direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×D4×Q8, C14.1192- 1+4, C4⋊2(Q8×C14), C4⋊Q8⋊14C14, C28⋊10(C2×Q8), (Q8×C28)⋊32C2, (C4×Q8)⋊12C14, C4.43(D4×C14), C22⋊2(Q8×C14), (D4×C28).25C2, (C4×D4).10C14, C28.404(C2×D4), C22⋊Q8⋊14C14, (C22×Q8)⋊7C14, C42.44(C2×C14), C14.63(C22×Q8), (C4×C28).285C22, (C2×C14).369C24, (C2×C28).676C23, C14.197(C22×D4), (D4×C14).334C22, C22.43(C23×C14), C23.43(C22×C14), (Q8×C14).275C22, C2.11(C7×2- 1+4), (C22×C28).455C22, (C22×C14).265C23, C2.9(Q8×C2×C14), (C7×C4⋊Q8)⋊35C2, (Q8×C2×C14)⋊19C2, (C2×C14)⋊8(C2×Q8), C2.21(D4×C2×C14), C4⋊C4.32(C2×C14), (C7×C22⋊Q8)⋊41C2, (C2×D4).80(C2×C14), (C2×Q8).62(C2×C14), C22⋊C4.20(C2×C14), (C7×C4⋊C4).397C22, (C2×C4).34(C22×C14), (C22×C4).67(C2×C14), (C7×C22⋊C4).153C22, SmallGroup(448,1332)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4×Q8
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, ce=ec, ede-1=d-1 >
Subgroups: 378 in 280 conjugacy classes, 182 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C7×Q8, C22×C14, D4×Q8, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, Q8×C14, Q8×C14, Q8×C14, D4×C28, Q8×C28, C7×C22⋊Q8, C7×C4⋊Q8, Q8×C2×C14, C7×D4×Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C24, C2×C14, C22×D4, C22×Q8, 2- 1+4, C7×D4, C7×Q8, C22×C14, D4×Q8, D4×C14, Q8×C14, C23×C14, D4×C2×C14, Q8×C2×C14, C7×2- 1+4, C7×D4×Q8
►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)
(1 102 45 91)(2 103 46 85)(3 104 47 86)(4 105 48 87)(5 99 49 88)(6 100 43 89)(7 101 44 90)(8 169 18 187)(9 170 19 188)(10 171 20 189)(11 172 21 183)(12 173 15 184)(13 174 16 185)(14 175 17 186)(22 167 223 180)(23 168 224 181)(24 162 218 182)(25 163 219 176)(26 164 220 177)(27 165 221 178)(28 166 222 179)(29 80 37 93)(30 81 38 94)(31 82 39 95)(32 83 40 96)(33 84 41 97)(34 78 42 98)(35 79 36 92)(50 106 70 126)(51 107 64 120)(52 108 65 121)(53 109 66 122)(54 110 67 123)(55 111 68 124)(56 112 69 125)(57 113 75 131)(58 114 76 132)(59 115 77 133)(60 116 71 127)(61 117 72 128)(62 118 73 129)(63 119 74 130)(134 190 154 210)(135 191 148 204)(136 192 149 205)(137 193 150 206)(138 194 151 207)(139 195 152 208)(140 196 153 209)(141 197 159 215)(142 198 160 216)(143 199 161 217)(144 200 155 211)(145 201 156 212)(146 202 157 213)(147 203 158 214)
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 25)(9 26)(10 27)(11 28)(12 22)(13 23)(14 24)(15 223)(16 224)(17 218)(18 219)(19 220)(20 221)(21 222)(36 46)(37 47)(38 48)(39 49)(40 43)(41 44)(42 45)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(78 91)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)(92 103)(93 104)(94 105)(95 99)(96 100)(97 101)(98 102)(106 119)(107 113)(108 114)(109 115)(110 116)(111 117)(112 118)(120 131)(121 132)(122 133)(123 127)(124 128)(125 129)(126 130)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)(204 215)(205 216)(206 217)(207 211)(208 212)(209 213)(210 214)
