direct product, abelian, monomial, 3-elementary
Aliases: C32×C21, SmallGroup(189,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32×C21 |
| C1 — C32×C21 |
| C1 — C32×C21 |
Generators and relations for C32×C21
G = < a,b,c | a3=b3=c21=1, ab=ba, ac=ca, bc=cb >
Subgroups: 56, all normal (4 characteristic)
C1, C3, C7, C32, C21, C33, C3×C21, C32×C21
Quotients: C1, C3, C7, C32, C21, C33, C3×C21, C32×C21
►Regular action on 189 points
Generators in S189
(1 129 68)(2 130 69)(3 131 70)(4 132 71)(5 133 72)(6 134 73)(7 135 74)(8 136 75)(9 137 76)(10 138 77)(11 139 78)(12 140 79)(13 141 80)(14 142 81)(15 143 82)(16 144 83)(17 145 84)(18 146 64)(19 147 65)(20 127 66)(21 128 67)(22 149 94)(23 150 95)(24 151 96)(25 152 97)(26 153 98)(27 154 99)(28 155 100)(29 156 101)(30 157 102)(31 158 103)(32 159 104)(33 160 105)(34 161 85)(35 162 86)(36 163 87)(37 164 88)(38 165 89)(39 166 90)(40 167 91)(41 168 92)(42 148 93)(43 180 121)(44 181 122)(45 182 123)(46 183 124)(47 184 125)(48 185 126)(49 186 106)(50 187 107)(51 188 108)(52 189 109)(53 169 110)(54 170 111)(55 171 112)(56 172 113)(57 173 114)(58 174 115)(59 175 116)(60 176 117)(61 177 118)(62 178 119)(63 179 120)
(1 60 25)(2 61 26)(3 62 27)(4 63 28)(5 43 29)(6 44 30)(7 45 31)(8 46 32)(9 47 33)(10 48 34)(11 49 35)(12 50 36)(13 51 37)(14 52 38)(15 53 39)(16 54 40)(17 55 41)(18 56 42)(19 57 22)(20 58 23)(21 59 24)(64 113 93)(65 114 94)(66 115 95)(67 116 96)(68 117 97)(69 118 98)(70 119 99)(71 120 100)(72 121 101)(73 122 102)(74 123 103)(75 124 104)(76 125 105)(77 126 85)(78 106 86)(79 107 87)(80 108 88)(81 109 89)(82 110 90)(83 111 91)(84 112 92)(127 174 150)(128 175 151)(129 176 152)(130 177 153)(131 178 154)(132 179 155)(133 180 156)(134 181 157)(135 182 158)(136 183 159)(137 184 160)(138 185 161)(139 186 162)(140 187 163)(141 188 164)(142 189 165)(143 169 166)(144 170 167)(145 171 168)(146 172 148)(147 173 149)
(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)
G:=sub<Sym(189)| (1,129,68)(2,130,69)(3,131,70)(4,132,71)(5,133,72)(6,134,73)(7,135,74)(8,136,75)(9,137,76)(10,138,77)(11,139,78)(12,140,79)(13,141,80)(14,142,81)(15,143,82)(16,144,83)(17,145,84)(18,146,64)(19,147,65)(20,127,66)(21,128,67)(22,149,94)(23,150,95)(24,151,96)(25,152,97)(26,153,98)(27,154,99)(28,155,100)(29,156,101)(30,157,102)(31,158,103)(32,159,104)(33,160,105)(34,161,85)(35,162,86)(36,163,87)(37,164,88)(38,165,89)(39,166,90)(40,167,91)(41,168,92)(42,148,93)(43,180,121)(44,181,122)(45,182,123)(46,183,124)(47,184,125)(48,185,126)(49,186,106)(50,187,107)(51,188,108)(52,189,109)(53,169,110)(54,170,111)(55,171,112)(56,172,113)(57,173,114)(58,174,115)(59,175,116)(60,176,117)(61,177,118)(62,178,119)(63,179,120), (1,60,25)(2,61,26)(3,62,27)(4,63,28)(5,43,29)(6,44,30)(7,45,31)(8,46,32)(9,47,33)(10,48,34)(11,49,35)(12,50,36)(13,51,37)(14,52,38)(15,53,39)(16,54,40)(17,55,41)(18,56,42)(19,57,22)(20,58,23)(21,59,24)(64,113,93)(65,114,94)(66,115,95)(67,116,96)(68,117,97)(69,118,98)(70,119,99)(71,120,100)(72,121,101)(73,122,102)(74,123,103)(75,124,104)(76,125,105)(77,126,85)(78,106,86)(79,107,87)(80,108,88)(81,109,89)(82,110,90)(83,111,91)(84,112,92)(127,174,150)(128,175,151)(129,176,152)(130,177,153)(131,178,154)(132,179,155)(133,180,156)(134,181,157)(135,182,158)(136,183,159)(137,184,160)(138,185,161)(139,186,162)(140,187,163)(141,188,164)(142,189,165)(143,169,166)(144,170,167)(145,171,168)(146,172,148)(147,173,149), (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)>;
G:=Group( (1,129,68)(2,130,69)(3,131,70)(4,132,71)(5,133,72)(6,134,73)(7,135,74)(8,136,75)(9,137,76)(10,138,77)(11,139,78)(12,140,79)(13,141,80)(14,142,81)(15,143,82)(16,144,83)(17,145,84)(18,146,64)(19,147,65)(20,127,66)(21,128,67)(22,149,94)(23,150,95)(24,151,96)(25,152,97)(26,153,98)(27,154,99)(28,155,100)(29,156,101)(30,157,102)(31,158,103)(32,159,104)(33,160,105)(34,161,85)(35,162,86)(36,163,87)(37,164,88)(38,165,89)(39,166,90)(40,167,91)(41,168,92)(42,148,93)(43,180,121)(44,181,122)(45,182,123)(46,183,124)(47,184,125)(48,185,126)(49,186,106)(50,187,107)(51,188,108)(52,189,109)(53,169,110)(54,170,111)(55,171,112)(56,172,113)(57,173,114)(58,174,115)(59,175,116)(60,176,117)(61,177,118)(62,178,119)(63,179,120), (1,60,25)(2,61,26)(3,62,27)(4,63,28)(5,43,29)(6,44,30)(7,45,31)(8,46,32)(9,47,33)(10,48,34)(11,49,35)(12,50,36)(13,51,37)(14,52,38)(15,53,39)(16,54,40)(17,55,41)(18,56,42)(19,57,22)(20,58,23)(21,59,24)(64,113,93)(65,114,94)(66,115,95)(67,116,96)(68,117,97)(69,118,98)(70,119,99)(71,120,100)(72,121,101)(73,122,102)(74,123,103)(75,124,104)(76,125,105)(77,126,85)(78,106,86)(79,107,87)(80,108,88)(81,109,89)(82,110,90)(83,111,91)(84,112,92)(127,174,150)(128,175,151)(129,176,152)(130,177,153)(131,178,154)(132,179,155)(133,180,156)(134,181,157)(135,182,158)(136,183,159)(137,184,160)(138,185,161)(139,186,162)(140,187,163)(141,188,164)(142,189,165)(143,169,166)(144,170,167)(145,171,168)(146,172,148)(147,173,149), (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) );
G=PermutationGroup([[(1,129,68),(2,130,69),(3,131,70),(4,132,71),(5,133,72),(6,134,73),(7,135,74),(8,136,75),(9,137,76),(10,138,77),(11,139,78),(12,140,79),(13,141,80),(14,142,81),(15,143,82),(16,144,83),(17,145,84),(18,146,64),(19,147,65),(20,127,66),(21,128,67),(22,149,94),(23,150,95),(24,151,96),(25,152,97),(26,153,98),(27,154,99),(28,155,100),(29,156,101),(30,157,102),(31,158,103),(32,159,104),(33,160,105),(34,161,85),(35,162,86),(36,163,87),(37,164,88),(38,165,89),(39,166,90),(40,167,91),(41,168,92),(42,148,93),(43,180,121),(44,181,122),(45,182,123),(46,183,124),(47,184,125),(48,185,126),(49,186,106),(50,187,107),(51,188,108),(52,189,109),(53,169,110),(54,170,111),(55,171,112),(56,172,113),(57,173,114),(58,174,115),(59,175,116),(60,176,117),(61,177,118),(62,178,119),(63,179,120)], [(1,60,25),(2,61,26),(3,62,27),(4,63,28),(5,43,29),(6,44,30),(7,45,31),(8,46,32),(9,47,33),(10,48,34),(11,49,35),(12,50,36),(13,51,37),(14,52,38),(15,53,39),(16,54,40),(17,55,41),(18,56,42),(19,57,22),(20,58,23),(21,59,24),(64,113,93),(65,114,94),(66,115,95),(67,116,96),(68,117,97),(69,118,98),(70,119,99),(71,120,100),(72,121,101),(73,122,102),(74,123,103),(75,124,104),(76,125,105),(77,126,85),(78,106,86),(79,107,87),(80,108,88),(81,109,89),(82,110,90),(83,111,91),(84,112,92),(127,174,150),(128,175,151),(129,176,152),(130,177,153),(131,178,154),(132,179,155),(133,180,156),(134,181,157),(135,182,158),(136,183,159),(137,184,160),(138,185,161),(139,186,162),(140,187,163),(141,188,164),(142,189,165),(143,169,166),(144,170,167),(145,171,168),(146,172,148),(147,173,149)], [(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)]])
C32×C21 is a maximal subgroup of
C33⋊D7
189 conjugacy classes
| class | 1 | 3A | ··· | 3Z | 7A | ··· | 7F | 21A | ··· | 21EZ |
| order | 1 | 3 | ··· | 3 | 7 | ··· | 7 | 21 | ··· | 21 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
189 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C7 | C21 |
| kernel | C32×C21 | C3×C21 | C33 | C32 |
| # reps | 1 | 26 | 6 | 156 |
Matrix representation of C32×C21 ►in GL3(𝔽43) generated by
| 1 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 1 |
| 6 | 0 | 0 |
| 0 | 36 | 0 |
| 0 | 0 | 36 |
| 23 | 0 | 0 |
| 0 | 14 | 0 |
| 0 | 0 | 6 |
G:=sub<GL(3,GF(43))| [1,0,0,0,36,0,0,0,1],[6,0,0,0,36,0,0,0,36],[23,0,0,0,14,0,0,0,6] >;
C32×C21 in GAP, Magma, Sage, TeX
C_3^2\times C_{21} % in TeX
G:=Group("C3^2xC21"); // GroupNames label
G:=SmallGroup(189,13);
// by ID
G=gap.SmallGroup(189,13);
# by ID
G:=PCGroup([4,-3,-3,-3,-7]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^21=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations