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## G = C7×Q8.D4order 448 = 26·7

### Direct product of C7 and Q8.D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×Q8.D4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — D4×C14 — C7×C4.4D4 — C7×Q8.D4
 Lower central C1 — C2 — C2×C4 — C7×Q8.D4
 Upper central C1 — C2×C14 — C4×C28 — C7×Q8.D4

Generators and relations for C7×Q8.D4
G = < a,b,c,d,e | a7=b4=d4=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=b2d-1 >

Subgroups: 202 in 112 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C56, C2×C28, C2×C28, C7×D4, C7×Q8, C7×Q8, C22×C14, Q8.D4, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×SD16, C7×Q16, D4×C14, Q8×C14, C7×D4⋊C4, C7×Q8⋊C4, C7×C4⋊C8, Q8×C28, C7×C4.4D4, C14×SD16, C14×Q16, C7×Q8.D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C4○D8, C8.C22, C7×D4, C22×C14, Q8.D4, D4×C14, C7×C4○D4, C7×C4⋊D4, C7×C4○D8, C7×C8.C22, C7×Q8.D4

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

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

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

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

133 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E ··· 4I 4J 7A ··· 7F 8A 8B 8C 8D 14A ··· 14R 14S ··· 14X 28A ··· 28X 28Y ··· 28BB 28BC ··· 28BH 56A ··· 56X order 1 2 2 2 2 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 8 2 2 2 2 4 ··· 4 8 1 ··· 1 4 4 4 4 1 ··· 1 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 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 C14 D4 D4 C4○D4 C4○D8 C7×D4 C7×D4 C7×C4○D4 C7×C4○D8 C8.C22 C7×C8.C22 kernel C7×Q8.D4 C7×D4⋊C4 C7×Q8⋊C4 C7×C4⋊C8 Q8×C28 C7×C4.4D4 C14×SD16 C14×Q16 Q8.D4 D4⋊C4 Q8⋊C4 C4⋊C8 C4×Q8 C4.4D4 C2×SD16 C2×Q16 C2×C28 C7×Q8 C28 C14 C2×C4 Q8 C4 C2 C14 C2 # reps 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 2 2 2 4 12 12 12 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
,
 112 0 0 0 0 112 0 0 0 0 0 112 0 0 1 0
,
 0 1 0 0 1 0 0 0 0 0 100 100 0 0 100 13
,
 0 15 0 0 15 0 0 0 0 0 15 0 0 0 0 15
,
 0 98 0 0 15 0 0 0 0 0 0 15 0 0 15 0
G:=sub<GL(4,GF(113))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[112,0,0,0,0,112,0,0,0,0,0,1,0,0,112,0],[0,1,0,0,1,0,0,0,0,0,100,100,0,0,100,13],[0,15,0,0,15,0,0,0,0,0,15,0,0,0,0,15],[0,15,0,0,98,0,0,0,0,0,0,15,0,0,15,0] >;

C7×Q8.D4 in GAP, Magma, Sage, TeX

C_7\times Q_8.D_4
% in TeX

G:=Group("C7xQ8.D4");
// GroupNames label

G:=SmallGroup(448,872);
// by ID

G=gap.SmallGroup(448,872);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,1968,2438,1192,14117,3547,124]);
// Polycyclic

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

׿
×
𝔽