direct product, metabelian, nilpotent (class 2), monomial
Aliases: Q8×C33, C6.13C62, C4.(C32×C6), C12.9(C3×C6), (C3×C12).19C6, C2.2(C3×C62), (C32×C12).7C2, (C32×C6).32C22, (C3×C6).37(C2×C6), SmallGroup(216,152)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C33
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 168, all normal (6 characteristic)
C1, C2, C3, C4, C6, Q8, C32, C12, C3×C6, C3×Q8, C33, C3×C12, C32×C6, Q8×C32, C32×C12, Q8×C33
Quotients: C1, C2, C3, C22, C6, Q8, C32, C2×C6, C3×C6, C3×Q8, C33, C62, C32×C6, Q8×C32, C3×C62, Q8×C33
►Regular action on 216 points
Generators in S216
(1 112 19)(2 109 20)(3 110 17)(4 111 18)(5 105 93)(6 106 94)(7 107 95)(8 108 96)(9 51 47)(10 52 48)(11 49 45)(12 50 46)(13 62 78)(14 63 79)(15 64 80)(16 61 77)(21 60 75)(22 57 76)(23 58 73)(24 59 74)(25 92 88)(26 89 85)(27 90 86)(28 91 87)(29 41 83)(30 42 84)(31 43 81)(32 44 82)(33 99 70)(34 100 71)(35 97 72)(36 98 69)(37 103 65)(38 104 66)(39 101 67)(40 102 68)(53 127 189)(54 128 190)(55 125 191)(56 126 192)(113 215 156)(114 216 153)(115 213 154)(116 214 155)(117 184 180)(118 181 177)(119 182 178)(120 183 179)(121 199 164)(122 200 161)(123 197 162)(124 198 163)(129 195 160)(130 196 157)(131 193 158)(132 194 159)(133 176 172)(134 173 169)(135 174 170)(136 175 171)(137 203 144)(138 204 141)(139 201 142)(140 202 143)(145 185 149)(146 186 150)(147 187 151)(148 188 152)(165 212 208)(166 209 205)(167 210 206)(168 211 207)
(1 87 11)(2 88 12)(3 85 9)(4 86 10)(5 63 35)(6 64 36)(7 61 33)(8 62 34)(13 71 96)(14 72 93)(15 69 94)(16 70 95)(17 89 47)(18 90 48)(19 91 45)(20 92 46)(21 43 67)(22 44 68)(23 41 65)(24 42 66)(25 50 109)(26 51 110)(27 52 111)(28 49 112)(29 103 73)(30 104 74)(31 101 75)(32 102 76)(37 58 83)(38 59 84)(39 60 81)(40 57 82)(53 186 209)(54 187 210)(55 188 211)(56 185 212)(77 99 107)(78 100 108)(79 97 105)(80 98 106)(113 172 195)(114 169 196)(115 170 193)(116 171 194)(117 144 199)(118 141 200)(119 142 197)(120 143 198)(121 180 203)(122 177 204)(123 178 201)(124 179 202)(125 152 207)(126 149 208)(127 150 205)(128 151 206)(129 156 176)(130 153 173)(131 154 174)(132 155 175)(133 160 215)(134 157 216)(135 158 213)(136 159 214)(137 164 184)(138 161 181)(139 162 182)(140 163 183)(145 165 192)(146 166 189)(147 167 190)(148 168 191)
(1 83 7)(2 84 8)(3 81 5)(4 82 6)(9 60 35)(10 57 36)(11 58 33)(12 59 34)(13 92 66)(14 89 67)(15 90 68)(16 91 65)(17 43 93)(18 44 94)(19 41 95)(20 42 96)(21 72 47)(22 69 48)(23 70 45)(24 71 46)(25 104 78)(26 101 79)(27 102 80)(28 103 77)(29 107 112)(30 108 109)(31 105 110)(32 106 111)(37 61 87)(38 62 88)(39 63 85)(40 64 86)(49 73 99)(50 74 100)(51 75 97)(52 76 98)(53 139 213)(54 140 214)(55 137 215)(56 138 216)(113 191 144)(114 192 141)(115 189 142)(116 190 143)(117 195 168)(118 196 165)(119 193 166)(120 194 167)(121 176 152)(122 173 149)(123 174 150)(124 175 151)(125 203 156)(126 204 153)(127 201 154)(128 202 155)(129 207 180)(130 208 177)(131 205 178)(132 206 179)(133 188 164)(134 185 161)(135 186 162)(136 187 163)(145 200 169)(146 197 170)(147 198 171)(148 199 172)(157 212 181)(158 209 182)(159 210 183)(160 211 184)
(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)
(1 191 3 189)(2 190 4 192)(5 115 7 113)(6 114 8 116)(9 166 11 168)(10 165 12 167)(13 175 15 173)(14 174 16 176)(17 127 19 125)(18 126 20 128)(21 178 23 180)(22 177 24 179)(25 187 27 185)(26 186 28 188)(29 137 31 139)(30 140 32 138)(33 195 35 193)(34 194 36 196)(37 199 39 197)(38 198 40 200)(41 203 43 201)(42 202 44 204)(45 207 47 205)(46 206 48 208)(49 211 51 209)(50 210 52 212)(53 112 55 110)(54 111 56 109)(57 118 59 120)(58 117 60 119)(61 172 63 170)(62 171 64 169)(65 121 67 123)(66 124 68 122)(69 130 71 132)(70 129 72 131)(73 184 75 182)(74 183 76 181)(77 133 79 135)(78 136 80 134)(81 142 83 144)(82 141 84 143)(85 146 87 148)(86 145 88 147)(89 150 91 152)(90 149 92 151)(93 154 95 156)(94 153 96 155)(97 158 99 160)(98 157 100 159)(101 162 103 164)(102 161 104 163)(105 213 107 215)(106 216 108 214)
G:=sub<Sym(216)| (1,112,19)(2,109,20)(3,110,17)(4,111,18)(5,105,93)(6,106,94)(7,107,95)(8,108,96)(9,51,47)(10,52,48)(11,49,45)(12,50,46)(13,62,78)(14,63,79)(15,64,80)(16,61,77)(21,60,75)(22,57,76)(23,58,73)(24,59,74)(25,92,88)(26,89,85)(27,90,86)(28,91,87)(29,41,83)(30,42,84)(31,43,81)(32,44,82)(33,99,70)(34,100,71)(35,97,72)(36,98,69)(37,103,65)(38,104,66)(39,101,67)(40,102,68)(53,127,189)(54,128,190)(55,125,191)(56,126,192)(113,215,156)(114,216,153)(115,213,154)(116,214,155)(117,184,180)(118,181,177)(119,182,178)(120,183,179)(121,199,164)(122,200,161)(123,197,162)(124,198,163)(129,195,160)(130,196,157)(131,193,158)(132,194,159)(133,176,172)(134,173,169)(135,174,170)(136,175,171)(137,203,144)(138,204,141)(139,201,142)(140,202,143)(145,185,149)(146,186,150)(147,187,151)(148,188,152)(165,212,208)(166,209,205)(167,210,206)(168,211,207), (1,87,11)(2,88,12)(3,85,9)(4,86,10)(5,63,35)(6,64,36)(7,61,33)(8,62,34)(13,71,96)(14,72,93)(15,69,94)(16,70,95)(17,89,47)(18,90,48)(19,91,45)(20,92,46)(21,43,67)(22,44,68)(23,41,65)(24,42,66)(25,50,109)(26,51,110)(27,52,111)(28,49,112)(29,103,73)(30,104,74)(31,101,75)(32,102,76)(37,58,83)(38,59,84)(39,60,81)(40,57,82)(53,186,209)(54,187,210)(55,188,211)(56,185,212)(77,99,107)(78,100,108)(79,97,105)(80,98,106)(113,172,195)(114,169,196)(115,170,193)(116,171,194)(117,144,199)(118,141,200)(119,142,197)(120,143,198)(121,180,203)(122,177,204)(123,178,201)(124,179,202)(125,152,207)(126,149,208)(127,150,205)(128,151,206)(129,156,176)(130,153,173)(131,154,174)(132,155,175)(133,160,215)(134,157,216)(135,158,213)(136,159,214)(137,164,184)(138,161,181)(139,162,182)(140,163,183)(145,165,192)(146,166,189)(147,167,190)(148,168,191), (1,83,7)(2,84,8)(3,81,5)(4,82,6)(9,60,35)(10,57,36)(11,58,33)(12,59,34)(13,92,66)(14,89,67)(15,90,68)(16,91,65)(17,43,93)(18,44,94)(19,41,95)(20,42,96)(21,72,47)(22,69,48)(23,70,45)(24,71,46)(25,104,78)(26,101,79)(27,102,80)(28,103,77)(29,107,112)(30,108,109)(31,105,110)(32,106,111)(37,61,87)(38,62,88)(39,63,85)(40,64,86)(49,73,99)(50,74,100)(51,75,97)(52,76,98)(53,139,213)(54,140,214)(55,137,215)(56,138,216)(113,191,144)(114,192,141)(115,189,142)(116,190,143)(117,195,168)(118,196,165)(119,193,166)(120,194,167)(121,176,152)(122,173,149)(123,174,150)(124,175,151)(125,203,156)(126,204,153)(127,201,154)(128,202,155)(129,207,180)(130,208,177)(131,205,178)(132,206,179)(133,188,164)(134,185,161)(135,186,162)(136,187,163)(145,200,169)(146,197,170)(147,198,171)(148,199,172)(157,212,181)(158,209,182)(159,210,183)(160,211,184), (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), (1,191,3,189)(2,190,4,192)(5,115,7,113)(6,114,8,116)(9,166,11,168)(10,165,12,167)(13,175,15,173)(14,174,16,176)(17,127,19,125)(18,126,20,128)(21,178,23,180)(22,177,24,179)(25,187,27,185)(26,186,28,188)(29,137,31,139)(30,140,32,138)(33,195,35,193)(34,194,36,196)(37,199,39,197)(38,198,40,200)(41,203,43,201)(42,202,44,204)(45,207,47,205)(46,206,48,208)(49,211,51,209)(50,210,52,212)(53,112,55,110)(54,111,56,109)(57,118,59,120)(58,117,60,119)(61,172,63,170)(62,171,64,169)(65,121,67,123)(66,124,68,122)(69,130,71,132)(70,129,72,131)(73,184,75,182)(74,183,76,181)(77,133,79,135)(78,136,80,134)(81,142,83,144)(82,141,84,143)(85,146,87,148)(86,145,88,147)(89,150,91,152)(90,149,92,151)(93,154,95,156)(94,153,96,155)(97,158,99,160)(98,157,100,159)(101,162,103,164)(102,161,104,163)(105,213,107,215)(106,216,108,214)>;
G:=Group( (1,112,19)(2,109,20)(3,110,17)(4,111,18)(5,105,93)(6,106,94)(7,107,95)(8,108,96)(9,51,47)(10,52,48)(11,49,45)(12,50,46)(13,62,78)(14,63,79)(15,64,80)(16,61,77)(21,60,75)(22,57,76)(23,58,73)(24,59,74)(25,92,88)(26,89,85)(27,90,86)(28,91,87)(29,41,83)(30,42,84)(31,43,81)(32,44,82)(33,99,70)(34,100,71)(35,97,72)(36,98,69)(37,103,65)(38,104,66)(39,101,67)(40,102,68)(53,127,189)(54,128,190)(55,125,191)(56,126,192)(113,215,156)(114,216,153)(115,213,154)(116,214,155)(117,184,180)(118,181,177)(119,182,178)(120,183,179)(121,199,164)(122,200,161)(123,197,162)(124,198,163)(129,195,160)(130,196,157)(131,193,158)(132,194,159)(133,176,172)(134,173,169)(135,174,170)(136,175,171)(137,203,144)(138,204,141)(139,201,142)(140,202,143)(145,185,149)(146,186,150)(147,187,151)(148,188,152)(165,212,208)(166,209,205)(167,210,206)(168,211,207), (1,87,11)(2,88,12)(3,85,9)(4,86,10)(5,63,35)(6,64,36)(7,61,33)(8,62,34)(13,71,96)(14,72,93)(15,69,94)(16,70,95)(17,89,47)(18,90,48)(19,91,45)(20,92,46)(21,43,67)(22,44,68)(23,41,65)(24,42,66)(25,50,109)(26,51,110)(27,52,111)(28,49,112)(29,103,73)(30,104,74)(31,101,75)(32,102,76)(37,58,83)(38,59,84)(39,60,81)(40,57,82)(53,186,209)(54,187,210)(55,188,211)(56,185,212)(77,99,107)(78,100,108)(79,97,105)(80,98,106)(113,172,195)(114,169,196)(115,170,193)(116,171,194)(117,144,199)(118,141,200)(119,142,197)(120,143,198)(121,180,203)(122,177,204)(123,178,201)(124,179,202)(125,152,207)(126,149,208)(127,150,205)(128,151,206)(129,156,176)(130,153,173)(131,154,174)(132,155,175)(133,160,215)(134,157,216)(135,158,213)(136,159,214)(137,164,184)(138,161,181)(139,162,182)(140,163,183)(145,165,192)(146,166,189)(147,167,190)(148,168,191), (1,83,7)(2,84,8)(3,81,5)(4,82,6)(9,60,35)(10,57,36)(11,58,33)(12,59,34)(13,92,66)(14,89,67)(15,90,68)(16,91,65)(17,43,93)(18,44,94)(19,41,95)(20,42,96)(21,72,47)(22,69,48)(23,70,45)(24,71,46)(25,104,78)(26,101,79)(27,102,80)(28,103,77)(29,107,112)(30,108,109)(31,105,110)(32,106,111)(37,61,87)(38,62,88)(39,63,85)(40,64,86)(49,73,99)(50,74,100)(51,75,97)(52,76,98)(53,139,213)(54,140,214)(55,137,215)(56,138,216)(113,191,144)(114,192,141)(115,189,142)(116,190,143)(117,195,168)(118,196,165)(119,193,166)(120,194,167)(121,176,152)(122,173,149)(123,174,150)(124,175,151)(125,203,156)(126,204,153)(127,201,154)(128,202,155)(129,207,180)(130,208,177)(131,205,178)(132,206,179)(133,188,164)(134,185,161)(135,186,162)(136,187,163)(145,200,169)(146,197,170)(147,198,171)(148,199,172)(157,212,181)(158,209,182)(159,210,183)(160,211,184), (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), (1,191,3,189)(2,190,4,192)(5,115,7,113)(6,114,8,116)(9,166,11,168)(10,165,12,167)(13,175,15,173)(14,174,16,176)(17,127,19,125)(18,126,20,128)(21,178,23,180)(22,177,24,179)(25,187,27,185)(26,186,28,188)(29,137,31,139)(30,140,32,138)(33,195,35,193)(34,194,36,196)(37,199,39,197)(38,198,40,200)(41,203,43,201)(42,202,44,204)(45,207,47,205)(46,206,48,208)(49,211,51,209)(50,210,52,212)(53,112,55,110)(54,111,56,109)(57,118,59,120)(58,117,60,119)(61,172,63,170)(62,171,64,169)(65,121,67,123)(66,124,68,122)(69,130,71,132)(70,129,72,131)(73,184,75,182)(74,183,76,181)(77,133,79,135)(78,136,80,134)(81,142,83,144)(82,141,84,143)(85,146,87,148)(86,145,88,147)(89,150,91,152)(90,149,92,151)(93,154,95,156)(94,153,96,155)(97,158,99,160)(98,157,100,159)(101,162,103,164)(102,161,104,163)(105,213,107,215)(106,216,108,214) );
G=PermutationGroup([[(1,112,19),(2,109,20),(3,110,17),(4,111,18),(5,105,93),(6,106,94),(7,107,95),(8,108,96),(9,51,47),(10,52,48),(11,49,45),(12,50,46),(13,62,78),(14,63,79),(15,64,80),(16,61,77),(21,60,75),(22,57,76),(23,58,73),(24,59,74),(25,92,88),(26,89,85),(27,90,86),(28,91,87),(29,41,83),(30,42,84),(31,43,81),(32,44,82),(33,99,70),(34,100,71),(35,97,72),(36,98,69),(37,103,65),(38,104,66),(39,101,67),(40,102,68),(53,127,189),(54,128,190),(55,125,191),(56,126,192),(113,215,156),(114,216,153),(115,213,154),(116,214,155),(117,184,180),(118,181,177),(119,182,178),(120,183,179),(121,199,164),(122,200,161),(123,197,162),(124,198,163),(129,195,160),(130,196,157),(131,193,158),(132,194,159),(133,176,172),(134,173,169),(135,174,170),(136,175,171),(137,203,144),(138,204,141),(139,201,142),(140,202,143),(145,185,149),(146,186,150),(147,187,151),(148,188,152),(165,212,208),(166,209,205),(167,210,206),(168,211,207)], [(1,87,11),(2,88,12),(3,85,9),(4,86,10),(5,63,35),(6,64,36),(7,61,33),(8,62,34),(13,71,96),(14,72,93),(15,69,94),(16,70,95),(17,89,47),(18,90,48),(19,91,45),(20,92,46),(21,43,67),(22,44,68),(23,41,65),(24,42,66),(25,50,109),(26,51,110),(27,52,111),(28,49,112),(29,103,73),(30,104,74),(31,101,75),(32,102,76),(37,58,83),(38,59,84),(39,60,81),(40,57,82),(53,186,209),(54,187,210),(55,188,211),(56,185,212),(77,99,107),(78,100,108),(79,97,105),(80,98,106),(113,172,195),(114,169,196),(115,170,193),(116,171,194),(117,144,199),(118,141,200),(119,142,