(1 50 34 74)(2 51 35 75)(3 52 29 76)(4 53 30 77)(5 54 31 71)(6 55 32 72)(7 56 33 73)(8 191 219 215)(9 192 220 216)(10 193 221 217)(11 194 222 211)(12 195 223 212)(13 196 224 213)(14 190 218 214)(15 208 22 201)(16 209 23 202)(17 210 24 203)(18 204 25 197)(19 205 26 198)(20 206 27 199)(21 207 28 200)(36 57 46 64)(37 58 47 65)(38 59 48 66)(39 60 49 67)(40 61 43 68)(41 62 44 69)(42 63 45 70)(78 130 102 106)(79 131 103 107)(80 132 104 108)(81 133 105 109)(82 127 99 110)(83 128 100 111)(84 129 101 112)(85 120 92 113)(86 121 93 114)(87 122 94 115)(88 123 95 116)(89 124 96 117)(90 125 97 118)(91 126 98 119)(134 162 158 186)(135 163 159 187)(136 164 160 188)(137 165 161 189)(138 166 155 183)(139 167 156 184)(140 168 157 185)(141 169 148 176)(142 170 149 177)(143 171 150 178)(144 172 151 179)(145 173 152 180)(146 174 153 181)(147 175 154 182)
(1 134 34 158)(2 135 35 159)(3 136 29 160)(4 137 30 161)(5 138 31 155)(6 139 32 156)(7 140 33 157)(8 131 219 107)(9 132 220 108)(10 133 221 109)(11 127 222 110)(12 128 223 111)(13 129 224 112)(14 130 218 106)(15 117 22 124)(16 118 23 125)(17 119 24 126)(18 113 25 120)(19 114 26 121)(20 115 27 122)(21 116 28 123)(36 141 46 148)(37 142 47 149)(38 143 48 150)(39 144 49 151)(40 145 43 152)(41 146 44 153)(42 147 45 154)(50 186 74 162)(51 187 75 163)(52 188 76 164)(53 189 77 165)(54 183 71 166)(55 184 72 167)(56 185 73 168)(57 176 64 169)(58 177 65 170)(59 178 66 171)(60 179 67 172)(61 180 68 173)(62 181 69 174)(63 182 70 175)(78 214 102 190)(79 215 103 191)(80 216 104 192)(81 217 105 193)(82 211 99 194)(83 212 100 195)(84 213 101 196)(85 204 92 197)(86 205 93 198)(87 206 94 199)(88 207 95 200)(89 208 96 201)(90 209 97 202)(91 210 98 203)
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), (1,102,45,91)(2,103,46,85)(3,104,47,86)(4,105,48,87)(5,99,49,88)(6,100,43,89)(7,101,44,90)(8,169,18,187)(9,170,19,188)(10,171,20,189)(11,172,21,183)(12,173,15,184)(13,174,16,185)(14,175,17,186)(22,167,223,180)(23,168,224,181)(24,162,218,182)(25,163,219,176)(26,164,220,177)(27,165,221,178)(28,166,222,179)(29,80,37,93)(30,81,38,94)(31,82,39,95)(32,83,40,96)(33,84,41,97)(34,78,42,98)(35,79,36,92)(50,106,70,126)(51,107,64,120)(52,108,65,121)(53,109,66,122)(54,110,67,123)(55,111,68,124)(56,112,69,125)(57,113,75,131)(58,114,76,132)(59,115,77,133)(60,116,71,127)(61,117,72,128)(62,118,73,129)(63,119,74,130)(134,190,154,210)(135,191,148,204)(136,192,149,205)(137,193,150,206)(138,194,151,207)(139,195,152,208)(140,196,153,209)(141,197,159,215)(142,198,160,216)(143,199,161,217)(144,200,155,211)(145,201,156,212)(146,202,157,213)(147,203,158,214), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(15,223)(16,224)(17,218)(18,219)(19,220)(20,221)(21,222)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,103)(93,104)(94,105)(95,99)(96,100)(97,101)(98,102)(106,119)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118)(120,131)(121,132)(122,133)(123,127)(124,128)(125,129)(126,130)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,215)(205,216)(206,217)(207,211)(208,212)(209,213)(210,214), (1,50,34,74)(2,51,35,75)(3,52,29,76)(4,53,30,77)(5,54,31,71)(6,55,32,72)(7,56,33,73)(8,191,219,215)(9,192,220,216)(10,193,221,217)(11,194,222,211)(12,195,223,212)(13,196,224,213)(14,190,218,214)(15,208,22,201)(16,209,23,202)(17,210,24,203)(18,204,25,197)(19,205,26,198)(20,206,27,199)(21,207,28,200)(36,57,46,64)(37,58,47,65)(38,59,48,66)(39,60,49,67)(40,61,43,68)(41,62,44,69)(42,63,45,70)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,120,92,113)(86,121,93,114)(87,122,94,115)(88,123,95,116)(89,124,96,117)(90,125,97,118)(91,126,98,119)(134,162,158,186)(135,163,159,187)(136,164,160,188)(137,165,161,189)(138,166,155,183)(139,167,156,184)(140,168,157,185)(141,169,148,176)(142,170,149,177)(143,171,150,178)(144,172,151,179)(145,173,152,180)(146,174,153,181)(147,175,154,182), (1,134,34,158)(2,135,35,159)(3,136,29,160)(4,137,30,161)(5,138,31,155)(6,139,32,156)(7,140,33,157)(8,131,219,107)(9,132,220,108)(10,133,221,109)(11,127,222,110)(12,128,223,111)(13,129,224,112)(14,130,218,106)(15,117,22,124)(16,118,23,125)(17,119,24,126)(18,113,25,120)(19,114,26,121)(20,115,27,122)(21,116,28,123)(36,141,46,148)(37,142,47,149)(38,143,48,150)(39,144,49,151)(40,145,43,152)(41,146,44,153)(42,147,45,154)(50,186,74,162)(51,187,75,163)(52,188,76,164)(53,189,77,165)(54,183,71,166)(55,184,72,167)(56,185,73,168)(57,176,64,169)(58,177,65,170)(59,178,66,171)(60,179,67,172)(61,180,68,173)(62,181,69,174)(63,182,70,175)(78,214,102,190)(79,215,103,191)(80,216,104,192)(81,217,105,193)(82,211,99,194)(83,212,100,195)(84,213,101,196)(85,204,92,197)(86,205,93,198)(87,206,94,199)(88,207,95,200)(89,208,96,201)(90,209,97,202)(91,210,98,203)>;
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), (1,102,45,91)(2,103,46,85)(3,104,47,86)(4,105,48,87)(5,99,49,88)(6,100,43,89)(7,101,44,90)(8,169,18,187)(9,170,19,188)(10,171,20,189)(11,172,21,183)(12,173,15,184)(13,174,16,185)(14,175,17,186)(22,167,223,180)(23,168,224,181)(24,162,218,182)(25,163,219,176)(26,164,220,177)(27,165,221,178)(28,166,222,179)(29,80,37,93)(30,81,38,94)(31,82,39,95)(32,83,40,96)(33,84,41,97)(34,78,42,98)(35,79,36,92)(50,106,70,126)(51,107,64,120)(52,108,65,121)(53,109,66,122)(54,110,67,123)(55,111,68,124)(56,112,69,125)(57,113,75,131)(58,114,76,132)(59,115,77,133)(60,116,71,127)(61,117,72,128)(62,118,73,129)(63,119,74,130)(134,190,154,210)(135,191,148,204)(136,192,149,205)(137,193,150,206)(138,194,151,207)(139,195,152,208)(140,196,153,209)(141,197,159,215)(142,198,160,216)(143,199,161,217)(144,200,155,211)(145,201,156,212)(146,202,157,213)(147,203,158,214), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(15,223)(16,224)(17,218)(18,219)(19,220)(20,221)(21,222)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(78,91)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90)(92,103)(93,104)(94,105)(95,99)(96,100)(97,101)(98,102)(106,119)(107,113)(108,114)(109,115)(110,116)(111,117)(112,118)(120,131)(121,132)(122,133)(123,127)(124,128)(125,129)(126,130)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,215)(205,216)(206,217)(207,211)(208,212)(209,213)(210,214), (1,50,34,74)(2,51,35,75)(3,52,29,76)(4,53,30,77)(5,54,31,71)(6,55,32,72)(7,56,33,73)(8,191,219,215)(9,192,220,216)(10,193,221,217)(11,194,222,211)(12,195,223,212)(13,196,224,213)(14,190,218,214)(15,208,22,201)(16,209,23,202)(17,210,24,203)(18,204,25,197)(19,205,26,198)(20,206,27,199)(21,207,28,200)(36,57,46,64)(37,58,47,65)(38,59,48