direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C32×D23, C69⋊2C6, C23⋊(C3×C6), (C3×C69)⋊3C2, SmallGroup(414,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C32×D23 |
Generators and relations for C32×D23
G = < a,b,c,d | a3=b3=c23=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C32×D23
►On 207 points
Generators in S207
(1 190 94)(2 191 95)(3 192 96)(4 193 97)(5 194 98)(6 195 99)(7 196 100)(8 197 101)(9 198 102)(10 199 103)(11 200 104)(12 201 105)(13 202 106)(14 203 107)(15 204 108)(16 205 109)(17 206 110)(18 207 111)(19 185 112)(20 186 113)(21 187 114)(22 188 115)(23 189 93)(24 157 126)(25 158 127)(26 159 128)(27 160 129)(28 161 130)(29 139 131)(30 140 132)(31 141 133)(32 142 134)(33 143 135)(34 144 136)(35 145 137)(36 146 138)(37 147 116)(38 148 117)(39 149 118)(40 150 119)(41 151 120)(42 152 121)(43 153 122)(44 154 123)(45 155 124)(46 156 125)(47 165 90)(48 166 91)(49 167 92)(50 168 70)(51 169 71)(52 170 72)(53 171 73)(54 172 74)(55 173 75)(56 174 76)(57 175 77)(58 176 78)(59 177 79)(60 178 80)(61 179 81)(62 180 82)(63 181 83)(64 182 84)(65 183 85)(66 184 86)(67 162 87)(68 163 88)(69 164 89)
(1 53 27)(2 54 28)(3 55 29)(4 56 30)(5 57 31)(6 58 32)(7 59 33)(8 60 34)(9 61 35)(10 62 36)(11 63 37)(12 64 38)(13 65 39)(14 66 40)(15 67 41)(16 68 42)(17 69 43)(18 47 44)(19 48 45)(20 49 46)(21 50 24)(22 51 25)(23 52 26)(70 126 114)(71 127 115)(72 128 93)(73 129 94)(74 130 95)(75 131 96)(76 132 97)(77 133 98)(78 134 99)(79 135 100)(80 136 101)(81 137 102)(82 138 103)(83 116 104)(84 117 105)(85 118 106)(86 119 107)(87 120 108)(88 121 109)(89 122 110)(90 123 111)(91 124 112)(92 125 113)(139 192 173)(140 193 174)(141 194 175)(142 195 176)(143 196 177)(144 197 178)(145 198 179)(146 199 180)(147 200 181)(148 201 182)(149 202 183)(150 203 184)(151 204 162)(152 205 163)(153 206 164)(154 207 165)(155 185 166)(156 186 167)(157 187 168)(158 188 169)(159 189 170)(160 190 171)(161 191 172)
(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)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 29)(25 28)(26 27)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(59 69)(60 68)(61 67)(62 66)(63 65)(70 75)(71 74)(72 73)(76 92)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(93 94)(95 115)(96 114)(97 113)(98 112)(99 111)(100 110)(101 109)(102 108)(103 107)(104 106)(116 118)(119 138)(120 137)(121 136)(122 135)(123 134)(124 133)(125 132)(126 131)(127 130)(128 129)(139 157)(140 156)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(158 161)(159 160)(162 179)(163 178)(164 177)(165 176)(166 175)(167 174)(168 173)(169 172)(170 171)(180 184)(181 183)(185 194)(186 193)(187 192)(188 191)(189 190)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)
G:=sub<Sym(207)| (1,190,94)(2,191,95)(3,192,96)(4,193,97)(5,194,98)(6,195,99)(7,196,100)(8,197,101)(9,198,102)(10,199,103)(11,200,104)(12,201,105)(13,202,106)(14,203,107)(15,204,108)(16,205,109)(17,206,110)(18,207,111)(19,185,112)(20,186,113)(21,187,114)(22,188,115)(23,189,93)(24,157,126)(25,158,127)(26,159,128)(27,160,129)(28,161,130)(29,139,131)(30,140,132)(31,141,133)(32,142,134)(33,143,135)(34,144,136)(35,145,137)(36,146,138)(37,147,116)(38,148,117)(39,149,118)(40,150,119)(41,151,120)(42,152,121)(43,153,122)(44,154,123)(45,155,124)(46,156,125)(47,165,90)(48,166,91)(49,167,92)(50,168,70)(51,169,71)(52,170,72)(53,171,73)(54,172,74)(55,173,75)(56,174,76)(57,175,77)(58,176,78)(59,177,79)(60,178,80)(61,179,81)(62,180,82)(63,181,83)(64,182,84)(65,183,85)(66,184,86)(67,162,87)(68,163,88)(69,164,89), (1,53,27)(2,54,28)(3,55,29)(4,56,30)(5,57,31)(6,58,32)(7,59,33)(8,60,34)(9,61,35)(10,62,36)(11,63,37)(12,64,38)(13,65,39)(14,66,40)(15,67,41)(16,68,42)(17,69,43)(18,47,44)(19,48,45)(20,49,46)(21,50,24)(22,51,25)(23,52,26)(70,126,114)(71,127,115)(72,128,93)(73,129,94)(74,130,95)(75,131,96)(76,132,97)(77,133,98)(78,134,99)(79,135,100)(80,136,101)(81,137,102)(82,138,103)(83,116,104)(84,117,105)(85,118,106)(86,119,107)(87,120,108)(88,121,109)(89,122,110)(90,123,111)(91,124,112)(92,125,113)(139,192,173)(140,193,174)(141,194,175)(142,195,176)(143,196,177)(144,197,178)(145,198,179)(146,199,180)(147,200,181)(148,201,182)(149,202,183)(150,203,184)(151,204,162)(152,205,163)(153,206,164)(154,207,165)(155,185,166)(156,186,167)(157,187,168)(158,188,169)(159,189,170)(160,190,171)(161,191,172), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,29)(25,28)(26,27)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(59,69)(60,68)(61,67)(62,66)(63,65)(70,75)(71,74)(72,73)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(93,94)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(116,118)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(139,157)(140,156)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(158,161)(159,160)(162,179)(163,178)(164,177)(165,176)(166,175)(167,174)(168,173)(169,172)(170,171)(180,184)(181,183)(185,194)(186,193)(187,192)(188,191)(189,190)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)>;
G:=Group( (1,190,94)(2,191,95)(3,192,96)(4,193,97)(5,194,98)(6,195,99)(7,196,100)(8,197,101)(9,198,102)(10,199,103)(11,200,104)(12,201,105)(13,202,106)(14,203,107)(15,204,108)(16,205,109)(17,206,110)(18,207,111)(19,185,112)(20,186,113)(21,187,114)(22,188,115)(23,189,93)(24,157,126)(25,158,127)(26,159,128)(27,160,129)(28,161,130)(29,139,131)(30,140,132)(31,141,133)(32,142,134)(33,143,135)(34,144,136)(35,145,137)(36,146,138)(37,147,116)(38,148,117)(39,149,118)(40,150,119)(41,151,120)(42,152,121)(43,153,122)(44,154,123)(45,155,124)(46,156,125)(47,165,90)(48,166,91)(49,167,92)(50,168,70)(51,169,71)(52,170,72)(53,171,73)(54,172,74)(55,173,75)(56,174,76)(57,175,77)(58,176,78)(59,177,79)(60,178,80)(61,179,81)(62,180,82)(63,181,83)(64,182,84)(65,183,85)(66,184,86)(67,162,87)(68,163,88)(69,164,89), (1,53,27)(2,54,28)(3,55,29)(4,56,30)(5,57,31)(6,58,32)(7,59,33)(8,60,34)(9,61,35)(10,62,36)(11,63,37)(12,64,38)(13,65,39)(14,66,40)(15,67,41)(16,68,42)(17,69,43)(18,47,44)(19,48,45)(20,49,46)(21,50,24)(22,51,25)(23,52,26)(70,126,114)(71,127,115)(72,128,93)(73,129,94)(74,130,95)(75,131,96)(76,132,97)(77,133,98)(78,134,99)(79,135,100)(80,136,101)(81,137,102)(82,138,103)(83,116,104)(84,117,105)(85,118,106)(86,119,107)(87,120,108)(88,121,109)(89,122,110)(90,123,111)(91,124,112)(92,125,113)(139,192,173)(140,193,174)(141,194,175)(142,195,176)(143,196,177)(144,197,178)(145,198,179)(146,199,180)(147,200,181)(148,201,182)(149,202,183)(150,203,184)(151,204,162)(152,205,163)(153,206,164)(154,207,165)(155,185,166)(156,186,167)(157,187,168)(158,188,169)(159,189,170)(160,190,171)(161,191,172), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,29)(25,28)(26,27)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(59,69)(60,68)(61,67)(62,66)(63,65)(70,75)(71,74)(72,73)(76,92)(77,91)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(93,94)(95,115)(96,114)(97,113)(98,112)(99,111)(100,110)(101,109)(102,108)(103,107)(104,106)(116,118)(119,138)(120,137)(121,136)(122,135)(123,134)(124,133)(125,132)(126,131)(127,130)(128,129)(139,157)(140,156)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(158,161)(159,160)(162,179)(163,178)(164,177)(165,176)(166,175)(167,174)(168,173)(169,172)(170,171)(180,184)(181,183)(185,194)(186,193)(187,192)(188,191)(189,190)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202) );
G=PermutationGroup([[(1,190,94),(2,191,95),(3,192,96),(4,193,97),(5,194,98),(6,195,99),(7,196,100),(8,197,101),(9,198,102),(10,199,103),(11,200,104),(12,201,105),(13,202,106),(14,203,107),(15,204,108),(16,205,109),(17,206,110),(18,207,111),(19,185,112),(20,186,113),(21,187,114),(22,188,115),(23,189,93),(24,157,126),(25,158,127),(26,159,128),(27,160,129),(28,161,130),(29,139,131),(30,140,132),(31,141,133),(32,142,134),(33,143,135),(34,144,136),(35,145,137),(36,146,138),(37,147,116),(38,148,117),(39,149,118),(40,150,119),(41,151,120),(42,152,121),(43,153,122),(44,154,123),(45,155,124),(46,156,125),(47,165,90),(48,166,91),(49,167,92),(50,168,70),(51,169,71),(52,170,72),(53,171,73),(54,172,74),(55,173,75),(56,174,76),(57,175,77),(58,176,78),(59,177,79),(60,178,80),(61,179,81),(62,180,82),(63,181,83),(64,182,84),(65,183,85),(66,184,86),(67,162,87),(68,163,88),(69,164,89)], [(1,53,27),(2,54,28),(3,55,29),(4,56,30),(5,57,31),(6,58,32),(7,59,33),(8,60,34),(9,61,35),(10,62,36),(11,63,37),(12,64,38),(13,65,39),(14,66,40),(15,67,41),(16,68,42),(17,69,43),(18,47,44),(19,48,45),(20,49,46),(21,50,24),(22,51,25),(23,52,26),(70,126,114),(71,127,115),(72,128,93),(73,129,94),(74,130,95),(75,131,96),(76,132,97),(77,133,98),(78,134,99),(79,135,100),(80,136,101),(81,137,102),(82,138,103),(83,116,104),(84,117,105),(85,118,106),(86,119,107),(87,120,108),(88,121,109),(89,122,110),(90,123,111),(91,124,112),(92,125,113),(139,192,173),(140,193,174),(141,194,175),(142,195,176),(143,196,177),(144,197,178),(145,198,179),(146,199,180),(147,200,181),(148,201,182),(149,202,183),(150,203,184),(151,204,162),(152,205,163),(153,206,164),(154,207,165),(155,185,166),(156,186,167),(157,187,168),(158,188,169),(159,189,170),(160,190,171),(161,191,172)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(24,29),(25,28),(26,27),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53),(59,69),(60,68),(61,67),(62,66),(63,65),(70,75),(71,74),(72,73),(76,92),(77,91),(78,90),(79,89),(80,88),(81,87),(82,86),(83,85),(93,94),(95,115),(96,114),(97,113),(98,112),(99,111),(100,110),(101,109),(102,108),(103,107),(104,106),(116,118),(119,138),(120,137),(121,136),(122,135),(123,134),(124,133),(125,132),(126,131),(127,130),(128,129),(139,157),(140,156),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(158,161),(159,160),(162,179),(163,178),(164,177),(165,176),(166,175),(167,174),(168,173),(169,172),(170,171),(180,184),(181,183),(185,194),(186,193),(187,192),(188,191),(189,190),(195,207),(196,206),(197,205),(198,204),(199,203),(200,202)]])
117 conjugacy classes
| class | 1 | 2 | 3A | ··· | 3H | 6A | ··· | 6H | 23A | ··· | 23K | 69A | ··· | 69CJ |
| order | 1 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 23 | ··· | 23 | 69 | ··· | 69 |
| size | 1 | 23 | 1 | ··· | 1 | 23 | ··· | 23 | 2 | ··· | 2 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C3 | C6 | D23 | C3×D23 |
| kernel | C32×D23 | C3×C69 | C3×D23 | C69 | C32 | C3 |
| # reps | 1 | 1 | 8 | 8 | 11 | 88 |
Matrix representation of C32×D23 ►in GL3(𝔽139) generated by
| 1 | 0 | 0 |
| 0 | 96 | 0 |
| 0 | 0 | 96 |
| 96 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 1 | 0 | 0 |
| 0 | 38 | 66 |
| 0 | 138 | 119 |
| 138 | 0 | 0 |
| 0 | 24 | 90 |
| 0 | 94 | 115 |
G:=sub<GL(3,GF(139))| [1,0,0,0,96,0,0,0,96],[96,0,0,0,42,0,0,0,42],[1,0,0,0,38,138,0,66,119],[138,0,0,0,24,94,0,90,115] >;
C32×D23 in GAP, Magma, Sage, TeX
C_3^2\times D_{23} % in TeX
G:=Group("C3^2xD23"); // GroupNames label
G:=SmallGroup(414,5);
// by ID
G=gap.SmallGroup(414,5);
# by ID
G:=PCGroup([4,-2,-3,-3,-23,6339]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^23=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export