metabelian, supersoluble, monomial
Aliases: C33⋊8Q8, C32⋊9Dic6, (C3×C6).62D6, C12.3(C3⋊S3), (C3×C12).11S3, C4.(C33⋊C2), (C32×C12).1C2, C33⋊5C4.2C2, C3⋊2(C32⋊4Q8), (C32×C6).26C22, C6.14(C2×C3⋊S3), C2.3(C2×C33⋊C2), SmallGroup(216,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C33⋊8Q8
G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, eae-1=a-1, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 636 in 168 conjugacy classes, 87 normal (7 characteristic)
C1, C2, C3, C4, C4, C6, Q8, C32, Dic3, C12, C3×C6, Dic6, C33, C3⋊Dic3, C3×C12, C32×C6, C32⋊4Q8, C33⋊5C4, C32×C12, C33⋊8Q8
Quotients: C1, C2, C22, S3, Q8, D6, C3⋊S3, Dic6, C2×C3⋊S3, C33⋊C2, C32⋊4Q8, C2×C33⋊C2, C33⋊8Q8
►Regular action on 216 points
Generators in S216
(1 168 74)(2 165 75)(3 166 76)(4 167 73)(5 22 71)(6 23 72)(7 24 69)(8 21 70)(9 110 47)(10 111 48)(11 112 45)(12 109 46)(13 61 49)(14 62 50)(15 63 51)(16 64 52)(17 68 138)(18 65 139)(19 66 140)(20 67 137)(25 117 207)(26 118 208)(27 119 205)(28 120 206)(29 78 90)(30 79 91)(31 80 92)(32 77 89)(33 106 96)(34 107 93)(35 108 94)(36 105 95)(37 57 98)(38 58 99)(39 59 100)(40 60 97)(41 85 101)(42 86 102)(43 87 103)(44 88 104)(53 199 185)(54 200 186)(55 197 187)(56 198 188)(81 177 159)(82 178 160)(83 179 157)(84 180 158)(113 204 192)(114 201 189)(115 202 190)(116 203 191)(121 211 142)(122 212 143)(123 209 144)(124 210 141)(125 216 145)(126 213 146)(127 214 147)(128 215 148)(129 172 151)(130 169 152)(131 170 149)(132 171 150)(133 174 154)(134 175 155)(135 176 156)(136 173 153)(161 194 181)(162 195 182)(163 196 183)(164 193 184)
(1 49 60)(2 50 57)(3 51 58)(4 52 59)(5 47 107)(6 48 108)(7 45 105)(8 46 106)(9 93 22)(10 94 23)(11 95 24)(12 96 21)(13 97 168)(14 98 165)(15 99 166)(16 100 167)(17 104 91)(18 101 92)(19 102 89)(20 103 90)(25 130 183)(26 131 184)(27 132 181)(28 129 182)(29 67 43)(30 68 44)(31 65 41)(32 66 42)(33 70 109)(34 71 110)(35 72 111)(36 69 112)(37 75 62)(38 76 63)(39 73 64)(40 74 61)(53 212 174)(54 209 175)(55 210 176)(56 211 173)(77 140 86)(78 137 87)(79 138 88)(80 139 85)(81 204 148)(82 201 145)(83 202 146)(84 203 147)(113 215 159)(114 216 160)(115 213 157)(116 214 158)(117 169 163)(118 170 164)(119 171 161)(120 172 162)(121 136 188)(122 133 185)(123 134 186)(124 135 187)(125 178 189)(126 179 190)(127 180 191)(128 177 192)(141 156 197)(142 153 198)(143 154 199)(144 155 200)(149 193 208)(150 194 205)(151 195 206)(152 196 207)
(1 22 43)(2 23 44)(3 24 41)(4 21 42)(5 103 74)(6 104 75)(7 101 76)(8 102 73)(9 29 49)(10 30 50)(11 31 51)(12 32 52)(13 110 78)(14 111 79)(15 112 80)(16 109 77)(17 37 108)(18 38 105)(19 39 106)(20 40 107)(25 173 202)(26 174 203)(27 175 204)(28 176 201)(33 140 100)(34 137 97)(35 138 98)(36 139 99)(45 92 63)(46 89 64)(47 90 61)(48 91 62)(53 147 131)(54 148 132)(55 145 129)(56 146 130)(57 94 68)(58 95 65)(59 96 66)(60 93 67)(69 85 166)(70 86 167)(71 87 168)(72 88 165)(81 181 209)(82 182 210)(83 183 211)(84 184 212)(113 205 134)(114 206 135)(115 207 136)(116 208 133)(117 153 190)(118 154 191)(119 155 192)(120 156 189)(121 157 196)(122 158 193)(123 159 194)(124 160 195)(125 172 197)(126 169 198)(127 170 199)(128 171 200)(141 178 162)(142 179 163)(143 180 164)(144 177 161)(149 185 214)(150 186 215)(151 187 216)(152 188 213)
(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)
(1 119 3 117)(2 118 4 120)(5 113 7 115)(6 116 8 114)(9 177 11 179)(10 180 12 178)(13 181 15 183)(14 184 16 182)(17 185 19 187)(18 188 20 186)(21 189 23 191)(22 192 24 190)(25 168 27 166)(26 167 28 165)(29 144 31 142)(30 143 32 141)(33 145 35 147)(34 148 36 146)(37 149 39 151)(38 152 40 150)(41 153 43 155)(42 156 44 154)(45 157 47 159)(46 160 48 158)(49 161 51 163)(50 164 52 162)(53 140 55 138)(54 139 56 137)(57 170 59 172)(58 169 60 171)(61 194 63 196)(62 193 64 195)(65 198 67 200)(66 197 68 199)(69 202 71 204)(70 201 72 203)(73 206 75 208)(74 205 76 207)(77 210 79 212)(78 209 80 211)(81 112 83 110)(82 111 84 109)(85 173 87 175)(86 176 88 174)(89 124 91 122)(90 123 92 121)(93 128 95 126)(94 127 96 125)(97 132 99 130)(98 131 100 129)(101 136 103 134)(102 135 104 133)(105 213 107 215)(106 216 108 214)
G:=sub<Sym(216)| (1,168,74)(2,165,75)(3,166,76)(4,167,73)(5,22,71)(6,23,72)(7,24,69)(8,21,70)(9,110,47)(10,111,48)(11,112,45)(12,109,46)(13,61,49)(14,62,50)(15,63,51)(16,64,52)(17,68,138)(18,65,139)(19,66,140)(20,67,137)(25,117,207)(26,118,208)(27,119,205)(28,120,206)(29,78,90)(30,79,91)(31,80,92)(32,77,89)(33,106,96)(34,107,93)(35,108,94)(36,105,95)(37,57,98)(38,58,99)(39,59,100)(40,60,97)(41,85,101)(42,86,102)(43,87,103)(44,88,104)(53,199,185)(54,200,186)(55,197,187)(56,198,188)(81,177,159)(82,178,160)(83,179,157)(84,180,158)(113,204,192)(114,201,189)(115,202,190)(116,203,191)(121,211,142)(122,212,143)(123,209,144)(124,210,141)(125,216,145)(126,213,146)(127,214,147)(128,215,148)(129,172,151)(130,169,152)(131,170,149)(132,171,150)(133,174,154)(134,175,155)(135,176,156)(136,173,153)(161,194,181)(162,195,182)(163,196,183)(164,193,184), (1,49,60)(2,50,57)(3,51,58)(4,52,59)(5,47,107)(6,48,108)(7,45,105)(8,46,106)(9,93,22)(10,94,23)(11,95,24)(12,96,21)(13,97,168)(14,98,165)(15,99,166)(16,100,167)(17,104,91)(18,101,92)(19,102,89)(20,103,90)(25,130,183)(26,131,184)(27,132,181)(28,129,182)(29,67,43)(30,68,44)(31,65,41)(32,66,42)(33,70,109)(34,71,110)(35,72,111)(36,69,112)(37,75,62)(38,76,63)(39,73,64)(40,74,61)(53,212,174)(54,209,175)(55,210,176)(56,211,173)(77,140,86)(78,137,87)(79,138,88)(80,139,85)(81,204,148)(82,201,145)(83,202,146)(84,203,147)(113,215,159)(114,216,160)(115,213,157)(116,214,158)(117,169,163)(118,170,164)(119,171,161)(120,172,162)(121,136,188)(122,133,185)(123,134,186)(124,135,187)(125,178,189)(126,179,190)(127,180,191)(128,177,192)(141,156,197)(142,153,198)(143,154,199)(144,155,200)(149,193,208)(150,194,205)(151,195,206)(152,196,207), (1,22,43)(2,23,44)(3,24,41)(4,21,42)(5,103,74)(6,104,75)(7,101,76)(8,102,73)(9,29,49)(10,30,50)(11,31,51)(12,32,52)(13,110,78)(14,111,79)(15,112,80)(16,109,77)(17,37,108)(18,38,105)(19,39,106)(20,40,107)(25,173,202)(26,174,203)(27,175,204)(28,176,201)(33,140,100)(34,137,97)(35,138,98)(36,139,99)(45,92,63)(46,89,64)(47,90,61)(48,91,62)(53,147,131)(54,148,132)(55,145,129)(56,146,130)(57,94,68)(58,95,65)(59,96,66)(60,93,67)(69,85,166)(70,86,167)(71,87,168)(72,88,165)(81,181,209)(82,182,210)(83,183,211)(84,184,212)(113,205,134)(114,206,135)(115,207,136)(116,208,133)(117,153,190)(118,154,191)(119,155,192)(120,156,189)(121,157,196)(122,158,193)(123,159,194)(124,160,195)(125,172,197)(126,169,198)(127,170,199)(128,171,200)(141,178,162)(142,179,163)(143,180,164)(144,177,161)(149,185,214)(150,186,215)(151,187,216)(152,188,213), (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), (1,119,3,117)(2,118,4,120)(5,113,7,115)(6,116,8,114)(9,177,11,179)(10,180,12,178)(13,181,15,183)(14,184,16,182)(17,185,19,187)(18,188,20,186)(21,189,23,191)(22,192,24,190)(25,168,27,166)(26,167,28,165)(29,144,31,142)(30,143,32,141)(33,145,35,147)(34,148,36,146)(37,149,39,151)(38,152,40,150)(41,153,43,155)(42,156,44,154)(45,157,47,159)(46,160,48,158)(49,161,51,163)(50,164,52,162)(53,140,55,138)(54,139,56,137)(57,170,59,172)(58,169,60,171)(61,194,63,196)(62,193,64,195)(65,198,67,200)(66,197,68,199)(69,202,71,204)(70,201,72,203)(73,206,75,208)(74,205,76,207)(77,210,79,212)(78,209,80,211)(81,112,83,110)(82,111,84,109)(85,173,87,175)(86,176,88,174)(89,124,91,122)(90,123,92,121)(93,128,95,126)(94,127,96,125)(97,132,99,130)(98,131,100,129)(101,136,103,134)(102,135,104,133)(105,213,107,215)(106,216,108,214)>;
G:=Group( (1,168,74)(2,165,75)(3,166,76)(4,167,73)(5,22,71)(6,23,72)(7,24,69)(8,21,70)(9,110,47)(10,111,48)(11,112,45)(12,109,46)(13,61,49)(14,62,50)(15,63,51)(16,64,52)(17,68,138)(18,65,139)(19,66,140)(20,67,137)(25,117,207)(26,118,208)(27,119,205)(28,120,206)(29,78,90)(30,79,91)(31,80,92)(32,77,89)(33,106,96)(34,107,93)(35,108,94)(36,105,95)(37,57,98)(38,58,99)(39,59,100)(40,60,97)(41,85,101)(42,86,102)(43,87,103)(44,88,104)(53,199,185)(54,200,186)(55,197,187)(56,198,188)(81,177,159)(82,178,160)(83,179,157)(84,180,158)(113,204,192)(114,201,189)(115,202,190)(116,203,191)(121,211,142)(122,212,143)(123,209,144)(124,210,141)(125,216,145)(126,213,146)(127,214,147)(128,215,148)(129,172,151)(130,169,152)(131,170,149)(132,171,150)(133,174,154)(134,175,155)(135,176,156)(136,173,153)(161,194,181)(162,195,182)(163,196,183)(164,193,184), (1,49,60)(2,50,57)(3,51,58)(4,52,59)(5,47,107)(6,48,108)(7,45,105)(8,46,106)(9,93,22)(10,94,23)(11,95,24)(12,96,21)(13,97,168)(14,98,165)(15,99,166)(16,100,167)(17,104,91)(18,101,92)(19,102,89)(20,103,90)(25,130,183)(26,131,184)(27,132,181)(28,129,182)(29,67,43)(30,68,44)(31,65,41)(32,66,42)(33,70,109)(34,71,110)(35,72,111)(36,69,112)(37,75,62)(38,76,63)(39,73,64)(40,74,61)(53,212,174)(54,209,175)(55,210,176)(56,211,173)(77,140,86)(78,137,87)(79,138,88)(80,139,85)(81,204,148)(82,201,145)(83,202,146)(84,203,147)(113,215,159)(114,216,160)(115,213,157)(116,214,158)(117,169,163)(118,170,164)(119,171,161)(120,172,162)(121,136,188)(122,133,185)(123,134,186)(124,135,187)(125,178,189)(126,179,190)(127,180,191)(128,177,192)(141,156,197)(142,153,198)(143,154,199)(144,155,200)(149,193,208)(150,194,205)(151,195,206)(152,196,207), (1,22,43)(2,23,44)(3,24,41)(4,21,42)(5,103,74)(6,104,75)(7,101,76)(8,102,73)(9,29,49)(10,30,50)(11,31,51)(12,32,52)(13,110,78)(14,111,79)(15,112,80)(16,109,77)(17,37,108)(18,38,105)(19,39,106)(20,40,107)(25,173,202)(26,174,203)(27,175,204)(28,176,201)(33,140,100)(34,137,97)(35,138,98)(36,139,99)(45,92,63)(46,89,64)(47,90,61)(48,91,62)(53,147,131)(54,148,132)(55,145,129)(56,146,130)(57,94,68)(58,95,65)(59,96,66)(60,93,