,66)(39,60,49,67)(40,61,43,68)(41,62,44,69)(42,63,45,70)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,120,92,113)(86,121,93,114)(87,122,94,115)(88,123,95,116)(89,124,96,117)(90,125,97,118)(91,126,98,119)(134,162,158,186)(135,163,159,187)(136,164,160,188)(137,165,161,189)(138,166,155,183)(139,167,156,184)(140,168,157,185)(141,169,148,176)(142,170,149,177)(143,171,150,178)(144,172,151,179)(145,173,152,180)(146,174,153,181)(147,175,154,182), (1,134,34,158)(2,135,35,159)(3,136,29,160)(4,137,30,161)(5,138,31,155)(6,139,32,156)(7,140,33,157)(8,131,219,107)(9,132,220,108)(10,133,221,109)(11,127,222,110)(12,128,223,111)(13,129,224,112)(14,130,218,106)(15,117,22,124)(16,118,23,125)(17,119,24,126)(18,113,25,120)(19,114,26,121)(20,115,27,122)(21,116,28,123)(36,141,46,148)(37,142,47,149)(38,143,48,150)(39,144,49,151)(40,145,43,152)(41,146,44,153)(42,147,45,154)(50,186,74,162)(51,187,75,163)(52,188,76,164)(53,189,77,165)(54,183,71,166)(55,184,72,167)(56,185,73,168)(57,176,64,169)(58,177,65,170)(59,178,66,171)(60,179,67,172)(61,180,68,173)(62,181,69,174)(63,182,70,175)(78,214,102,190)(79,215,103,191)(80,216,104,192)(81,217,105,193)(82,211,99,194)(83,212,100,195)(84,213,101,196)(85,204,92,197)(86,205,93,198)(87,206,94,199)(88,207,95,200)(89,208,96,201)(90,209,97,202)(91,210,98,203) );
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)], [(1,102,45,91),(2,103,46,85),(3,104,47,86),(4,105,48,87),(5,99,49,88),(6,100,43,89),(7,101,44,90),(8,169,18,187),(9,170,19,188),(10,171,20,189),(11,172,21,183),(12,173,15,184),(13,174,16,185),(14,175,17,186),(22,167,223,180),(23,168,224,181),(24,162,218,182),(25,163,219,176),(26,164,220,177),(27,165,221,178),(28,166,222,179),(29,80,37,93),(30,81,38,94),(31,82,39,95),(32,83,40,96),(33,84,41,97),(34,78,42,98),(35,79,36,92),(50,106,70,126),(51,107,64,120),(52,108,65,121),(53,109,66,122),(54,110,67,123),(55,111,68,124),(56,112,69,125),(57,113,75,131),(58,114,76,132),(59,115,77,133),(60,116,71,127),(61,117,72,128),(62,118,73,129),(63,119,74,130),(134,190,154,210),(135,191,148,204),(136,192,149,205),(137,193,150,206),(138,194,151,207),(139,195,152,208),(140,196,153,209),(141,197,159,215),(142,198,160,216),(143,199,161,217),(144,200,155,211),(145,201,156,212),(146,202,157,213),(147,203,158,214)], [(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,25),(9,26),(10,27),(11,28),(12,22),(13,23),(14,24),(15,223),(16,224),(17,218),(18,219),(19,220),(20,221),(21,222),(36,46),(37,47),(38,48),(39,49),(40,43),(41,44),(42,45),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(78,91),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90),(92,103),(93,104),(94,105),(95,99),(96,100),(97,101),(98,102),(106,119),(107,113),(108,114),(109,115),(110,116),(111,117),(112,118),(120,131),(121,132),(122,133),(123,127),(124,128),(125,129),(126,130),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(162,186),(163,187),(164,188),(165,189),(166,183),(167,184),(168,185),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(190,203),(191,197),(192,198),(193,199),(194,200),(195,201),(196,202),(204,215),(205,216),(206,217),(207,211),(208,212),(209,213),(210,214)], [(1,50,34,74),(2,51,35,75),(3,52,29,76),(4,53,30,77),(5,54,31,71),(6,55,32,72),(7,56,33,73),(8,191,219,215),(9,192,220,216),(10,193,221,217),(11,194,222,211),(12,195,223,212),(13,196,224,213),(14,190,218,214),(15,208,22,201),(16,209,23,202),(17,210,24,203),(18,204,25,197),(19,205,26,198),(20,206,27,199),(21,207,28,200),(36,57,46,64),(37,58,47,65),(38,59,48,66),(39,60,49,67),(40,61,43,68),(41,62,44,69),(42,63,45,70),(78,130,102,106),(79,131,103,107),(80,132,104,108),(81,133,105,109),(82,127,99,110),(83,128,100,111),(84,129,101,112),(85,120,92,113),(86,121,93,114),(87,122,94,115),(88,123,95,116),(89,124,96,117),(90,125,97,118),(91,126,98,119),(134,162,158,186),(135,163,159,187),(136,164,160,188),(137,165,161,189),(138,166,155,183),(139,167,156,184),(140,168,157,185),(141,169,148,176),(142,170,149,177),(143,171,150,178),(144,172,151,179),(145,173,152,180),(146,174,153,181),(147,175,154,182)], [(1,134,34,158),(2,135,35,159),(3,136,29,160),(4,137,30,161),(5,138,31,155),(6,139,32,156),(7,140,33,157),(8,131,219,107),(9,132,220,108),(10,133,221,109),(11,127,222,110),(12,128,223,111),(13,129,224,112),(14,130,218,106),(15,117,22,124),(16,118,23,125),(17,119,24,126),(18,113,25,120),(19,114,26,121),(20,115,27,122),(21,116,28,123),(36,141,46,148),(37,142,47,149),(38,143,48,150),(39,144,49,151),(40,145,43,152),(41,146,44,153),(42,147,45,154),(50,186,74,162),(51,187,75,163),(52,188,76,164),(53,189,77,165),(54,183,71,166),(55,184,72,167),(56,185,73,168),(57,176,64,169),(58,177,65,170),(59,178,66,171),(60,179,67,172),(61,180,68,173),(62,181,69,174),(63,182,70,175),(78,214,102,190),(79,215,103,191),(80,216,104,192),(81,217,105,193),(82,211,99,194),(83,212,100,195),(84,213,101,196),(85,204,92,197),(86,205,93,198),(87,206,94,199),(88,207,95,200),(89,208,96,201),(90,209,97,202),(91,210,98,203)]])
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 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | Q8 | D4 | C7×Q8 | C7×D4 | 2- 1+4 | C7×2- 1+4 |
| kernel | C7×D4×Q8 | D4×C28 | Q8×C28 | C7×C22⋊Q8 | C7×C4⋊Q8 | Q8×C2×C14 | D4×Q8 | C4×D4 | C4×Q8 | C22⋊Q8 | C4⋊Q8 | C22×Q8 | C7×D4 | C7×Q8 | D4 | Q8 | C14 | C2 |
| # reps | 1 | 3 | 1 | 6 | 3 | 2 | 6 | 18 | 6 | 36 | 18 | 12 | 4 | 4 | 24 | 24 | 1 | 6 |
Matrix representation of C7×D4×Q8 ►in GL4(𝔽29) generated by
| 25 | 0 | 0 | 0 |
| 0 | 25 | 0 | 0 |
| 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 28 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 28 | 28 |
| 28 | 27 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
G:=sub<GL(4,GF(29))| [25,0,0,0,0,25,0,0,0,0,7,0,0,0,0,7],[1,0,0,0,0,1,0,0,0,0,1,28,0,0,2,28],[28,0,0,0,0,28,0,0,0,0,1,28,0,0,0,28],[28,1,0,0,27,1,0,0,0,0,1,0,0,0,0,1],[17,12,0,0,0,12,0,0,0,0,28,0,0,0,0,28] >;
C7×D4×Q8 in GAP, Magma, Sage, TeX
C_7\times D_4\times Q_8
% in TeX
G:=Group("C7xD4xQ8"); // GroupNames label
G:=SmallGroup(448,1332);
// by ID
G=gap.SmallGroup(448,1332);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,792,4790,1690,416]);
// 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,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations