metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (Q8×C14)⋊6C4, (C2×Q8)⋊4Dic7, (C2×C28).193D4, C28.208(C2×D4), Q8.5(C2×Dic7), (C22×Q8).2D7, Q8⋊Dic7⋊37C2, C28.82(C22×C4), (C2×Q8).166D14, C28.34(C22⋊C4), (C2×C28).475C23, (C22×C4).154D14, (C22×C14).199D4, C7⋊4(C23.38D4), C23.87(C7⋊D4), C4.11(C23.D7), C4.12(C22×Dic7), C2.5(C28.C23), C4⋊Dic7.353C22, (Q8×C14).201C22, C14.101(C8.C22), (C22×C28).201C22, C22.21(C23.D7), C23.21D14.19C2, (Q8×C2×C14).2C2, C4.92(C2×C7⋊D4), (C7×Q8).23(C2×C4), (C2×C28).120(C2×C4), (C2×C14).559(C2×D4), C14.78(C2×C22⋊C4), (C2×C7⋊C8).174C22, (C2×C4).25(C2×Dic7), C22.94(C2×C7⋊D4), C2.14(C2×C23.D7), (C2×C4).198(C7⋊D4), (C2×C4).561(C22×D7), (C2×C4.Dic7).27C2, (C2×C14).115(C22⋊C4), SmallGroup(448,759)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (Q8×C14)⋊6C4
G = < a,b,c,d | a14=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=a7b-1c >
Subgroups: 436 in 150 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, Dic7, C28, C28, C28, C2×C14, C2×C14, C2×C14, Q8⋊C4, C42⋊C2, C2×M4(2), C22×Q8, C7⋊C8, C2×Dic7, C2×C28, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×C14, C23.38D4, C2×C7⋊C8, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C22×C28, C22×C28, Q8×C14, Q8×C14, Q8⋊Dic7, C2×C4.Dic7, C23.21D14, Q8×C2×C14, (Q8×C14)⋊6C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C2×C22⋊C4, C8.C22, C2×Dic7, C7⋊D4, C22×D7, C23.38D4, C23.D7, C22×Dic7, C2×C7⋊D4, C28.C23, C2×C23.D7, (Q8×C14)⋊6C4
►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 41 31 23)(2 42 32 24)(3 36 33 25)(4 37 34 26)(5 38 35 27)(6 39 29 28)(7 40 30 22)(8 91 103 107)(9 85 104 108)(10 86 105 109)(11 87 99 110)(12 88 100 111)(13 89 101 112)(14 90 102 106)(15 61 65 96)(16 62 66 97)(17 63 67 98)(18 57 68 92)(19 58 69 93)(20 59 70 94)(21 60 64 95)(43 81 54 77)(44 82 55 71)(45 83 56 72)(46 84 50 73)(47 78 51 74)(48 79 52 75)(49 80 53 76)(113 191 120 184)(114 192 121 185)(115 193 122 186)(116 194 123 187)(117 195 124 188)(118 196 125 189)(119 183 126 190)(127 215 134 222)(128 216 135 223)(129 217 136 224)(130 218 137 211)(131 219 138 212)(132 220 139 213)(133 221 140 214)(141 160 148 167)(142 161 149 168)(143 162 150 155)(144 163 151 156)(145 164 152 157)(146 165 153 158)(147 166 154 159)(169 201 176 208)(170 202 177 209)(171 203 178 210)(172 204 179 197)(173 205 180 198)(174 206 181 199)(175 207 182 200)
(1 79 31 75)(2 80 32 76)(3 81 33 77)(4 82 34 71)(5 83 35 72)(6 84 29 73)(7 78 30 74)(8 15 103 65)(9 16 104 66)(10 17 105 67)(11 18 99 68)(12 19 100 69)(13 20 101 70)(14 21 102 64)(22 51 40 47)(23 52 41 48)(24 53 42 49)(25 54 36 43)(26 55 37 44)(27 56 38 45)(28 50 39 46)(57 87 92 110)(58 88 93 111)(59 89 94 112)(60 90 95 106)(61 91 96 107)(62 85 97 108)(63 86 98 109)(113 156 120 163)(114 157 121 164)(115 158 122 165)(116 159 123 166)(117 160 124 167)(118 161 125 168)(119 162 126 155)(127 202 134 209)(128 203 135 210)(129 204 136 197)(130 205 137 198)(131 206 138 199)(132 207 139 200)(133 208 140 201)(141 188 148 195)(142 189 149 196)(143 190 150 183)(144 191 151 184)(145 192 152 185)(146 193 153 186)(147 194 154 187)(169 221 176 214)(170 222 177 215)(171 223 178 216)(172 224 179 217)(173 211 180 218)(174 212 181 219)(175 213 182 220)
