direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C3⋊D33, C6⋊D33, C66⋊1S3, C3⋊2D66, C33⋊6D6, C32⋊6D22, C22⋊(C3⋊S3), (C3×C66)⋊1C2, (C3×C6)⋊2D11, (C3×C33)⋊6C22, C11⋊2(C2×C3⋊S3), SmallGroup(396,29)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C33 — C2×C3⋊D33 |
Generators and relations for C2×C3⋊D33
G = < a,b,c,d | a2=b3=c33=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 816 in 60 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C3, C22, S3, C6, C32, C11, D6, C3⋊S3, C3×C6, D11, C22, C33, C2×C3⋊S3, D22, D33, C66, C3×C33, D66, C3⋊D33, C3×C66, C2×C3⋊D33
Quotients: C1, C2, C22, S3, D6, C3⋊S3, D11, C2×C3⋊S3, D22, D33, D66, C3⋊D33, C2×C3⋊D33
►On 198 points
Generators in S198
(1 129)(2 130)(3 131)(4 132)(5 100)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 128)(34 160)(35 161)(36 162)(37 163)(38 164)(39 165)(40 133)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 141)(49 142)(50 143)(51 144)(52 145)(53 146)(54 147)(55 148)(56 149)(57 150)(58 151)(59 152)(60 153)(61 154)(62 155)(63 156)(64 157)(65 158)(66 159)(67 184)(68 185)(69 186)(70 187)(71 188)(72 189)(73 190)(74 191)(75 192)(76 193)(77 194)(78 195)(79 196)(80 197)(81 198)(82 166)(83 167)(84 168)(85 169)(86 170)(87 171)(88 172)(89 173)(90 174)(91 175)(92 176)(93 177)(94 178)(95 179)(96 180)(97 181)(98 182)(99 183)
(1 71 37)(2 72 38)(3 73 39)(4 74 40)(5 75 41)(6 76 42)(7 77 43)(8 78 44)(9 79 45)(10 80 46)(11 81 47)(12 82 48)(13 83 49)(14 84 50)(15 85 51)(16 86 52)(17 87 53)(18 88 54)(19 89 55)(20 90 56)(21 91 57)(22 92 58)(23 93 59)(24 94 60)(25 95 61)(26 96 62)(27 97 63)(28 98 64)(29 99 65)(30 67 66)(31 68 34)(32 69 35)(33 70 36)(100 192 134)(101 193 135)(102 194 136)(103 195 137)(104 196 138)(105 197 139)(106 198 140)(107 166 141)(108 167 142)(109 168 143)(110 169 144)(111 170 145)(112 171 146)(113 172 147)(114 173 148)(115 174 149)(116 175 150)(117 176 151)(118 177 152)(119 178 153)(120 179 154)(121 180 155)(122 181 156)(123 182 157)(124 183 158)(125 184 159)(126 185 160)(127 186 161)(128 187 162)(129 188 163)(130 189 164)(131 190 165)(132 191 133)
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)(41 99)(42 98)(43 97)(44 96)(45 95)(46 94)(47 93)(48 92)(49 91)(50 90)(51 89)(52 88)(53 87)(54 86)(55 85)(56 84)(57 83)(58 82)(59 81)(60 80)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(100 124)(101 123)(102 122)(103 121)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(125 132)(126 131)(127 130)(128 129)(133 184)(134 183)(135 182)(136 181)(137 180)(138 179)(139 178)(140 177)(141 176)(142 175)(143 174)(144 173)(145 172)(146 171)(147 170)(148 169)(149 168)(150 167)(151 166)(152 198)(153 197)(154 196)(155 195)(156 194)(157 193)(158 192)(159 191)(160 190)(161 189)(162 188)(163 187)(164 186)(165 185)
G:=sub<Sym(198)| (1,129)(2,130)(3,131)(4,132)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,160)(35,161)(36,162)(37,163)(38,164)(39,165)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,144)(52,145)(53,146)(54,147)(55,148)(56,149)(57,150)(58,151)(59,152)(60,153)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,184)(68,185)(69,186)(70,187)(71,188)(72,189)(73,190)(74,191)(75,192)(76,193)(77,194)(78,195)(79,196)(80,197)(81,198)(82,166)(83,167)(84,168)(85,169)(86,170)(87,171)(88,172)(89,173)(90,174)(91,175)(92,176)(93,177)(94,178)(95,179)(96,180)(97,181)(98,182)(99,183), (1,71,37)(2,72,38)(3,73,39)(4,74,40)(5,75,41)(6,76,42)(7,77,43)(8,78,44)(9,79,45)(10,80,46)(11,81,47)(12,82,48)(13,83,49)(14,84,50)(15,85,51)(16,86,52)(17,87,53)(18,88,54)(19,89,55)(20,90,56)(21,91,57)(22,92,58)(23,93,59)(24,94,60)(25,95,61)(26,96,62)(27,97,63)(28,98,64)(29,99,65)(30,67,66)(31,68,34)(32,69,35)(33,70,36)(100,192,134)(101,193,135)(102,194,136)(103,195,137)(104,196,138)(105,197,139)(106,198,140)(107,166,141)(108,167,142)(109,168,143)(110,169,144)(111,170,145)(112,171,146)(113,172,147)(114,173,148)(115,174,149)(116,175,150)(117,176,151)(118,177,152)(119,178,153)(120,179,154)(121,180,155)(122,181,156)(123,182,157)(124,183,158)(125,184,159)(126,185,160)(127,186,161)(128,187,162)(129,188,163)(130,189,164)(131,190,165)(132,191,133), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,99)(42,98)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(100,124)(101,123)(102,122)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(125,132)(126,131)(127,130)(128,129)(133,184)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,171)(147,170)(148,169)(149,168)(150,167)(151,166)(152,198)(153,197)(154,196)(155,195)(156,194)(157,193)(158,192)(159,191)(160,190)(161,189)(162,188)(163,187)(164,186)(165,185)>;
G:=Group( (1,129)(2,130)(3,131)(4,132)(5,100)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,160)(35,161)(36,162)(37,163)(38,164)(39,165)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(49,142)(50,143)(51,144)(52,145)(53,146)(54,147)(55,148)(56,149)(57,150)(58,151)(59,152)(60,153)(61,154)(62,155)(63,156)(64,157)(65,158)(66,159)(67,184)(68,185)(69,186)(70,187)(71,188)(72,189)(73,190)(74,191)(75,192)(76,193)(77,194)(78,195)(79,196)(80,197)(81,198)(82,166)(83,167)(84,168)(85,169)(86,170)(87,171)(88,172)(89,173)(90,174)(91,175)(92,176)(93,177)(94,178)(95,179)(96,180)(97,181)(98,182)(99,183), (1,71,37)(2,72,38)(3,73,39)(4,74,40)(5,75,41)(6,76,42)(7,77,43)(8,78,44)(9,79,45)(10,80,46)(11,81,47)(12,82,48)(13,83,49)(14,84,50)(15,85,51)(16,86,52)(17,87,53)(18,88,54)(19,89,55)(20,90,56)(21,91,57)(22,92,58)(23,93,59)(24,94,60)(25,95,61)(26,96,62)(27,97,63)(28,98,64)(29,99,65)(30,67,66)(31,68,34)(32,69,35)(33,70,36)(100,192,134)(101,193,135)(102,194,136)(103,195,137)(104,196,138)(105,197,139)(106,198,140)(107,166,141)(108,167,142)(109,168,143)(110,169,144)(111,170,145)(112,171,146)(113,172,147)(114,173,148)(115,174,149)(116,175,150)(117,176,151)(118,177,152)(119,178,153)(120,179,154)(121,180,155)(122,181,156)(123,182,157)(124,183,158)(125,184,159)(126,185,160)(127,186,161)(128,187,162)(129,188,163)(130,189,164)(131,190,165)(132,191,133), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,99)(42,98)(43,97)(44,96)(45,95)(46,94)(47,93)(48,92)(49,91)(50,90)(51,89)(52,88)(53,87)(54,86)(55,85)(56,84)(57,83)(58,82)(59,81)(60,80)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(100,124)(101,123)(102,122)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(125,132)(126,131)(127,130)(128,129)(133,184)(134,183)(135,182)(136,181)(137,180)(138,179)(139,178)(140,177)(141,176)(142,175)(143,174)(144,173)(145,172)(146,171)(147,170)(148,169)(149,168)(150,167)(151,166)(152,198)(153,197)(154,196)(155,195)(156,194)(157,193)(158,192)(159,191)(160,190)(161,189)(162,188)(163,187)(164,186)(165,185) );
G=PermutationGroup([[(1,129),(2,130),(3,131),(4,132),(5,100),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,121),(27,122),(28,123),(29,124),(30,125),(31,126),(32,127),(33,128),(34,160),(35,161),(36,162),(37,163),(38,164),(39,165),(40,133),(41,134),(42,135),(43,136),(44,137),(45,138),(46,139),(47,140),(48,141),(49,142),(50,143),(51,144),(52,145),(53,146),(54,147),(55,148),(56,149),(57,150),(58,151),(59,152),(60,153),(61,154),(62,155),(63,156),(64,157),(65,158),(66,159),(67,184),(68,185),(69,186),(70,187),(71,188),(72,189),(73,190),(74,191),(75,192),(76,193),(77,194),(78,195),(79,196),(80,197),(81,198),(82,166),(83,167),(84,168),(85,169),(86,170),(87,171),(88,172),(89,173),(90,174),(91,175),(92,176),(93,177),(94,178),(95,179),(96,180),(97,181),(98,182),(99,183)], [(1,71,37),(2,72,38),(3,73,39),(4,74,40),(5,75,41),(6,76,42),(7,77,43),(8,78,44),(9,79,45),(10,80,46),(11,81,47),(12,82,48),(13,83,49),(14,84,50),(15,85,51),(16,86,52),(17,87,53),(18,88,54),(19,89,55),(20,90,56),(21,91,57),(22,92,58),(23,93,59),(24,94,60),(25,95,61),(26,96,62),(27,97,63),(28,98,64),(29,99,65),(30,67,66),(31,68,34),(32,69,35),(33,70,36),(100,192,134),(101,193,135),(102,194,136),(103,195,137),(104,196,138),(105,197,139),(106,198,140),(107,166,141),(108,167,142),(109,168,143),(110,169,144),(111,170,145),(112,171,146),(113,172,147),(114,173,148),(115,174,149),(116,175,150),(117,176,151),(118,177,152),(119,178,153),(120,179,154),(121,180,155),(122,181,156),(123,182,157),(124,183,158),(125,184,159),(126,185,160),(127,186,161),(128,187,162),(129,188,163),(130,189,164),(131,190,165),(132,191,133)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(34,73),(35,72),(36,71),(37,70),(38,69),(39,68),(40,67),(41,99),(42,98),(43,97),(44,96),(45,95),(46,94),(47,93),(48,92),(49,91),(50,90),(51,89),(52,88),(53,87),(54,86),(55,85),(56,84),(57,83),(58,82),(59,81),(60,80),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(100,124),(101,123),(102,122),(103,121),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(125,132),(126,131),(127,130),(128,129),(133,184),(134,183),(135,182),(136,181),(137,180),(138,179),(139,178),(140,177),(141,176),(142,175),(143,174),(144,173),(145,172),(146,171),(147,170),(148,169),(149,168),(150,167),(151,166),(152,198),(153,197),(154,196),(155,195),(156,194),(157,193),(158,192),(159,191),(160,190),(161,189),(162,188),(163,187),(164,186),(165,185)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 6A | 6B | 6C | 6D | 11A | ··· | 11E | 22A | ··· | 22E | 33A | ··· | 33AN | 66A | ··· | 66AN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 11 | ··· | 11 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 |
| size | 1 | 1 | 99 | 99 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | S3 | D6 | D11 | D22 | D33 | D66 |
| kernel | C2×C3⋊D33 | C3⋊D33 | C3×C66 | C66 | C33 | C3×C6 | C32 | C6 | C3 |
| # reps | 1 | 2 | 1 | 4 | 4 | 5 | 5 | 40 | 40 |
Matrix representation of C2×C3⋊D33 ►in GL4(𝔽67) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 0 |
| 0 | 0 | 0 | 66 |
| 42 | 24 | 0 | 0 |
| 28 | 24 | 0 | 0 |
| 0 | 0 | 49 | 21 |
| 0 | 0 | 46 | 17 |
| 12 | 23 | 0 | 0 |
| 38 | 45 | 0 | 0 |
| 0 | 0 | 7 | 26 |
| 0 | 0 | 41 | 28 |
| 55 | 2 | 0 | 0 |
| 29 | 12 | 0 | 0 |
| 0 | 0 | 7 | 26 |
| 0 | 0 | 60 | 60 |
G:=sub<GL(4,GF(67))| [1,0,0,0,0,1,0,0,0,0,66,0,0,0,0,66],[42,28,0,0,24,24,0,0,0,0,49,46,0,0,21,17],[12,38,0,0,23,45,0,0,0,0,7,41,0,0,26,28],[55,29,0,0,2,12,0,0,0,0,7,60,0,0,26,60] >;
C2×C3⋊D33 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes D_{33} % in TeX
G:=Group("C2xC3:D33"); // GroupNames label
G:=SmallGroup(396,29);
// by ID
G=gap.SmallGroup(396,29);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-11,122,483,9004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^3=c^33=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations