)(48 135)(49 136)(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)(92 157)(93 158)(94 159)(95 160)(96 161)(97 155)(98 156)(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)
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), (1,79,77,95)(2,80,71,96)(3,81,72,97)(4,82,73,98)(5,83,74,92)(6,84,75,93)(7,78,76,94)(8,18,222,33)(9,19,223,34)(10,20,224,35)(11,21,218,29)(12,15,219,30)(13,16,220,31)(14,17,221,32)(22,46,37,214)(23,47,38,215)(24,48,39,216)(25,49,40,217)(26,43,41,211)(27,44,42,212)(28,45,36,213)(50,67,59,85)(51,68,60,86)(52,69,61,87)(53,70,62,88)(54,64,63,89)(55,65,57,90)(56,66,58,91)(99,128,137,153)(100,129,138,154)(101,130,139,148)(102,131,140,149)(103,132,134,150)(104,133,135,151)(105,127,136,152)(106,141,115,123)(107,142,116,124)(108,143,117,125)(109,144,118,126)(110,145,119,120)(111,146,113,121)(112,147,114,122)(155,184,193,209)(156,185,194,210)(157,186,195,204)(158,187,196,205)(159,188,190,206)(160,189,191,207)(161,183,192,208)(162,197,171,179)(163,198,172,180)(164,199,173,181)(165,200,174,182)(166,201,175,176)(167,202,169,177)(168,203,170,178), (1,123,77,141)(2,124,71,142)(3,125,72,143)(4,126,73,144)(5,120,74,145)(6,121,75,146)(7,122,76,147)(8,204,222,186)(9,205,223,187)(10,206,224,188)(11,207,218,189)(12,208,219,183)(13,209,220,184)(14,210,221,185)(15,192,30,161)(16,193,31,155)(17,194,32,156)(18,195,33,157)(19,196,34,158)(20,190,35,159)(21,191,29,160)(22,167,37,169)(23,168,38,170)(24,162,39,171)(25,163,40,172)(26,164,41,173)(27,165,42,174)(28,166,36,175)(43,181,211,199)(44,182,212,200)(45,176,213,201)(46,177,214,202)(47,178,215,203)(48,179,216,197)(49,180,217,198)(50,104,59,135)(51,105,60,136)(52,99,61,137)(53,100,62,138)(54,101,63,139)(55,102,57,140)(56,103,58,134)(64,148,89,130)(65,149,90,131)(66,150,91,132)(67,151,85,133)(68,152,86,127)(69,153,87,128)(70,154,88,129)(78,114,94,112)(79,115,95,106)(80,116,96,107)(81,117,97,108)(82,118,98,109)(83,119,92,110)(84,113,93,111), (1,197,85,207)(2,198,86,208)(3,199,87,209)(4,200,88,210)(5,201,89,204)(6,202,90,205)(7,203,91,206)(8,119,213,101)(9,113,214,102)(10,114,215,103)(11,115,216,104)(12,116,217,105)(13,117,211,99)(14,118,212,100)(15,142,25,152)(16,143,26,153)(17,144,27,154)(18,145,28,148)(19,146,22,149)(20,147,23,150)(21,141,24,151)(29,123,39,133)(30,124,40,127)(31,125,41,128)(32,126,42,129)(33,120,36,130)(34,121,37,131)(35,122,38,132)(43,137,220,108)(44,138,221,109)(45,139,222,110)(46,140,223,111)(47,134,224,112)(48,135,218,106)(49,136,219,107)(50,191,79,162)(51,192,80,163)(52,193,81,164)(53,194,82,165)(54,195,83,166)(55,196,84,167)(56,190,78,168)(57,158,93,169)(58,159,94,170)(59,160,95,171)(60,161,96,172)(61,155,97,173)(62,156,98,174)(63,157,92,175)(64,186,74,176)(65,187,75,177)(66,188,76,178)(67,189,77,179)(68,183,71,180)(69,184,72,181)(70,185,73,182), (1,207)(2,208)(3,209)(4,210)(5,204)(6,205)(7,206)(8,119)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,141)(22,149)(23,150)(24,151)(25,152)(26,153)(27,154)(28,148)(29,123)(30,124)(31,125)(32,126)(33,120)(34,121)(35,122)(36,130)(37,131)(38,132)(39,133)(40,127)(41,128)(42,129)(43,137)(44,138)(45,139)(46,140)(47,134)(48,135)(49,136)(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)(92,157)(93,158)(94,159)(95,160)(96,161)(97,155)(98,156)(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)>;
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), (1,79,77,95)(2,80,71,96)(3,81,72,97)(4,82,73,98)(5,83,74,92)(6,84,75,93)(7,78,76,94)(8,18,222,33)(9,19,223,34)(10,20,224,35)(11,21,218,29)(12,15,219,30)(13,16,220,31)(14,17,221,32)(22,46,37,214)(23,47,38,215)(24,48,39,216)(25,49,40,217)(26,43,41,211)(27,44,42,212)(28,45,36,213)(50,67,59,85)(51,68,60,86)(52,69,61,87)(53,70,62,88)(54,64,63,89)(55,65,57,90)(56,66,58,91)(99,128,137,153)(100,129,138,154)(101,130,139,148)(102,131,140,149)(103,132,134,150)(104,133,135,151)(105,127,136,152)(106,141,115,123)(107,142,116,124)(108,143,117,125)(109,144,118,126)(110,145,119,120)(111,146,113,121)(112,147,114,122)(155,184,193,209)(156,185,194,210)(157,186,195,204)(158,187,196,205)(159,188,190,206)(160,189,191,207)(161,183,192,208)(162,197,171,179)(163,198,172,180)(164,199,173,181)(165,200,174,182)(166,201,175,176)(167,202,169,177)(168,203,170,178), (1,123,77,141)(2,124,71,142)(3,125,72,143)(4,126,73,144)(5,120,74,145)(6,121,75,146)(7,122,76,147)(8,204,222,186)(9,205,223,187)(10,206,224,188)(11,207,218,189)(12,208,219,183)(13,209,220,184)(14,210,221,185)(15,192,30,161)(16,193,31,155)(17,194,32,156)(18,195,33,157)(19,196,34,158)(20,190,35,159)(21,191,29,160)(22,167,37,169)(23,168,38,170)(24,162,39,171)(25,163,40,172)(26,164,41,173)(27,165,42,174)(28,166,36,175)(43,181,211,199)(44,182,212,200)(45,176,213,201)(46,177,214,202)(47,178,215,203)(48,179,216,197)(49,180,217,198)(50,104,59,135)(51,105,60,136)(52,99,61,137)(53,100,62,138)(54,101,63,139)(55,102,57,140)(56,103,58,134)(64,148,89,130)(65,149,90,131)(66,150,91,132)(67,151,85,133)(68,152,86,127)(69,153,87,128)(70,154,88,129)(78,114,94,112)(79,115,95,106)(80,116,96,107)(81,117,97,108)(82,118,98,109)(83,119,92,110)(84,113,93,111), (1,197,85,207)(2,198,86,208)(3,199,87,209)(4,200,88,210)(5,201,89,204)(6,202,90,205)(7,203,91,206)(8,119,213,101)(9,113,214,102)(10,114,215,103)(11,115,216,104)(12,116,217,105)(13,117,211,99)(14,118,212,100)(15,142,25,152)(16,143,26,153)(17,144,27,154)(18,145,28,148)(19,146,22,149)(20,147,23,150)(21,141,24,151)(29,123,39,133)(30,124,40,127)(31,125,41,128)(32,126,42,129)(33,120,36,130)(34,121,37,131)(35,122,38,132)(43,137,220,108)(44,138,221,109)(45,139,222,110)(46,140,223,111)(47,134,224,112)(48,135,218,106)(49,136,219,107)(50,191,79,162)(51,192,80,163)(52,193,81,164)(53,194,82,165)(54,195,83,166)(55,196,84,167)(56,190,78,168)(57,158,93,169)(58,159,94,170)(59,160,95,171)(60,161,96,172)(61,155,97,173)(62,156,98,174)(63,157,92,175)(64,186,74,176)(65,187,75,177)(66,188,76,178)(67,189,77,179)(68,183,71,180)(69,184,72,181)(70,185,73,182), (1,207)(2,208)(3,209)(4,210)(5,204)(6,205)(7,206)(8,119)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,142)(16,143)(17,144)(18,145)(19,146)(20,147)(21,141)(22,149)(23,150)(24,151)(25,152)(26,153)(27,154)(28,148)(29,123)(30,124)(31,125)(32,126)(33,120)(34,121)(35,122)(36,130)(37,131)(38,132)(39,133)(40,127)(41,128)(42,129)(43,137)(44,138)(45,139)(46,140)(47,134)(48,135)(49,136)(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)(92,157)(93,158)(94,159)(95,160)(96,161)(97,155)(98,156)(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) );
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)], [(1,79,77,95),(2,80,71,96),(3,81,72,97),(4,82,73,98),(5,83,74,92),(6,84,75,93),(7,78,76,94),(8,18,222,33),(9,19,223,34),(10,20,224,35),(11,21,218,29),(12,15,219,30),(13,16,220,31),(14,17,221,32),(22,46,37,214),(23,47,38,215),(24,48,39,216),(25,49,40,217),(26,43,41,211),(27,44,42,212),(28,45,36,213),(50,67,59,85),(51,68,60,86),(52,69,61,87),(53,70,62,88),(54,64,63,89),(55,65,57,90),(56,66,58,91),(99,128,137,153),(100,129,138,154),(101,130,139,148),(102,131,140,149),(103,132,134,150),(104,133,135,151),(105,127,136,152),(106,141,115,123),(107,142,116,124),(108,143,117,125),(109,144,118,126),(110,145,119,120),(111,146,113,121),(112,147,114,122),(155,184,193,209),(156,185,194,210),(157,186,195,204),(158,187,196,205),(159,188,190,206),(160,189,191,207),(161,183,192,208),(162,197,171,179),(163,198,172,180),(164,199,173,181),(165,200,174,182),(166,201,175,176),(167,202,169,177),(168,203,170,178)], [(1,123,77,141),(2,124,71,142),(3,125,72,143),(4,126,73,144),(5,120,74,145),(6,121,75,146),(7,122,76,147),(8,204,222,186),(9,205,223,187),(10,206,224,188),(11,207,218,189),(12,208,219,183),(13,209,220,184),(14,210,221,185),(15,192,30,161),(16,193,31,155),(17,194,32,156),(18,195,33,157),(19,196,34,158),(20,190,35,159),(21,191,29,160),(22,167,37,169),(23,168,38,170),(24,162,39,171),(25,163,40,172),(26,164,41,173),(27,165,42,174),(28,166,36,175),(43,181,211,199),(44,182,212,200),(45,176,213,201),(46,177,214,202),(47,178,215,203),(48,179,216,197),(49,180,217,198),(50,104,59,135),(51,105,60,136),(52,99,61,137),(53,100,62,138),(54,101,63,139),(55,102,57,140),(56,103,58,134),(64,148,89,130),(65,149,90,131),(66,150,91,132),(67,151,85,133),(68,152,86,127),(69,153,87,128),(70,154,88,129),(78,114,94,112),(79,115,95,106),(80,116,96,107),(81,117,97,108),(82,118,98,109),(83,119,92,110),(84,113,93,111)], [(1,197,85,207),(2,198,86,208),(3,199,87,209),(4,200,88,210),(5,201,89,204),(6,202,90,205),(7,203,91,206),(8,119,213,101),(9,113,214,102),(10,114,215,103),(11,115,216,104),(12,116,217,105),(13,117,211,99),(14,118,212,100),(15,142,25,152),(16,143,26,153),(17,144,27,154),(18,145,28,148),(19,146,22,149),(20,147,23,150),(21,141,24,151),(29,123,39,133),(30,124,40,127),(31,125,41,128),(32,126,42,129),(33,120,36,130),(34,121,37,131),(35,122,38,132),(43,137,220,108),(44,138,221,109),(45,139,222,110),(46,140,223,111),(47,134,224,112),(48,135,218,106),(49,136,219,107),(50,191,79,162),(51,192,80,163),(52,193,81,164),(53,194,82,165),(54,195,83,166),(55,196,84,167),(56,190,78,168),(57,158,93,169),(58,159,94,170),(59,160,95,171),(60,161,96,172),(61,155,97,173),(62,156,98,174),(63,157,92,175),(64,186,74,176),(65,187,75,177),(66,188,76,178),(67,189,77,179),(68,183,71,180),(69,184,72,181),(70,185,73,182)], [(1,207),(2,208),(3,209),(4,210),(5,204),(6,205),(7,206),(8,119),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,142),(16,143),(17,144),(18,145),(19,146),(20,147),(21,141),(22,149),(23,150),(24,151),(25,152),(26,153),(27,154),(28,148),(29,123),(30,124),(31,125),(32,126),(33,120),(34,121),(35,122),(36,130),(37,131),(38,132),(39,133),(40,127),(41,128),(42,129),(43,137),(44,138),(45,139),(46,140),(47,134),(48,135),(49,136),(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),(92,157),(93,158),(94,159),(95,160),(96,161),(97,155),(98,156),(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)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | ··· | 4G | 4H | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AJ | 28A | ··· | 28L | 28M | ··· | 28AP | 28AQ | ··· | 28AV | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | ··· | 4 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 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 | D4 | D4 | SD16 | C7×D4 | C7×D4 | C7×D4 | C7×SD16 | C8.C22 | C7×C8.C22 |
| kernel | C7×Q8⋊D4 | C7×C22⋊C8 | C7×Q8⋊C4 | C7×C4⋊D4 | C14×SD16 | Q8×C2×C14 | Q8⋊D4 | C22⋊C8 | Q8⋊C4 | C4⋊D4 | C2×SD16 | C22×Q8 | C2×C28 | C7×Q8 | C22×C14 | C2×C14 | C2×C4 | Q8 | C23 | C22 | C14 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 6 | 6 | 12 | 6 | 12 | 6 | 1 | 4 | 1 | 4 | 6 | 24 | 6 | 24 | 1 | 6 |
Matrix representation of C7×Q8⋊D4 ►in GL4(𝔽113) generated by
| 30 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 111 |
| 0 | 0 | 1 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 47 | 59 |
| 0 | 0 | 20 | 66 |
| 0 | 1 | 0 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 0 | 24 | 44 |
| 0 | 0 | 46 | 89 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 24 | 44 |
| 0 | 0 | 46 | 89 |
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,111,112],[112,0,0,0,0,112,0,0,0,0,47,20,0,0,59,66],[0,112,0,0,1,0,0,0,0,0,24,46,0,0,44,89],[0,1,0,0,1,0,0,0,0,0,24,46,0,0,44,89] >;
C7×Q8⋊D4 in GAP, Magma, Sage, TeX
C_7\times Q_8\rtimes D_4
% in TeX
G:=Group("C7xQ8:D4"); // GroupNames label
G:=SmallGroup(448,856);
// by ID
G=gap.SmallGroup(448,856);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1576,2438,9804,4911,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=d^4=e^2=1,c^2=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=e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations