direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×D4⋊3Q8, C14.1652+ 1+4, D4⋊3(C7×Q8), C4⋊Q8⋊15C14, (C7×D4)⋊10Q8, (C4×Q8)⋊15C14, (Q8×C28)⋊35C2, C4.18(Q8×C14), (D4×C28).27C2, (C4×D4).12C14, C22⋊Q8⋊17C14, C28.124(C2×Q8), C22.6(Q8×C14), C42.48(C2×C14), C42.C2⋊10C14, C28.325(C4○D4), C14.64(C22×Q8), (C2×C14).374C24, (C2×C28).680C23, (C4×C28).289C22, (D4×C14).335C22, C23.45(C22×C14), C22.48(C23×C14), (Q8×C14).184C22, C2.17(C7×2+ 1+4), (C22×C28).459C22, (C22×C14).268C23, (C7×C4⋊Q8)⋊36C2, (C2×C4⋊C4)⋊23C14, (C14×C4⋊C4)⋊50C2, C2.10(Q8×C2×C14), C4.37(C7×C4○D4), C4⋊C4.74(C2×C14), C2.27(C14×C4○D4), (C7×C22⋊Q8)⋊44C2, (C2×C14).55(C2×Q8), (C2×D4).81(C2×C14), C14.246(C2×C4○D4), (C2×Q8).64(C2×C14), (C7×C42.C2)⋊27C2, (C7×C4⋊C4).400C22, C22⋊C4.24(C2×C14), (C2×C4).36(C22×C14), (C22×C4).70(C2×C14), (C7×C22⋊C4).156C22, SmallGroup(448,1337)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4⋊3Q8
G = < a,b,c,d,e | a7=b4=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b2c, ede-1=d-1 >
Subgroups: 314 in 228 conjugacy classes, 166 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, D4⋊3Q8, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, Q8×C14, C14×C4⋊C4, D4×C28, D4×C28, Q8×C28, C7×C22⋊Q8, C7×C42.C2, C7×C4⋊Q8, C7×D4⋊3Q8
Quotients: C1, C2, C22, C7, Q8, C23, C14, C2×Q8, C4○D4, C24, C2×C14, C22×Q8, C2×C4○D4, 2+ 1+4, C7×Q8, C22×C14, D4⋊3Q8, Q8×C14, C7×C4○D4, C23×C14, Q8×C2×C14, C14×C4○D4, C7×2+ 1+4, C7×D4⋊3Q8
►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 94 35 91)(2 95 29 85)(3 96 30 86)(4 97 31 87)(5 98 32 88)(6 92 33 89)(7 93 34 90)(8 177 21 174)(9 178 15 175)(10 179 16 169)(11 180 17 170)(12 181 18 171)(13 182 19 172)(14 176 20 173)(22 167 223 188)(23 168 224 189)(24 162 218 183)(25 163 219 184)(26 164 220 185)(27 165 221 186)(28 166 222 187)(36 82 47 103)(37 83 48 104)(38 84 49 105)(39 78 43 99)(40 79 44 100)(41 80 45 101)(42 81 46 102)(50 106 71 127)(51 107 72 128)(52 108 73 129)(53 109 74 130)(54 110 75 131)(55 111 76 132)(56 112 77 133)(57 113 67 123)(58 114 68 124)(59 115 69 125)(60 116 70 126)(61 117 64 120)(62 118 65 121)(63 119 66 122)(134 190 155 211)(135 191 156 212)(136 192 157 213)(137 193 158 214)(138 194 159 215)(139 195 160 216)(140 196 161 217)(141 197 151 207)(142 198 152 208)(143 199 153 209)(144 200 154 210)(145 201 148 204)(146 202 149 205)(147 203 150 206)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 224)(9 218)(10 219)(11 220)(12 221)(13 222)(14 223)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 43)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 72)(58 73)(59 74)(60 75)(61 76)(62 77)(63 71)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 104)(93 105)(94 99)(95 100)(96 101)(97 102)(98 103)(106 119)(107 113)(108 114)(109 115)(110 116)(111 117)(112 118)(120 132)(121 133)(122 127)(123 128)(124 129)(125 130)(126 131)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 155)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 183)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)(204 216)(205 217)(206 211)(207 212)(208 213)(209 214)(210 215)
(1 50 39 66)(2 51 40 67)(3 52 41 68)(4 53 42 69)(5 54 36 70)(6 55 37 64)(7 56 38 65)(8 217 23 202)(9 211 24 203)(10 212 25 197)(11 213 26 198)(12 214 27 199)(13 215 28 200)(14 216 22 201)(15 190 218 206)(16 191 219 207)(17 192 220 208)(18 193 221 209)(19 194 222 210)(20 195 223 204)(21 196 224 205)(29 72 44 57)(30 73 45 58)(31 74 46 59)(32 75 47 60)(33 76 48 61)(34 77 49 62)(35 71 43 63)(78 122 94 106)(79 123 95 107)(80 124 96 108)(81 125 97 109)(82 126 98 110)(83 120 92 111)(84 121 93 112)(85 128 100 113)(86 129 101 114)(87 130 102 115)(88 131 103 116)(89 132 104 117)(90 133 105 118)(91 127 99 119)(134 162 150 178)(135 163 151 179)(136 164 152 180)(137 165 153 181)(138 166 154 182)(139 167 148 176)(140 168 149 177)(141 169 156 184)(142 170 157 185)(143 171 158 186)(144 172 159 187)(145 173 160 188)(146 174 161 189)(147 175 155 183)
(1 211 39 203)(2 212 40 197)(3 213 41 198)(4 214 42 199)(5 215 36 200)(6 216 37 201)(7 217 38 202)(8 65 23 56)(9 66 24 50)(10 67 25 51)(11 68 26 52)(12 69 27 53)(13 70 28 54)(14 64 22 55)(15 63 218 71)(16 57 219 72)(17 58 220 73)(18 59 221 74)(19 60 222 75)(20 61 223 76)(21 62 224 77)(29 191 44 207)(30 192 45 208)(31 193 46 209)(32 194 47 210)(33 195 48 204)(34 196 49 205)(35 190 43 206)(78 150 94 134)(79 151 95 135)(80 152 96 136)(81 153 97 137)(82 154 98 138)(83 148 92 139)(84 149 93 140)(85 156 100 141)(86 157 101 142)(87 158 102 143)(88 159 103 144)(89 160 104 145)(90 161 105 146)(91 155 99 147)(106 178 122 162)(107 179 123 163)(108 180 124 164)(109 181 125 165)(110 182 126 166)(111 176 120 167)(112 177 121 168)(113 184 128 169)(114 185 129 170)(115 186 130 171)(116 187 131 172)(117 188 132 173)(118 189 133 174)(119 183 127 175)
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), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,177,21,174)(9,178,15,175)(10,179,16,169)(11,180,17,170)(12,181,18,171)(13,182,19,172)(14,176,20,173)(22,167,223,188)(23,168,224,189)(24,162,218,183)(25,163,219,184)(26,164,220,185)(27,165,221,186)(28,166,222,187)(36,82,47,103)(37,83,48,104)(38,84,49,105)(39,78,43,99)(40,79,44,100)(41,80,45,101)(42,81,46,102)(50,106,71,127)(51,107,72,128)(52,108,73,129)(53,109,74,130)(54,110,75,131)(55,111,76,132)(56,112,77,133)(57,113,67,123)(58,114,68,124)(59,115,69,125)(60,116,70,126)(61,117,64,120)(62,118,65,121)(63,119,66,122)(134,190,155,211)(135,191,156,212)(136,192,157,213)(137,193,158,214)(138,194,159,215)(139,195,160,216)(140,196,161,217)(141,197,151,207)(142,198,152,208)(143,199,153,209)(144,200,154,210)(145,201,148,204)(146,202,149,205)(147,203,150,206), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,224)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,104)(93,105)(94,99)(95,100)(96,101)(97,102)(98,103)(106,119)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118)(120,132)(121,133)(122,127)(123,128)(124,129)(125,130)(126,131)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,216)(205,217)(206,211)(207,212)(208,213)(209,214)(210,215), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,217,23,202)(9,211,24,203)(10,212,25,197)(11,213,26,198)(12,214,27,199)(13,215,28,200)(14,216,22,201)(15,190,218,206)(16,191,219,207)(17,192,220,208)(18,193,221,209)(19,194,222,210)(20,195,223,204)(21,196,224,205)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,128,100,113)(86,129,101,114)(87,130,102,115)(88,131,103,116)(89,132,104,117)(90,133,105,118)(91,127,99,119)(134,162,150,178)(135,163,151,179)(136,164,152,180)(137,165,153,181)(138,166,154,182)(139,167,148,176)(140,168,149,177)(141,169,156,184)(142,170,157,185)(143,171,158,186)(144,172,159,187)(145,173,160,188)(146,174,161,189)(147,175,155,183), (1,211,39,203)(2,212,40,197)(3,213,41,198)(4,214,42,199)(5,215,36,200)(6,216,37,201)(7,217,38,202)(8,65,23,56)(9,66,24,50)(10,67,25,51)(11,68,26,52)(12,69,27,53)(13,70,28,54)(14,64,22,55)(15,63,218,71)(16,57,219,72)(17,58,220,73)(18,59,221,74)(19,60,222,75)(20,61,223,76)(21,62,224,77)(29,191,44,207)(30,192,45,208)(31,193,46,209)(32,194,47,210)(33,195,48,204)(34,196,49,205)(35,190,43,206)(78,150,94,134)(79,151,95,135)(80,152,96,136)(81,153,97,137)(82,154,98,138)(83,148,92,139)(84,149,93,140)(85,156,100,141)(86,157,101,142)(87,158,102,143)(88,159,103,144)(89,160,104,145)(90,161,105,146)(91,155,99,147)(106,178,122,162)(107,179,123,163)(108,180,124,164)(109,181,125,165)(110,182,126,166)(111,176,120,167)(112,177,121,168)(113,184,128,169)(114,185,129,170)(115,186,130,171)(116,187,131,172)(117,188,132,173)(118,189,133,174)(119,183,127,175)>;
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), (1,94,35,91)(2,95,29,85)(3,96,30,86)(4,97,31,87)(5,98,32,88)(6,92,33,89)(7,93,34,90)(8,177,21,174)(9,178,15,175)(10,179,16,169)(11,180,17,170)(12,181,18,171)(13,182,19,172)(14,176,20,173)(22,167,223,188)(23,168,224,189)(24,162,218,183)(25,163,219,184)(26,164,220,185)(27,165,221,186)(28,166,222,187)(36,82,47,103)(37,83,48,104)(38,84,49,105)(39,78,43,99)(40,79,44,100)(41,80,45,101)(42,81,46,102)(50,106,71,127)(51,107,72,128)(52,108,73,129)(53,109,74,130)(54,110,75,131)(55,111,76,132)(56,112,77,133)(57,113,67,123)(58,114,68,124)(59,115,69,125)(60,116,70,126)(61,117,64,120)(62,118,65,121)(63,119,66,122)(134,190,155,211)(135,191,156,212)(136,192,157,213)(137,193,158,214)(138,194,159,215)(139,195,160,216)(140,196,161,217)(141,197,151,207)(142,198,152,208)(143,199,153,209)(144,200,154,210)(145,201,148,204)(146,202,149,205)(147,203,150,206), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,224)(9,218)(10,219)(11,220)(12,221)(13,222)(14,223)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,43)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,72)(58,73)(59,74)(60,75)(61,76)(62,77)(63,71)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,104)(93,105)(94,99)(95,100)(96,101)(97,102)(98,103)(106,119)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118)(120,132)(121,133)(122,127)(123,128)(124,129)(125,130)(126,131)(134,150)(135,151)(136,152)(137,153)(138,154)(139,148)(140,149)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,178)(163,179)(164,180)(165,181)(166,182)(167,176)(168,177)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,216)(205,217)(206,211)(207,212)(208,213)(209,214)(210,215), (1,50,39,66)(2,51,40,67)(3,52,41,68)(4,53,42,69)(5,54,36,70)(6,55,37,64)(7,56,38,65)(8,217,23,202)(9,211,24,203)(10,212,25,197)(11,213,26,198)(12,214,27,199)(13,215,28,200)(14,216,22,201)(15,190,218,206)(16,191,219,207)(17,192,220,208)(18,193,221,209)(19,194,222,210)(20,195,223,204)(21,196,224,205)(29,72,44,57)(30,73,45,58)(31,74,46,59)(32,75,47,60)(33,76,48,61)(34,77,49,62)(35,71,43,63)(78,122,94,106)(79,123,95,107)(80,124,96,108)(81,125,97,109)(82,126,98,110)(83,120,92,111)(84,121,93,112)(85,128,100,113)(86,129,101,114)(87,130,102,115)(88,131,103,116)(89,132,104,117)(90,133,105,118)(91,127,99,119)(134,162,150,178)(135,163,151,179)(136,164,152,180)(137,165,153,181)(138,166,154,182)(139,167,148,176)(140,168,149,177)(141,169,156,184)(142,170,157,185)(143,171,158,186)(144,172,159,187)(145,173,160,188)(146,174,161,189)(147,175,155,183), (1,211,39,203)(2,212,40,197)(3,213,41,198)(4,214,42,199)(5,215,36,200)(6,216,37,201)(7,217,38,202)(8,65,23,56)(9,66,24,50)(10,67,25,51)(11,68,26,52)(12,69,27,53)(13,70,28,54)(14,64,22,55)(15,63,218,71)(16,57,219,72)(17,58,220,73)(18,59,221,74)(19,60,222,75)(20,61,223,76)(21,62,224,77)(29,191,44,207)(30,192,45,208)(31,193,46,209)(32,194,47,210)(33,195,48,204)(34,196,49,205)(35,190,43,206)(78,150,94,134)(79,151,95,135)(80,152,96,136)(81,153,97,137)(82,154,98,138)(83,148,92,139)(84,149,93,140)(85,156,100,141)(86,157,101,142)(87,158,102,143)(88,159,103,144)(89,160,104,145)(90,161,105,146)(91,155,99,147)(106,178,122,162)(107,179,123,163)(108,180,124,164)(109,181,125,165)(110,182,126,166)(111,176,120,167)(112,177,121,168)(113,184,128,169)(114,185,129,170)(115,186,130,171)(116,187,131,172)(117,188,132,173)(118,189,133,174)(119,183,127,175) );
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)], [(1,94,35,91),(2,95,29,85),(3,96,30,86),(4,97,31,87),(5,98,32,88),(6,92,33,89),(7,93,34,90),(8,177,21,174),(9,178,15,175),(10,179,16,169),(11,180,17,170),(12,181,18,171),(13,182,19,172),(14,176,20,173),(22,167,223,188),(23,168,224,189),(24,162,218,183),(25,163,219,184),(26,164,220,185),(27,165,221,186),(28,166,222,187),(36,82,47,103),(37,83,48,104),(38,84,49,105),(39,78,43,99),(40,79,44,100),(41,80,45,101),(42,81,46,102),(50,106,71,127),(51,107,72,128),(52,108,73,129),(53,109,74,130),(54,110,75,131),(55,111,76,132),(56,112,77,133),(57,113,67,123),(58,114,68,124),(59,115,69,125),(60,116,70,126),(61,117,64,120),(62,118,65,121),(63,119,66,122),(134,190,155,211),(135,191,156,212),(136,192,157,213),(137,193,158,214),(138,194,159,215),(139,195,160,216),(140,196,161,217),(141,197,151,207),(142,198,152,208),(143,199,153,209),(144,200,154,210),(145,201,148,204),(146,202,149,205),(147,203,150,206)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,224),(9,218),(10,219),(11,220),(12,221),(13,222),(14,223),(15,24),(16,25),(17,26),(18,27),(19,28),(20,22),(21,23),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,43),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,72),(58,73),(59,74),(60,75),(61,76),(62,77),(63,71),(78,91),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(92,104),(93,105),(94,99),(95,100),(96,101),(97,102),(98,103),(106,119),(107,113),(108,114),(109,115),(110,116),(111,117),(112,118),(120,132),(121,133),(122,127),(123,128),(124,129),(125,130),(126,131),(134,150),(135,151),(136,152),(137,153),(138,154),(139,148),(140,149),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,155),(162,178),(163,179),(164,180),(165,181),(166,182),(167,176),(168,177),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,183),(190,203),(191,197),(192,198),(193,199),(194,200),(195,201),(196,202),(204,216),(205,217),(206,211),(207,212),(208,213),(209,214),(210,215)], [(1,50,39,66),(2,51,40,67),(3,52,41,68),(4,53,42,69),(5,54,36,70),(6,55,37,64),(7,56,38,65),(8,217,23,202),(9,211,24,203),(10,212,25,197),(11,213,26,198),(12,214,27,199),(13,215,28,200),(14,216,22,201),(15,190,218,206),(16,191,219,207),(17,192,220,208),(18,193,221,209),(19,194,222,210),(20,195,223,204),(21,196,224,205),(29,72,44,57),(30,73,45,58),(31,74,46,59),(32,75,47,60),(33,76,48,61),(34,77,49,62),(35,71,43,63),(78,122,94,106),(79,123,95,107),(80,124,96,108),(81,125,97,109),(82,126,98,110),(83,120,92,111),(84,121,93,112),(85,128,100,113),(86,129,101,114),(87,130,102,115),(88,131,103,116),(89,132,104,117),(90,133,105,118),(91,127,99,119),(134,162,150,178),(135,163,151,179),(136,164,152,180),(137,165,153,181),(138,166,154,182),(139,167,148,176),(140,168,149,177),(141,169,156,184),(142,170,157,185),(143,171,158,186),(144,172,159,187),(145,173,160,188),(146,174,161,189),(147,175,155,183)], [(1,211,39,203),(2,212,40,197),(3,213,41,198),(4,214,42,199),(5,215,36,200),(6,216,37,201),(7,217,38,202),(8,65,23,56),(9,66,24,50),(10,67,25,51),(11,68,26,52),(12,69,27,53),(13,70,28,54),(14,64,22,55),(15,63,218,71),(16,57,219,72),(17,58,220,73),(18,59,221,74),(19,60,222,75),(20,61,223,76),(21,62,224,77),(29,191,44,207),(30,192,45,208),(31,193,46,209),(32,194,47,210),(33,195,48,204),(34,196,49,205),(35,190,43,206),(78,150,94,134),(79,151,95,135),(80,152,96,136),(81,153,97,137),(82,154,98,138),(83,148,92,139),(84,149,93,140),(85,156,100,141),(86,157,101,142),(87,158,102,143),(88,159,103,144),(89,160,104,145),(90,161,105,146),(91,155,99,147),(106,178,122,162),(107,179,123,163),(108,180,124,164),(109,181,125,165),(110,182,126,166),(111,176,120,167),(112,177,121,168),(113,184,128,169),(114,185,129,170),(115,186,130,171),(116,187,131,172),(117,188,132,173),(118,189,133,174),(119,183,127,175)]])
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4H | 4I | ··· | 4Q | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AP | 28A | ··· | 28AV | 28AW | ··· | 28CX |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | Q8 | C4○D4 | C7×Q8 | C7×C4○D4 | 2+ 1+4 | C7×2+ 1+4 |
| kernel | C7×D4⋊3Q8 | C14×C4⋊C4 | D4×C28 | Q8×C28 | C7×C22⋊Q8 | C7×C42.C2 | C7×C4⋊Q8 | D4⋊3Q8 | C2×C4⋊C4 | C4×D4 | C4×Q8 | C22⋊Q8 | C42.C2 | C4⋊Q8 | C7×D4 | C28 | D4 | C4 | C14 | C2 |
| # reps | 1 | 2 | 3 | 1 | 6 | 2 | 1 | 6 | 12 | 18 | 6 | 36 | 12 | 6 | 4 | 4 | 24 | 24 | 1 | 6 |
Matrix representation of C7×D4⋊3Q8 ►in GL4(𝔽29) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 0 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
| 0 | 17 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 15 | 21 |
| 0 | 0 | 21 | 14 |
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,7,0,0,0,0,7],[0,1,0,0,28,0,0,0,0,0,28,0,0,0,0,28],[1,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,0,28,0,0,1,0],[0,12,0,0,17,0,0,0,0,0,15,21,0,0,21,14] >;
C7×D4⋊3Q8 in GAP, Magma, Sage, TeX
C_7\times D_4\rtimes_3Q_8
% in TeX
G:=Group("C7xD4:3Q8"); // GroupNames label
G:=SmallGroup(448,1337);
// by ID
G=gap.SmallGroup(448,1337);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,2360,4790,1690,416]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
// generators/relations