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## G = C7×D4.Q8order 448 = 26·7

### Direct product of C7 and D4.Q8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×D4.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C7×C4⋊C4 — C7×C42.C2 — C7×D4.Q8
 Lower central C1 — C2 — C2×C4 — C7×D4.Q8
 Upper central C1 — C2×C14 — C4×C28 — C7×D4.Q8

Generators and relations for C7×D4.Q8
G = < a,b,c,d,e | a7=b4=c2=d4=1, e2=b2d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >

Subgroups: 186 in 102 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C28, C28, C2×C14, C2×C14, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C56, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, D4.Q8, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C22×C28, D4×C14, C7×D4⋊C4, C7×C4⋊C8, C7×C4.Q8, C7×C2.D8, D4×C28, C7×C42.C2, C7×D4.Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C4○D4, C2×C14, C22⋊Q8, C4○D8, C8⋊C22, C7×D4, C7×Q8, C22×C14, D4.Q8, D4×C14, Q8×C14, C7×C4○D4, C7×C22⋊Q8, C7×C4○D8, C7×C8⋊C22, C7×D4.Q8

Smallest permutation representation of 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 83 77 85)(2 84 71 86)(3 78 72 87)(4 79 73 88)(5 80 74 89)(6 81 75 90)(7 82 76 91)(8 21 39 29)(9 15 40 30)(10 16 41 31)(11 17 42 32)(12 18 36 33)(13 19 37 34)(14 20 38 35)(22 211 220 43)(23 212 221 44)(24 213 222 45)(25 214 223 46)(26 215 224 47)(27 216 218 48)(28 217 219 49)(50 95 59 70)(51 96 60 64)(52 97 61 65)(53 98 62 66)(54 92 63 67)(55 93 57 68)(56 94 58 69)(99 117 121 108)(100 118 122 109)(101 119 123 110)(102 113 124 111)(103 114 125 112)(104 115 126 106)(105 116 120 107)(127 142 152 140)(128 143 153 134)(129 144 154 135)(130 145 148 136)(131 146 149 137)(132 147 150 138)(133 141 151 139)(155 164 177 173)(156 165 178 174)(157 166 179 175)(158 167 180 169)(159 168 181 170)(160 162 182 171)(161 163 176 172)(183 196 208 198)(184 190 209 199)(185 191 210 200)(186 192 204 201)(187 193 205 202)(188 194 206 203)(189 195 207 197)
(1 95)(2 96)(3 97)(4 98)(5 92)(6 93)(7 94)(8 218)(9 219)(10 220)(11 221)(12 222)(13 223)(14 224)(15 217)(16 211)(17 212)(18 213)(19 214)(20 215)(21 216)(22 41)(23 42)(24 36)(25 37)(26 38)(27 39)(28 40)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 83)(51 84)(52 78)(53 79)(54 80)(55 81)(56 82)(57 90)(58 91)(59 85)(60 86)(61 87)(62 88)(63 89)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(99 153)(100 154)(101 148)(102 149)(103 150)(104 151)(105 152)(106 139)(107 140)(108 134)(109 135)(110 136)(111 137)(112 138)(113 146)(114 147)(115 141)(116 142)(117 143)(118 144)(119 145)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)(155 199)(156 200)(157 201)(158 202)(159 203)(160 197)(161 198)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(169 187)(170 188)(171 189)(172 183)(173 184)(174 185)(175 186)(176 196)(177 190)(178 191)(179 192)(180 193)(181 194)(182 195)
(1 106 50 151)(2 107 51 152)(3 108 52 153)(4 109 53 154)(5 110 54 148)(6 111 55 149)(7 112 56 150)(8 171 218 189)(9 172 219 183)(10 173 220 184)(11 174 221 185)(12 175 222 186)(13 169 223 187)(14 170 224 188)(15 161 49 196)(16 155 43 190)(17 156 44 191)(18 157 45 192)(19 158 46 193)(20 159 47 194)(21 160 48 195)(22 209 41 164)(23 210 42 165)(24 204 36 166)(25 205 37 167)(26 206 38 168)(27 207 39 162)(28 208 40 163)(29 182 216 197)(30 176 217 198)(31 177 211 199)(32 178 212 200)(33 179 213 201)(34 180 214 202)(35 181 215 203)(57 131 75 113)(58 132 76 114)(59 133 77 115)(60 127 71 116)(61 128 72 117)(62 129 73 118)(63 130 74 119)(64 142 86 120)(65 143 87 121)(66 144 88 122)(67 145 89 123)(68 146 90 124)(69 147 91 125)(70 141 85 126)(78 99 97 134)(79 100 98 135)(80 101 92 136)(81 102 93 137)(82 103 94 138)(83 104 95 139)(84 105 96 140)
(1 171 59 207)(2 172 60 208)(3 173 61 209)(4 174 62 210)(5 175 63 204)(6 169 57 205)(7 170 58 206)(8 151 27 115)(9 152 28 116)(10 153 22 117)(11 154 23 118)(12 148 24 119)(13 149 25 113)(14 150 26 114)(15 142 217 105)(16 143 211 99)(17 144 212 100)(18 145 213 101)(19 146 214 102)(20 147 215 103)(21 141 216 104)(29 139 48 126)(30 140 49 120)(31 134 43 121)(32 135 44 122)(33 136 45 123)(34 137 46 124)(35 138 47 125)(36 130 222 110)(37 131 223 111)(38 132 224 112)(39 133 218 106)(40 127 219 107)(41 128 220 108)(42 129 221 109)(50 189 77 162)(51 183 71 163)(52 184 72 164)(53 185 73 165)(54 186 74 166)(55 187 75 167)(56 188 76 168)(64 196 84 176)(65 190 78 177)(66 191 79 178)(67 192 80 179)(68 193 81 180)(69 194 82 181)(70 195 83 182)(85 160 95 197)(86 161 96 198)(87 155 97 199)(88 156 98 200)(89 157 92 201)(90 158 93 202)(91 159 94 203)

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), (1,83,77,85)(2,84,71,86)(3,78,72,87)(4,79,73,88)(5,80,74,89)(6,81,75,90)(7,82,76,91)(8,21,39,29)(9,15,40,30)(10,16,41,31)(11,17,42,32)(12,18,36,33)(13,19,37,34)(14,20,38,35)(22,211,220,43)(23,212,221,44)(24,213,222,45)(25,214,223,46)(26,215,224,47)(27,216,218,48)(28,217,219,49)(50,95,59,70)(51,96,60,64)(52,97,61,65)(53,98,62,66)(54,92,63,67)(55,93,57,68)(56,94,58,69)(99,117,121,108)(100,118,122,109)(101,119,123,110)(102,113,124,111)(103,114,125,112)(104,115,126,106)(105,116,120,107)(127,142,152,140)(128,143,153,134)(129,144,154,135)(130,145,148,136)(131,146,149,137)(132,147,150,138)(133,141,151,139)(155,164,177,173)(156,165,178,174)(157,166,179,175)(158,167,180,169)(159,168,181,170)(160,162,182,171)(161,163,176,172)(183,196,208,198)(184,190,209,199)(185,191,210,200)(186,192,204,201)(187,193,205,202)(188,194,206,203)(189,195,207,197), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,218)(9,219)(10,220)(11,221)(12,222)(13,223)(14,224)(15,217)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(22,41)(23,42)(24,36)(25,37)(26,38)(27,39)(28,40)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,83)(51,84)(52,78)(53,79)(54,80)(55,81)(56,82)(57,90)(58,91)(59,85)(60,86)(61,87)(62,88)(63,89)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(99,153)(100,154)(101,148)(102,149)(103,150)(104,151)(105,152)(106,139)(107,140)(108,134)(109,135)(110,136)(111,137)(112,138)(113,146)(114,147)(115,141)(116,142)(117,143)(118,144)(119,145)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(155,199)(156,200)(157,201)(158,202)(159,203)(160,197)(161,198)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(176,196)(177,190)(178,191)(179,192)(180,193)(181,194)(182,195), (1,106,50,151)(2,107,51,152)(3,108,52,153)(4,109,53,154)(5,110,54,148)(6,111,55,149)(7,112,56,150)(8,171,218,189)(9,172,219,183)(10,173,220,184)(11,174,221,185)(12,175,222,186)(13,169,223,187)(14,170,224,188)(15,161,49,196)(16,155,43,190)(17,156,44,191)(18,157,45,192)(19,158,46,193)(20,159,47,194)(21,160,48,195)(22,209,41,164)(23,210,42,165)(24,204,36,166)(25,205,37,167)(26,206,38,168)(27,207,39,162)(28,208,40,163)(29,182,216,197)(30,176,217,198)(31,177,211,199)(32,178,212,200)(33,179,213,201)(34,180,214,202)(35,181,215,203)(57,131,75,113)(58,132,76,114)(59,133,77,115)(60,127,71,116)(61,128,72,117)(62,129,73,118)(63,130,74,119)(64,142,86,120)(65,143,87,121)(66,144,88,122)(67,145,89,123)(68,146,90,124)(69,147,91,125)(70,141,85,126)(78,99,97,134)(79,100,98,135)(80,101,92,136)(81,102,93,137)(82,103,94,138)(83,104,95,139)(84,105,96,140), (1,171,59,207)(2,172,60,208)(3,173,61,209)(4,174,62,210)(5,175,63,204)(6,169,57,205)(7,170,58,206)(8,151,27,115)(9,152,28,116)(10,153,22,117)(11,154,23,118)(12,148,24,119)(13,149,25,113)(14,150,26,114)(15,142,217,105)(16,143,211,99)(17,144,212,100)(18,145,213,101)(19,146,214,102)(20,147,215,103)(21,141,216,104)(29,139,48,126)(30,140,49,120)(31,134,43,121)(32,135,44,122)(33,136,45,123)(34,137,46,124)(35,138,47,125)(36,130,222,110)(37,131,223,111)(38,132,224,112)(39,133,218,106)(40,127,219,107)(41,128,220,108)(42,129,221,109)(50,189,77,162)(51,183,71,163)(52,184,72,164)(53,185,73,165)(54,186,74,166)(55,187,75,167)(56,188,76,168)(64,196,84,176)(65,190,78,177)(66,191,79,178)(67,192,80,179)(68,193,81,180)(69,194,82,181)(70,195,83,182)(85,160,95,197)(86,161,96,198)(87,155,97,199)(88,156,98,200)(89,157,92,201)(90,158,93,202)(91,159,94,203)>;

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), (1,83,77,85)(2,84,71,86)(3,78,72,87)(4,79,73,88)(5,80,74,89)(6,81,75,90)(7,82,76,91)(8,21,39,29)(9,15,40,30)(10,16,41,31)(11,17,42,32)(12,18,36,33)(13,19,37,34)(14,20,38,35)(22,211,220,43)(23,212,221,44)(24,213,222,45)(25,214,223,46)(26,215,224,47)(27,216,218,48)(28,217,219,49)(50,95,59,70)(51,96,60,64)(52,97,61,65)(53,98,62,66)(54,92,63,67)(55,93,57,68)(56,94,58,69)(99,117,121,108)(100,118,122,109)(101,119,123,110)(102,113,124,111)(103,114,125,112)(104,115,126,106)(105,116,120,107)(127,142,152,140)(128,143,153,134)(129,144,154,135)(130,145,148,136)(131,146,149,137)(132,147,150,138)(133,141,151,139)(155,164,177,173)(156,165,178,174)(157,166,179,175)(158,167,180,169)(159,168,181,170)(160,162,182,171)(161,163,176,172)(183,196,208,198)(184,190,209,199)(185,191,210,200)(186,192,204,201)(187,193,205,202)(188,194,206,203)(189,195,207,197), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,218)(9,219)(10,220)(11,221)(12,222)(13,223)(14,224)(15,217)(16,211)(17,212)(18,213)(19,214)(20,215)(21,216)(22,41)(23,42)(24,36)(25,37)(26,38)(27,39)(28,40)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,83)(51,84)(52,78)(53,79)(54,80)(55,81)(56,82)(57,90)(58,91)(59,85)(60,86)(61,87)(62,88)(63,89)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(99,153)(100,154)(101,148)(102,149)(103,150)(104,151)(105,152)(106,139)(107,140)(108,134)(109,135)(110,136)(111,137)(112,138)(113,146)(114,147)(115,141)(116,142)(117,143)(118,144)(119,145)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(155,199)(156,200)(157,201)(158,202)(159,203)(160,197)(161,198)(162,207)(163,208)(164,209)(165,210)(166,204)(167,205)(168,206)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(176,196)(177,190)(178,191)(179,192)(180,193)(181,194)(182,195), (1,106,50,151)(2,107,51,152)(3,108,52,153)(4,109,53,154)(5,110,54,148)(6,111,55,149)(7,112,56,150)(8,171,218,189)(9,172,219,183)(10,173,220,184)(11,174,221,185)(12,175,222,186)(13,169,223,187)(14,170,224,188)(15,161,49,196)(16,155,43,190)(17,156,44,191)(18,157,45,192)(19,158,46,193)(20,159,47,194)(21,160,48,195)(22,209,41,164)(23,210,42,165)(24,204,36,166)(25,205,37,167)(26,206,38,168)(27,207,39,162)(28,208,40,163)(29,182,216,197)(30,176,217,198)(31,177,211,199)(32,178,212,200)(33,179,213,201)(34,180,214,202)(35,181,215,203)(57,131,75,113)(58,132,76,114)(59,133,77,115)(60,127,71,116)(61,128,72,117)(62,129,73,118)(63,130,74,119)(64,142,86,120)(65,143,87,121)(66,144,88,122)(67,145,89,123)(68,146,90,124)(69,147,91,125)(70,141,85,126)(78,99,97,134)(79,100,98,135)(80,101,92,136)(81,102,93,137)(82,103,94,138)(83,104,95,139)(84,105,96,140), (1,171,59,207)(2,172,60,208)(3,173,61,209)(4,174,62,210)(5,175,63,204)(6,169,57,205)(7,170,58,206)(8,151,27,115)(9,152,28,116)(10,153,22,117)(11,154,23,118)(12,148,24,119)(13,149,25,113)(14,150,26,114)(15,142,217,105)(16,143,211,99)(17,144,212,100)(18,145,213,101)(19,146,214,102)(20,147,215,103)(21,141,216,104)(29,139,48,126)(30,140,49,120)(31,134,43,121)(32,135,44,122)(33,136,45,123)(34,137,46,124)(35,138,47,125)(36,130,222,110)(37,131,223,111)(38,132,224,112)(39,133,218,106)(40,127,219,107)(41,128,220,108)(42,129,221,109)(50,189,77,162)(51,183,71,163)(52,184,72,164)(53,185,73,165)(54,186,74,166)(55,187,75,167)(56,188,76,168)(64,196,84,176)(65,190,78,177)(66,191,79,178)(67,192,80,179)(68,193,81,180)(69,194,82,181)(70,195,83,182)(85,160,95,197)(86,161,96,198)(87,155,97,199)(88,156,98,200)(89,157,92,201)(90,158,93,202)(91,159,94,203) );

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)], [(1,83,77,85),(2,84,71,86),(3,78,72,87),(4,79,73,88),(5,80,74,89),(6,81,75,90),(7,82,76,91),(8,21,39,29),(9,15,40,30),(10,16,41,31),(11,17,42,32),(12,18,36,33),(13,19,37,34),(14,20,38,35),(22,211,220,43),(23,212,221,44),(24,213,222,45),(25,214,223,46),(26,215,224,47),(27,216,218,48),(28,217,219,49),(50,95,59,70),(51,96,60,64),(52,97,61,65),(53,98,62,66),(54,92,63,67),(55,93,57,68),(56,94,58,69),(99,117,121,108),(100,118,122,109),(101,119,123,110),(102,113,124,111),(103,114,125,112),(104,115,126,106),(105,116,120,107),(127,142,152,140),(128,143,153,134),(129,144,154,135),(130,145,148,136),(131,146,149,137),(132,147,150,138),(133,141,151,139),(155,164,177,173),(156,165,178,174),(157,166,179,175),(158,167,180,169),(159,168,181,170),(160,162,182,171),(161,163,176,172),(183,196,208,198),(184,190,209,199),(185,191,210,200),(186,192,204,201),(187,193,205,202),(188,194,206,203),(189,195,207,197)], [(1,95),(2,96),(3,97),(4,98),(5,92),(6,93),(7,94),(8,218),(9,219),(10,220),(11,221),(12,222),(13,223),(14,224),(15,217),(16,211),(17,212),(18,213),(19,214),(20,215),(21,216),(22,41),(23,42),(24,36),(25,37),(26,38),(27,39),(28,40),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(50,83),(51,84),(52,78),(53,79),(54,80),(55,81),(56,82),(57,90),(58,91),(59,85),(60,86),(61,87),(62,88),(63,89),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(99,153),(100,154),(101,148),(102,149),(103,150),(104,151),(105,152),(106,139),(107,140),(108,134),(109,135),(110,136),(111,137),(112,138),(113,146),(114,147),(115,141),(116,142),(117,143),(118,144),(119,145),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133),(155,199),(156,200),(157,201),(158,202),(159,203),(160,197),(161,198),(162,207),(163,208),(164,209),(165,210),(166,204),(167,205),(168,206),(169,187),(170,188),(171,189),(172,183),(173,184),(174,185),(175,186),(176,196),(177,190),(178,191),(179,192),(180,193),(181,194),(182,195)], [(1,106,50,151),(2,107,51,152),(3,108,52,153),(4,109,53,154),(5,110,54,148),(6,111,55,149),(7,112,56,150),(8,171,218,189),(9,172,219,183),(10,173,220,184),(11,174,221,185),(12,175,222,186),(13,169,223,187),(14,170,224,188),(15,161,49,196),(16,155,43,190),(17,156,44,191),(18,157,45,192),(19,158,46,193),(20,159,47,194),(21,160,48,195),(22,209,41,164),(23,210,42,165),(24,204,36,166),(25,205,37,167),(26,206,38,168),(27,207,39,162),(28,208,40,163),(29,182,216,197),(30,176,217,198),(31,177,211,199),(32,178,212,200),(33,179,213,201),(34,180,214,202),(35,181,215,203),(57,131,75,113),(58,132,76,114),(59,133,77,115),(60,127,71,116),(61,128,72,117),(62,129,73,118),(63,130,74,119),(64,142,86,120),(65,143,87,121),(66,144,88,122),(67,145,89,123),(68,146,90,124),(69,147,91,125),(70,141,85,126),(78,99,97,134),(79,100,98,135),(80,101,92,136),(81,102,93,137),(82,103,94,138),(83,104,95,139),(84,105,96,140)], [(1,171,59,207),(2,172,60,208),(3,173,61,209),(4,174,62,210),(5,175,63,204),(6,169,57,205),(7,170,58,206),(8,151,27,115),(9,152,28,116),(10,153,22,117),(11,154,23,118),(12,148,24,119),(13,149,25,113),(14,150,26,114),(15,142,217,105),(16,143,211,99),(17,144,212,100),(18,145,213,101),(19,146,214,102),(20,147,215,103),(21,141,216,104),(29,139,48,126),(30,140,49,120),(31,134,43,121),(32,135,44,122),(33,136,45,123),(34,137,46,124),(35,138,47,125),(36,130,222,110),(37,131,223,111),(38,132,224,112),(39,133,218,106),(40,127,219,107),(41,128,220,108),(42,129,221,109),(50,189,77,162),(51,183,71,163),(52,184,72,164),(53,185,73,165),(54,186,74,166),(55,187,75,167),(56,188,76,168),(64,196,84,176),(65,190,78,177),(66,191,79,178),(67,192,80,179),(68,193,81,180),(69,194,82,181),(70,195,83,182),(85,160,95,197),(86,161,96,198),(87,155,97,199),(88,156,98,200),(89,157,92,201),(90,158,93,202),(91,159,94,203)]])

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 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 D4 Q8 C4○D4 C4○D8 C7×D4 C7×Q8 C7×C4○D4 C7×C4○D8 C8⋊C22 C7×C8⋊C22 kernel C7×D4.Q8 C7×D4⋊C4 C7×C4⋊C8 C7×C4.Q8 C7×C2.D8 D4×C28 C7×C42.C2 D4.Q8 D4⋊C4 C4⋊C8 C4.Q8 C2.D8 C4×D4 C42.C2 C2×C28 C7×D4 C28 C14 C2×C4 D4 C4 C2 C14 C2 # reps 1 2 1 1 1 1 1 6 12 6 6 6 6 6 2 2 2 4 12 12 12 24 1 6

Matrix representation of C7×D4.Q8 in GL4(𝔽113) generated by

 1 0 0 0 0 1 0 0 0 0 30 0 0 0 0 30
,
 0 1 0 0 112 0 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 112 0 0 0 0 112
,
 15 0 0 0 0 15 0 0 0 0 1 111 0 0 1 112
,
 82 31 0 0 31 31 0 0 0 0 63 103 0 0 58 50
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,30,0,0,0,0,30],[0,112,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,112,0,0,0,0,112],[15,0,0,0,0,15,0,0,0,0,1,1,0,0,111,112],[82,31,0,0,31,31,0,0,0,0,63,58,0,0,103,50] >;

C7×D4.Q8 in GAP, Magma, Sage, TeX

C_7\times D_4.Q_8
% in TeX

G:=Group("C7xD4.Q8");
// GroupNames label

G:=SmallGroup(448,886);
// by ID

G=gap.SmallGroup(448,886);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,813,1968,2438,14117,3547,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=1,e^2=b^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,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^-1>;
// generators/relations

׿
×
𝔽