(1 212 19 146)(2 218 20 152)(3 224 21 144)(4 216 15 150)(5 222 16 142)(6 214 17 148)(7 220 18 154)(8 126 71 203)(9 118 72 209)(10 124 73 201)(11 116 74 207)(12 122 75 199)(13 114 76 205)(14 120 77 197)(22 132 92 166)(23 138 93 158)(24 130 94 164)(25 136 95 156)(26 128 96 162)(27 134 97 168)(28 140 98 160)(29 221 67 141)(30 213 68 147)(31 219 69 153)(32 211 70 145)(33 217 64 151)(34 223 65 143)(35 215 66 149)(36 129 60 163)(37 135 61 155)(38 127 62 161)(39 133 63 167)(40 139 57 159)(41 131 58 165)(42 137 59 157)(43 172 90 184)(44 178 91 190)(45 170 85 196)(46 176 86 188)(47 182 87 194)(48 174 88 186)(49 180 89 192)(50 169 109 195)(51 175 110 187)(52 181 111 193)(53 173 112 185)(54 179 106 191)(55 171 107 183)(56 177 108 189)(78 200 99 123)(79 206 100 115)(80 198 101 121)(81 204 102 113)(82 210 103 119)(83 202 104 125)(84 208 105 117)
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), (1,41,31,23)(2,42,32,24)(3,36,33,25)(4,37,34,26)(5,38,35,27)(6,39,29,28)(7,40,30,22)(8,91,103,107)(9,85,104,108)(10,86,105,109)(11,87,99,110)(12,88,100,111)(13,89,101,112)(14,90,102,106)(15,61,65,96)(16,62,66,97)(17,63,67,98)(18,57,68,92)(19,58,69,93)(20,59,70,94)(21,60,64,95)(43,81,54,77)(44,82,55,71)(45,83,56,72)(46,84,50,73)(47,78,51,74)(48,79,52,75)(49,80,53,76)(113,191,120,184)(114,192,121,185)(115,193,122,186)(116,194,123,187)(117,195,124,188)(118,196,125,189)(119,183,126,190)(127,215,134,222)(128,216,135,223)(129,217,136,224)(130,218,137,211)(131,219,138,212)(132,220,139,213)(133,221,140,214)(141,160,148,167)(142,161,149,168)(143,162,150,155)(144,163,151,156)(145,164,152,157)(146,165,153,158)(147,166,154,159)(169,201,176,208)(170,202,177,209)(171,203,178,210)(172,204,179,197)(173,205,180,198)(174,206,181,199)(175,207,182,200), (1,79,31,75)(2,80,32,76)(3,81,33,77)(4,82,34,71)(5,83,35,72)(6,84,29,73)(7,78,30,74)(8,15,103,65)(9,16,104,66)(10,17,105,67)(11,18,99,68)(12,19,100,69)(13,20,101,70)(14,21,102,64)(22,51,40,47)(23,52,41,48)(24,53,42,49)(25,54,36,43)(26,55,37,44)(27,56,38,45)(28,50,39,46)(57,87,92,110)(58,88,93,111)(59,89,94,112)(60,90,95,106)(61,91,96,107)(62,85,97,108)(63,86,98,109)(113,156,120,163)(114,157,121,164)(115,158,122,165)(116,159,123,166)(117,160,124,167)(118,161,125,168)(119,162,126,155)(127,202,134,209)(128,203,135,210)(129,204,136,197)(130,205,137,198)(131,206,138,199)(132,207,139,200)(133,208,140,201)(141,188,148,195)(142,189,149,196)(143,190,150,183)(144,191,151,184)(145,192,152,185)(146,193,153,186)(147,194,154,187)(169,221,176,214)(170,222,177,215)(171,223,178,216)(172,224,179,217)(173,211,180,218)(174,212,181,219)(175,213,182,220), (1,212,19,146)(2,218,20,152)(3,224,21,144)(4,216,15,150)(5,222,16,142)(6,214,17,148)(7,220,18,154)(8,126,71,203)(9,118,72,209)(10,124,73,201)(11,116,74,207)(12,122,75,199)(13,114,76,205)(14,120,77,197)(22,132,92,166)(23,138,93,158)(24,130,94,164)(25,136,95,156)(26,128,96,162)(27,134,97,168)(28,140,98,160)(29,221,67,141)(30,213,68,147)(31,219,69,153)(32,211,70,145)(33,217,64,151)(34,223,65,143)(35,215,66,149)(36,129,60,163)(37,135,61,155)(38,127,62,161)(39,133,63,167)(40,139,57,159)(41,131,58,165)(42,137,59,157)(43,172,90,184)(44,178,91,190)(45,170,85,196)(46,176,86,188)(47,182,87,194)(48,174,88,186)(49,180,89,192)(50,169,109,195)(51,175,110,187)(52,181,111,193)(53,173,112,185)(54,179,106,191)(55,171,107,183)(56,177,108,189)(78,200,99,123)(79,206,100,115)(80,198,101,121)(81,204,102,113)(82,210,103,119)(83,202,104,125)(84,208,105,117)>;
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), (1,41,31,23)(2,42,32,24)(3,36,33,25)(4,37,34,26)(5,38,35,27)(6,39,29,28)(7,40,30,22)(8,91,103,107)(9,85,104,108)(10,86,105,109)(11,87,99,110)(12,88,100,111)(13,89,101,112)(14,90,102,106)(15,61,65,96)(16,62,66,97)(17,63,67,98)(18,57,68,92)(19,58,69,93)(20,59,70,94)(21,60,64,95)(43,81,54,77)(44,82,55,71)(45,83,56,72)(46,84,50,73)(47,78,51,74)(48,79,52,75)(49,80,53,76)(113,191,120,184)(114,192,121,185)(115,193,122,186)(116,194,123,187)(117,195,124,188)(118,196,125,189)(119,183,126,190)(127,215,134,222)(128,216,135,223)(129,217,136,224)(130,218,137,211)(131,219,138,212)(132,220,139,213)(133,221,140,214)(141,160,148,167)(142,161,149,168)(143,162,150,155)(144,163,151,156)(145,164,152,157)(146,165,153,158)(147,166,154,159)(169,201,176,208)(170,202,177,209)(171,203,178,210)(172,204,179,197)(173,205,180,198)(174,206,181,199)(175,207,182,200), (1,79,31,75)(2,80,32,76)(3,81,33,77)(4,82,34,71)(5,83,35,72)(6,84,29,73)(7,78,30,74)(8,15,103,65)(9,16,104,66)(10,17,105,67)(11,18,99,68)(12,19,100,69)(13,20,101,70)(14,21,102,64)(22,51,40,47)(23,52,41,48)(24,53,42,49)(25,54,36,43)(26,55,37,44)(27,56,38,45)(28,50,39,46)(57,87,92,110)(58,88,93,111)(59,89,94,112)(60,90,95,106)(61,91,96,107)(62,85,97,108)(63,86,98,109)(113,156,120,163)(114,157,121,164)(115,158,122,165)(116,159,123,166)(117,160,124,167)(118,161,125,168)(119,162,126,155)(127,202,134,209)(128,203,135,210)(129,204,136,197)(130,205,137,198)(131,206,138,199)(132,207,139,200)(133,208,140,201)(141,188,148,195)(142,189,149,196)(143,190,150,183)(144,191,151,184)(145,192,152,185)(146,193,153,186)(147,194,154,187)(169,221,176,214)(170,222,177,215)(171,223,178,216)(172,224,179,217)(173,211,180,218)(174,212,181,219)(175,213,182,220), (1,212,19,146)(2,218,20,152)(3,224,21,144)(4,216,15,150)(5,222,16,142)(6,214,17,148)(7,220,18,154)(8,126,71,203)(9,118,72,209)(10,124,73,201)(11,116,74,207)(12,122,75,199)(13,114,76,205)(14,120,77,197)(22,132,92,166)(23,138,93,158)(24,130,94,164)(25,136,95,156)(26,128,96,162)(27,134,97,168)(28,140,98,160)(29,221,67,141)(30,213,68,147)(31,219,69,153)(32,211,70,145)(33,217,64,151)(34,223,65,143)(35,215,66,149)(36,129,60,163)(37,135,61,155)(38,127,62,161)(39,133,63,167)(40,139,57,159)(41,131,58,165)(42,137,59,157)(43,172,90,184)(44,178,91,190)(45,170,85,196)(46,176,86,188)(47,182,87,194)(48,174,88,186)(49,180,89,192)(50,169,109,195)(51,175,110,187)(52,181,111,193)(53,173,112,185)(54,179,106,191)(55,171,107,183)(56,177,108,189)(78,200,99,123)(79,206,100,115)(80,198,101,121)(81,204,102,113)(82,210,103,119)(83,202,104,125)(84,208,105,117) );
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)], [(1,41,31,23),(2,42,32,24),(3,36,33,25),(4,37,34,26),(5,38,35,27),(6,39,29,28),(7,40,30,22),(8,91,103,107),(9,85,104,108),(10,86,105,109),(11,87,99,110),(12,88,100,111),(13,89,101,112),(14,90,102,106),(15,61,65,96),(16,62,66,97),(17,63,67,98),(18,57,68,92),(19,58,69,93),(20,59,70,94),(21,60,64,95),(43,81,54,77),(44,82,55,71),(45,83,56,72),(46,84,50,73),(47,78,51,74),(48,79,52,75),(49,80,53,76),(113,191,120,184),(114,192,121,185),(115,193,122,186),(116,194,123,187),(117,195,124,188),(118,196,125,189),(119,183,126,190),(127,215,134,222),(128,216,135,223),(129,217,136,224),(130,218,137,211),(131,219,138,212),(132,220,139,213),(133,221,140,214),(141,160,148,167),(142,161,149,168),(143,162,150,155),(144,163,151,156),(145,164,152,157),(146,165,153,158),(147,166,154,159),(169,201,176,208),(170,202,177,209),(171,203,178,210),(172,204,179,197),(173,205,180,198),(174,206,181,199),(175,207,182,200)], [(1,79,31,75),(2,80,32,76),(3,81,33,77),(4,82,34,71),(5,83,35,72),(6,84,29,73),(7,78,30,74),(8,15,103,65),(9,16,104,66),(10,17,105,67),(11,18,99,68),(12,19,100,69),(13,20,101,70),(14,21,102,64),(22,51,40,47),(23,52,41,48),(24,53,42,49),(25,54,36,43),(26,55,37,44),(27,56,38,45),(28,50,39,46),(57,87,92,110),(58,88,93,111),(59,89,94,112),(60,90,95,106),(61,91,96,107),(62,85,97,108),(63,86,98,109),(113,156,120,163),(114,157,121,164),(115,158,122,165),(116,159,123,166),(117,160,124,167),(118,161,125,168),(119,162,126,155),(127,202,134,209),(128,203,135,210),(129,204,136,197),(130,205,137,198),(131,206,138,199),(132,207,139,200),(133,208,140,201),(141,188,148,195),(142,189,149,196),(143,190,150,183),(144,191,151,184),(145,192,152,185),(146,193,153,186),(147,194,154,187),(169,221,176,214),(170,222,177,215),(171,223,178,216),(172,224,179,217),(173,211,180,218),(174,212,181,219),(175,213,182,220)], [(1,212,19,146),(2,218,20,152),(3,224,21,144),(4,216,15,150),(5,222,16,142),(6,214,17,148),(7,220,18,154),(8,126,71,203),(9,118,72,209),(10,124,73,201),(11,116,74,207),(12,122,75,199),(13,114,76,205),(14,120,77,197),(22,132,92,166),(23,138,93,158),(24,130,94,164),(25,136,95,156),(26,128,96,162),(27,134,97,168),(28,140,98,160),(29,221,67,141),(30,213,68,147),(31,219,69,153),(32,211,70,145),(33,217,64,151),(34,223,65,143),(35,215,66,149),(36,129,60,163),(37,135,61,155),(38,127,62,161),(39,133,63,167),(40,139,57,159),(41,131,58,165),(42,137,59,157),(43,172,90,184),(44,178,91,190),(45,170,85,196),(46,176,86,188),(47,182,87,194),(48,174,88,186),(49,180,89,192),(50,169,109,195),(51,175,110,187),(52,181,111,193),(53,173,112,185),(54,179,106,191),(55,171,107,183),(56,177,108,189),(78,200,99,123),(79,206,100,115),(80,198,101,121),(81,204,102,113),(82,210,103,119),(83,202,104,125),(84,208,105,117)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14U | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | D14 | Dic7 | D14 | C7⋊D4 | C7⋊D4 | C8.C22 | C28.C23 |
| kernel | (Q8×C14)⋊6C4 | Q8⋊Dic7 | C2×C4.Dic7 | C23.21D14 | Q8×C2×C14 | Q8×C14 | C2×C28 | C22×C14 | C22×Q8 | C22×C4 | C2×Q8 | C2×Q8 | C2×C4 | C23 | C14 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 3 | 3 | 12 | 6 | 18 | 6 | 2 | 12 |
Matrix representation of (Q8×C14)⋊6C4 ►in GL8(𝔽113)
| 24 | 103 | 0 | 0 | 0 | 0 | 0 | 0 |
| 10 | 10 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 98 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 111 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 | 0 | 1 |
| 0 | 0 | 0 | 0 | 15 | 98 | 112 | 0 |
| 29 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 106 | 84 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 88 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 97 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 95 | 44 | 60 |
| 0 | 0 | 0 | 0 | 0 | 95 | 60 | 69 |
| 2 | 78 | 0 | 0 | 0 | 0 | 0 | 0 |
| 13 | 111 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 87 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 0 | 111 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 112 | 1 |
| 0 | 0 | 0 | 0 | 112 | 0 | 98 | 0 |
| 0 | 0 | 0 | 0 | 112 | 1 | 98 | 0 |
G:=sub<GL(8,GF(113))| [24,10,0,0,0,0,0,0,103,10,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,98,98,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,15,0,0,0,0,111,112,98,98,0,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0],[29,106,0,0,0,0,0,0,7,84,0,0,0,0,0,0,0,0,61,3,0,0,0,0,0,0,3,52,0,0,0,0,0,0,0,0,16,60,18,0,0,0,0,0,88,97,95,95,0,0,0,0,0,0,44,60,0,0,0,0,0,0,60,69],[2,13,0,0,0,0,0,0,78,111,0,0,0,0,0,0,0,0,67,87,0,0,0,0,0,0,51,46,0,0,0,0,0,0,0,0,15,0,112,112,0,0,0,0,0,0,0,1,0,0,0,0,111,112,98,98,0,0,0,0,0,1,0,0] >;
(Q8×C14)⋊6C4 in GAP, Magma, Sage, TeX
(Q_8\times C_{14})\rtimes_6C_4 % in TeX
G:=Group("(Q8xC14):6C4"); // GroupNames label
G:=SmallGroup(448,759);
// by ID
G=gap.SmallGroup(448,759);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,422,387,184,1684,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^14=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^7*b^-1*c>;
// generators/relations