67)(69,85,166)(70,86,167)(71,87,168)(72,88,165)(81,181,209)(82,182,210)(83,183,211)(84,184,212)(113,205,134)(114,206,135)(115,207,136)(116,208,133)(117,153,190)(118,154,191)(119,155,192)(120,156,189)(121,157,196)(122,158,193)(123,159,194)(124,160,195)(125,172,197)(126,169,198)(127,170,199)(128,171,200)(141,178,162)(142,179,163)(143,180,164)(144,177,161)(149,185,214)(150,186,215)(151,187,216)(152,188,213), (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), (1,119,3,117)(2,118,4,120)(5,113,7,115)(6,116,8,114)(9,177,11,179)(10,180,12,178)(13,181,15,183)(14,184,16,182)(17,185,19,187)(18,188,20,186)(21,189,23,191)(22,192,24,190)(25,168,27,166)(26,167,28,165)(29,144,31,142)(30,143,32,141)(33,145,35,147)(34,148,36,146)(37,149,39,151)(38,152,40,150)(41,153,43,155)(42,156,44,154)(45,157,47,159)(46,160,48,158)(49,161,51,163)(50,164,52,162)(53,140,55,138)(54,139,56,137)(57,170,59,172)(58,169,60,171)(61,194,63,196)(62,193,64,195)(65,198,67,200)(66,197,68,199)(69,202,71,204)(70,201,72,203)(73,206,75,208)(74,205,76,207)(77,210,79,212)(78,209,80,211)(81,112,83,110)(82,111,84,109)(85,173,87,175)(86,176,88,174)(89,124,91,122)(90,123,92,121)(93,128,95,126)(94,127,96,125)(97,132,99,130)(98,131,100,129)(101,136,103,134)(102,135,104,133)(105,213,107,215)(106,216,108,214) );
G=PermutationGroup([[(1,168,74),(2,165,75),(3,166,76),(4,167,73),(5,22,71),(6,23,72),(7,24,69),(8,21,70),(9,110,47),(10,111,48),(11,112,45),(12,109,46),(13,61,49),(14,62,50),(15,63,51),(16,64,52),(17,68,138),(18,65,139),(19,66,140),(20,67,137),(25,117,207),(26,118,208),(27,119,205),(28,120,206),(29,78,90),(30,79,91),(31,80,92),(32,77,89),(33,106,96),(34,107,93),(35,108,94),(36,105,95),(37,57,98),(38,58,99),(39,59,100),(40,60,97),(41,85,101),(42,86,102),(43,87,103),(44,88,104),(53,199,185),(54,200,186),(55,197,187),(56,198,188),(81,177,159),(82,178,160),(83,179,157),(84,180,158),(113,204,192),(114,201,189),(115,202,190),(116,203,191),(121,211,142),(122,212,143),(123,209,144),(124,210,141),(125,216,145),(126,213,146),(127,214,147),(128,215,148),(129,172,151),(130,169,152),(131,170,149),(132,171,150),(133,174,154),(134,175,155),(135,176,156),(136,173,153),(161,194,181),(162,195,182),(163,196,183),(164,193,184)], [(1,49,60),(2,50,57),(3,51,58),(4,52,59),(5,47,107),(6,48,108),(7,45,105),(8,46,106),(9,93,22),(10,94,23),(11,95,24),(12,96,21),(13,97,168),(14,98,165),(15,99,166),(16,100,167),(17,104,91),(18,101,92),(19,102,89),(20,103,90),(25,130,183),(26,131,184),(27,132,181),(28,129,182),(29,67,43),(30,68,44),(31,65,41),(32,66,42),(33,70,109),(34,71,110),(35,72,111),(36,69,112),(37,75,62),(38,76,63),(39,73,64),(40,74,61),(53,212,174),(54,209,175),(55,210,176),(56,211,173),(77,140,86),(78,137,87),(79,138,88),(80,139,85),(81,204,148),(82,201,145),(83,202,146),(84,203,147),(113,215,159),(114,216,160),(115,213,157),(116,214,158),(117,169,163),(118,170,164),(119,171,161),(120,172,162),(121,136,188),(122,133,185),(123,134,186),(124,135,187),(125,178,189),(126,179,190),(127,180,191),(128,177,192),(141,156,197),(142,153,198),(143,154,199),(144,155,200),(149,193,208),(150,194,205),(151,195,206),(152,196,207)], [(1,22,43),(2,23,44),(3,24,41),(4,21,42),(5,103,74),(6,104,75),(7,101,76),(8,102,73),(9,29,49),(10,30,50),(11,31,51),(12,32,52),(13,110,78),(14,111,79),(15,112,80),(16,109,77),(17,37,108),(18,38,105),(19,39,106),(20,40,107),(25,173,202),(26,174,203),(27,175,204),(28,176,201),(33,140,100),(34,137,97),(35,138,98),(36,139,99),(45,92,63),(46,89,64),(47,90,61),(48,91,62),(53,147,131),(54,148,132),(55,145,129),(56,146,130),(57,94,68),(58,95,65),(59,96,66),(60,93,67),(69,85,166),(70,86,167),(71,87,168),(72,88,165),(81,181,209),(82,182,210),(83,183,211),(84,184,212),(113,205,134),(114,206,135),(115,207,136),(116,208,133),(117,153,190),(118,154,191),(119,155,192),(120,156,189),(121,157,196),(122,158,193),(123,159,194),(124,160,195),(125,172,197),(126,169,198),(127,170,199),(128,171,200),(141,178,162),(142,179,163),(143,180,164),(144,177,161),(149,185,214),(150,186,215),(151,187,216),(152,188,213)], [(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)], [(1,119,3,117),(2,118,4,120),(5,113,7,115),(6,116,8,114),(9,177,11,179),(10,180,12,178),(13,181,15,183),(14,184,16,182),(17,185,19,187),(18,188,20,186),(21,189,23,191),(22,192,24,190),(25,168,27,166),(26,167,28,165),(29,144,31,142),(30,143,32,141),(33,145,35,147),(34,148,36,146),(37,149,39,151),(38,152,40,150),(41,153,43,155),(42,156,44,154),(45,157,47,159),(46,160,48,158),(49,161,51,163),(50,164,52,162),(53,140,55,138),(54,139,56,137),(57,170,59,172),(58,169,60,171),(61,194,63,196),(62,193,64,195),(65,198,67,200),(66,197,68,199),(69,202,71,204),(70,201,72,203),(73,206,75,208),(74,205,76,207),(77,210,79,212),(78,209,80,211),(81,112,83,110),(82,111,84,109),(85,173,87,175),(86,176,88,174),(89,124,91,122),(90,123,92,121),(93,128,95,126),(94,127,96,125),(97,132,99,130),(98,131,100,129),(101,136,103,134),(102,135,104,133),(105,213,107,215),(106,216,108,214)]])
C33⋊8Q8 is a maximal subgroup of
C33⋊14SD16 C33⋊16SD16 C33⋊7Q16 C33⋊8Q16 C33⋊21SD16 C33⋊12Q16 C33⋊24SD16 C33⋊15Q16 S3×C32⋊4Q8 (C3×D12)⋊S3 C3⋊S3×Dic6 C12.57S32 C62.160D6 C62.100D6 Q8×C33⋊C2
C33⋊8Q8 is a maximal quotient of
C62.146D6 C62.147D6
57 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3M | 4A | 4B | 4C | 6A | ··· | 6M | 12A | ··· | 12Z |
| order | 1 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | ··· | 2 | 2 | 54 | 54 | 2 | ··· | 2 | 2 | ··· | 2 |
57 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | Q8 | D6 | Dic6 |
| kernel | C33⋊8Q8 | C33⋊5C4 | C32×C12 | C3×C12 | C33 | C3×C6 | C32 |
| # reps | 1 | 2 | 1 | 13 | 1 | 13 | 26 |
Matrix representation of C33⋊8Q8 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 6 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 9 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 0 | 3 | 10 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[3,6,0,0,0,0,7,10,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,9,0,0,0,0,11,2,0,0,0,0,0,0,8,0,0,0,0,0,8,5,0,0,0,0,0,0,3,3,0,0,0,0,6,10] >;
C33⋊8Q8 in GAP, Magma, Sage, TeX
C_3^3\rtimes_8Q_8
% in TeX
G:=Group("C3^3:8Q8"); // GroupNames label
G:=SmallGroup(216,145);
// by ID
G=gap.SmallGroup(216,145);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,31,387,1444,5189]);
// 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,e*a*e^-1=a^-1,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations