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G = C32×C21order 189 = 33·7

Abelian group of type [3,3,21]

direct product, abelian, monomial, 3-elementary

Aliases: C32×C21, SmallGroup(189,13)

Series: Derived Chief Lower central Upper central

C1 — C32×C21
C1C7C21C3×C21 — 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 [×13], C7, C32 [×13], C21 [×13], C33, C3×C21 [×13], C32×C21
Quotients: C1, C3 [×13], C7, C32 [×13], C21 [×13], C33, C3×C21 [×13], C32×C21

Smallest permutation representation of C32×C21
Regular action on 189 points
Generators in S189
(1 133 80)(2 134 81)(3 135 82)(4 136 83)(5 137 84)(6 138 64)(7 139 65)(8 140 66)(9 141 67)(10 142 68)(11 143 69)(12 144 70)(13 145 71)(14 146 72)(15 147 73)(16 127 74)(17 128 75)(18 129 76)(19 130 77)(20 131 78)(21 132 79)(22 168 99)(23 148 100)(24 149 101)(25 150 102)(26 151 103)(27 152 104)(28 153 105)(29 154 85)(30 155 86)(31 156 87)(32 157 88)(33 158 89)(34 159 90)(35 160 91)(36 161 92)(37 162 93)(38 163 94)(39 164 95)(40 165 96)(41 166 97)(42 167 98)(43 174 108)(44 175 109)(45 176 110)(46 177 111)(47 178 112)(48 179 113)(49 180 114)(50 181 115)(51 182 116)(52 183 117)(53 184 118)(54 185 119)(55 186 120)(56 187 121)(57 188 122)(58 189 123)(59 169 124)(60 170 125)(61 171 126)(62 172 106)(63 173 107)
(1 52 42)(2 53 22)(3 54 23)(4 55 24)(5 56 25)(6 57 26)(7 58 27)(8 59 28)(9 60 29)(10 61 30)(11 62 31)(12 63 32)(13 43 33)(14 44 34)(15 45 35)(16 46 36)(17 47 37)(18 48 38)(19 49 39)(20 50 40)(21 51 41)(64 122 103)(65 123 104)(66 124 105)(67 125 85)(68 126 86)(69 106 87)(70 107 88)(71 108 89)(72 109 90)(73 110 91)(74 111 92)(75 112 93)(76 113 94)(77 114 95)(78 115 96)(79 116 97)(80 117 98)(81 118 99)(82 119 100)(83 120 101)(84 121 102)(127 177 161)(128 178 162)(129 179 163)(130 180 164)(131 181 165)(132 182 166)(133 183 167)(134 184 168)(135 185 148)(136 186 149)(137 187 150)(138 188 151)(139 189 152)(140 169 153)(141 170 154)(142 171 155)(143 172 156)(144 173 157)(145 174 158)(146 175 159)(147 176 160)
(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)| (1,133,80)(2,134,81)(3,135,82)(4,136,83)(5,137,84)(6,138,64)(7,139,65)(8,140,66)(9,141,67)(10,142,68)(11,143,69)(12,144,70)(13,145,71)(14,146,72)(15,147,73)(16,127,74)(17,128,75)(18,129,76)(19,130,77)(20,131,78)(21,132,79)(22,168,99)(23,148,100)(24,149,101)(25,150,102)(26,151,103)(27,152,104)(28,153,105)(29,154,85)(30,155,86)(31,156,87)(32,157,88)(33,158,89)(34,159,90)(35,160,91)(36,161,92)(37,162,93)(38,163,94)(39,164,95)(40,165,96)(41,166,97)(42,167,98)(43,174,108)(44,175,109)(45,176,110)(46,177,111)(47,178,112)(48,179,113)(49,180,114)(50,181,115)(51,182,116)(52,183,117)(53,184,118)(54,185,119)(55,186,120)(56,187,121)(57,188,122)(58,189,123)(59,169,124)(60,170,125)(61,171,126)(62,172,106)(63,173,107), (1,52,42)(2,53,22)(3,54,23)(4,55,24)(5,56,25)(6,57,26)(7,58,27)(8,59,28)(9,60,29)(10,61,30)(11,62,31)(12,63,32)(13,43,33)(14,44,34)(15,45,35)(16,46,36)(17,47,37)(18,48,38)(19,49,39)(20,50,40)(21,51,41)(64,122,103)(65,123,104)(66,124,105)(67,125,85)(68,126,86)(69,106,87)(70,107,88)(71,108,89)(72,109,90)(73,110,91)(74,111,92)(75,112,93)(76,113,94)(77,114,95)(78,115,96)(79,116,97)(80,117,98)(81,118,99)(82,119,100)(83,120,101)(84,121,102)(127,177,161)(128,178,162)(129,179,163)(130,180,164)(131,181,165)(132,182,166)(133,183,167)(134,184,168)(135,185,148)(136,186,149)(137,187,150)(138,188,151)(139,189,152)(140,169,153)(141,170,154)(142,171,155)(143,172,156)(144,173,157)(145,174,158)(146,175,159)(147,176,160), (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( (1,133,80)(2,134,81)(3,135,82)(4,136,83)(5,137,84)(6,138,64)(7,139,65)(8,140,66)(9,141,67)(10,142,68)(11,143,69)(12,144,70)(13,145,71)(14,146,72)(15,147,73)(16,127,74)(17,128,75)(18,129,76)(19,130,77)(20,131,78)(21,132,79)(22,168,99)(23,148,100)(24,149,101)(25,150,102)(26,151,103)(27,152,104)(28,153,105)(29,154,85)(30,155,86)(31,156,87)(32,157,88)(33,158,89)(34,159,90)(35,160,91)(36,161,92)(37,162,93)(38,163,94)(39,164,95)(40,165,96)(41,166,97)(42,167,98)(43,174,108)(44,175,109)(45,176,110)(46,177,111)(47,178,112)(48,179,113)(49,180,114)(50,181,115)(51,182,116)(52,183,117)(53,184,118)(54,185,119)(55,186,120)(56,187,121)(57,188,122)(58,189,123)(59,169,124)(60,170,125)(61,171,126)(62,172,106)(63,173,107), (1,52,42)(2,53,22)(3,54,23)(4,55,24)(5,56,25)(6,57,26)(7,58,27)(8,59,28)(9,60,29)(10,61,30)(11,62,31)(12,63,32)(13,43,33)(14,44,34)(15,45,35)(16,46,36)(17,47,37)(18,48,38)(19,49,39)(20,50,40)(21,51,41)(64,122,103)(65,123,104)(66,124,105)(67,125,85)(68,126,86)(69,106,87)(70,107,88)(71,108,89)(72,109,90)(73,110,91)(74,111,92)(75,112,93)(76,113,94)(77,114,95)(78,115,96)(79,116,97)(80,117,98)(81,118,99)(82,119,100)(83,120,101)(84,121,102)(127,177,161)(128,178,162)(129,179,163)(130,180,164)(131,181,165)(132,182,166)(133,183,167)(134,184,168)(135,185,148)(136,186,149)(137,187,150)(138,188,151)(139,189,152)(140,169,153)(141,170,154)(142,171,155)(143,172,156)(144,173,157)(145,174,158)(146,175,159)(147,176,160), (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([(1,133,80),(2,134,81),(3,135,82),(4,136,83),(5,137,84),(6,138,64),(7,139,65),(8,140,66),(9,141,67),(10,142,68),(11,143,69),(12,144,70),(13,145,71),(14,146,72),(15,147,73),(16,127,74),(17,128,75),(18,129,76),(19,130,77),(20,131,78),(21,132,79),(22,168,99),(23,148,100),(24,149,101),(25,150,102),(26,151,103),(27,152,104),(28,153,105),(29,154,85),(30,155,86),(31,156,87),(32,157,88),(33,158,89),(34,159,90),(35,160,91),(36,161,92),(37,162,93),(38,163,94),(39,164,95),(40,165,96),(41,166,97),(42,167,98),(43,174,108),(44,175,109),(45,176,110),(46,177,111),(47,178,112),(48,179,113),(49,180,114),(50,181,115),(51,182,116),(52,183,117),(53,184,118),(54,185,119),(55,186,120),(56,187,121),(57,188,122),(58,189,123),(59,169,124),(60,170,125),(61,171,126),(62,172,106),(63,173,107)], [(1,52,42),(2,53,22),(3,54,23),(4,55,24),(5,56,25),(6,57,26),(7,58,27),(8,59,28),(9,60,29),(10,61,30),(11,62,31),(12,63,32),(13,43,33),(14,44,34),(15,45,35),(16,46,36),(17,47,37),(18,48,38),(19,49,39),(20,50,40),(21,51,41),(64,122,103),(65,123,104),(66,124,105),(67,125,85),(68,126,86),(69,106,87),(70,107,88),(71,108,89),(72,109,90),(73,110,91),(74,111,92),(75,112,93),(76,113,94),(77,114,95),(78,115,96),(79,116,97),(80,117,98),(81,118,99),(82,119,100),(83,120,101),(84,121,102),(127,177,161),(128,178,162),(129,179,163),(130,180,164),(131,181,165),(132,182,166),(133,183,167),(134,184,168),(135,185,148),(136,186,149),(137,187,150),(138,188,151),(139,189,152),(140,169,153),(141,170,154),(142,171,155),(143,172,156),(144,173,157),(145,174,158),(146,175,159),(147,176,160)], [(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···3Z7A···7F21A···21EZ
order13···37···721···21
size11···11···11···1

189 irreducible representations

dim1111
type+
imageC1C3C7C21
kernelC32×C21C3×C21C33C32
# reps1266156

Matrix representation of C32×C21 in GL3(𝔽43) generated by

100
0360
001
,
600
0360
0036
,
2300
0140
006
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

׿
×
𝔽