197),(120,143,198),(121,180,203),(122,177,204),(123,178,201),(124,179,202),(125,152,207),(126,149,208),(127,150,205),(128,151,206),(129,156,176),(130,153,173),(131,154,174),(132,155,175),(133,160,215),(134,157,216),(135,158,213),(136,159,214),(137,164,184),(138,161,181),(139,162,182),(140,163,183),(145,165,192),(146,166,189),(147,167,190),(148,168,191)], [(1,83,7),(2,84,8),(3,81,5),(4,82,6),(9,60,35),(10,57,36),(11,58,33),(12,59,34),(13,92,66),(14,89,67),(15,90,68),(16,91,65),(17,43,93),(18,44,94),(19,41,95),(20,42,96),(21,72,47),(22,69,48),(23,70,45),(24,71,46),(25,104,78),(26,101,79),(27,102,80),(28,103,77),(29,107,112),(30,108,109),(31,105,110),(32,106,111),(37,61,87),(38,62,88),(39,63,85),(40,64,86),(49,73,99),(50,74,100),(51,75,97),(52,76,98),(53,139,213),(54,140,214),(55,137,215),(56,138,216),(113,191,144),(114,192,141),(115,189,142),(116,190,143),(117,195,168),(118,196,165),(119,193,166),(120,194,167),(121,176,152),(122,173,149),(123,174,150),(124,175,151),(125,203,156),(126,204,153),(127,201,154),(128,202,155),(129,207,180),(130,208,177),(131,205,178),(132,206,179),(133,188,164),(134,185,161),(135,186,162),(136,187,163),(145,200,169),(146,197,170),(147,198,171),(148,199,172),(157,212,181),(158,209,182),(159,210,183),(160,211,184)], [(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)], [(1,191,3,189),(2,190,4,192),(5,115,7,113),(6,114,8,116),(9,166,11,168),(10,165,12,167),(13,175,15,173),(14,174,16,176),(17,127,19,125),(18,126,20,128),(21,178,23,180),(22,177,24,179),(25,187,27,185),(26,186,28,188),(29,137,31,139),(30,140,32,138),(33,195,35,193),(34,194,36,196),(37,199,39,197),(38,198,40,200),(41,203,43,201),(42,202,44,204),(45,207,47,205),(46,206,48,208),(49,211,51,209),(50,210,52,212),(53,112,55,110),(54,111,56,109),(57,118,59,120),(58,117,60,119),(61,172,63,170),(62,171,64,169),(65,121,67,123),(66,124,68,122),(69,130,71,132),(70,129,72,131),(73,184,75,182),(74,183,76,181),(77,133,79,135),(78,136,80,134),(81,142,83,144),(82,141,84,143),(85,146,87,148),(86,145,88,147),(89,150,91,152),(90,149,92,151),(93,154,95,156),(94,153,96,155),(97,158,99,160),(98,157,100,159),(101,162,103,164),(102,161,104,163),(105,213,107,215),(106,216,108,214)]])
Q8×C33 is a maximal subgroup of
C33⋊27SD16 C33⋊15Q16 (Q8×C33)⋊C2
135 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3Z | 4A | 4B | 4C | 6A | ··· | 6Z | 12A | ··· | 12BZ |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | - | |||
| image | C1 | C2 | C3 | C6 | Q8 | C3×Q8 |
| kernel | Q8×C33 | C32×C12 | Q8×C32 | C3×C12 | C33 | C32 |
| # reps | 1 | 3 | 26 | 78 | 1 | 26 |
Matrix representation of Q8×C33 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 9 | 3 |
| 0 | 0 | 3 | 4 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,12,0,0,0,0,9,3,0,0,3,4] >;
Q8×C33 in GAP, Magma, Sage, TeX
Q_8\times C_3^3
% in TeX
G:=Group("Q8xC3^3"); // GroupNames label
G:=SmallGroup(216,152);
// by ID
G=gap.SmallGroup(216,152);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-2,648,1321,655]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations