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G = C3×C63order 189 = 33·7

Abelian group of type [3,63]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C63, SmallGroup(189,9)

Series: Derived Chief Lower central Upper central

C1 — C3×C63
C1C3C21C63 — C3×C63
C1 — C3×C63
C1 — C3×C63

Generators and relations for C3×C63
 G = < a,b | a3=b63=1, ab=ba >


Smallest permutation representation of C3×C63
Regular action on 189 points
Generators in S189
(1 67 158)(2 68 159)(3 69 160)(4 70 161)(5 71 162)(6 72 163)(7 73 164)(8 74 165)(9 75 166)(10 76 167)(11 77 168)(12 78 169)(13 79 170)(14 80 171)(15 81 172)(16 82 173)(17 83 174)(18 84 175)(19 85 176)(20 86 177)(21 87 178)(22 88 179)(23 89 180)(24 90 181)(25 91 182)(26 92 183)(27 93 184)(28 94 185)(29 95 186)(30 96 187)(31 97 188)(32 98 189)(33 99 127)(34 100 128)(35 101 129)(36 102 130)(37 103 131)(38 104 132)(39 105 133)(40 106 134)(41 107 135)(42 108 136)(43 109 137)(44 110 138)(45 111 139)(46 112 140)(47 113 141)(48 114 142)(49 115 143)(50 116 144)(51 117 145)(52 118 146)(53 119 147)(54 120 148)(55 121 149)(56 122 150)(57 123 151)(58 124 152)(59 125 153)(60 126 154)(61 64 155)(62 65 156)(63 66 157)
(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)| (1,67,158)(2,68,159)(3,69,160)(4,70,161)(5,71,162)(6,72,163)(7,73,164)(8,74,165)(9,75,166)(10,76,167)(11,77,168)(12,78,169)(13,79,170)(14,80,171)(15,81,172)(16,82,173)(17,83,174)(18,84,175)(19,85,176)(20,86,177)(21,87,178)(22,88,179)(23,89,180)(24,90,181)(25,91,182)(26,92,183)(27,93,184)(28,94,185)(29,95,186)(30,96,187)(31,97,188)(32,98,189)(33,99,127)(34,100,128)(35,101,129)(36,102,130)(37,103,131)(38,104,132)(39,105,133)(40,106,134)(41,107,135)(42,108,136)(43,109,137)(44,110,138)(45,111,139)(46,112,140)(47,113,141)(48,114,142)(49,115,143)(50,116,144)(51,117,145)(52,118,146)(53,119,147)(54,120,148)(55,121,149)(56,122,150)(57,123,151)(58,124,152)(59,125,153)(60,126,154)(61,64,155)(62,65,156)(63,66,157), (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( (1,67,158)(2,68,159)(3,69,160)(4,70,161)(5,71,162)(6,72,163)(7,73,164)(8,74,165)(9,75,166)(10,76,167)(11,77,168)(12,78,169)(13,79,170)(14,80,171)(15,81,172)(16,82,173)(17,83,174)(18,84,175)(19,85,176)(20,86,177)(21,87,178)(22,88,179)(23,89,180)(24,90,181)(25,91,182)(26,92,183)(27,93,184)(28,94,185)(29,95,186)(30,96,187)(31,97,188)(32,98,189)(33,99,127)(34,100,128)(35,101,129)(36,102,130)(37,103,131)(38,104,132)(39,105,133)(40,106,134)(41,107,135)(42,108,136)(43,109,137)(44,110,138)(45,111,139)(46,112,140)(47,113,141)(48,114,142)(49,115,143)(50,116,144)(51,117,145)(52,118,146)(53,119,147)(54,120,148)(55,121,149)(56,122,150)(57,123,151)(58,124,152)(59,125,153)(60,126,154)(61,64,155)(62,65,156)(63,66,157), (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([(1,67,158),(2,68,159),(3,69,160),(4,70,161),(5,71,162),(6,72,163),(7,73,164),(8,74,165),(9,75,166),(10,76,167),(11,77,168),(12,78,169),(13,79,170),(14,80,171),(15,81,172),(16,82,173),(17,83,174),(18,84,175),(19,85,176),(20,86,177),(21,87,178),(22,88,179),(23,89,180),(24,90,181),(25,91,182),(26,92,183),(27,93,184),(28,94,185),(29,95,186),(30,96,187),(31,97,188),(32,98,189),(33,99,127),(34,100,128),(35,101,129),(36,102,130),(37,103,131),(38,104,132),(39,105,133),(40,106,134),(41,107,135),(42,108,136),(43,109,137),(44,110,138),(45,111,139),(46,112,140),(47,113,141),(48,114,142),(49,115,143),(50,116,144),(51,117,145),(52,118,146),(53,119,147),(54,120,148),(55,121,149),(56,122,150),(57,123,151),(58,124,152),(59,125,153),(60,126,154),(61,64,155),(62,65,156),(63,66,157)], [(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)])

C3×C63 is a maximal subgroup of   C3⋊D63

189 conjugacy classes

class 1 3A···3H7A···7F9A···9R21A···21AV63A···63DD
order13···37···79···921···2163···63
size11···11···11···11···11···1

189 irreducible representations

dim11111111
type+
imageC1C3C3C7C9C21C21C63
kernelC3×C63C63C3×C21C3×C9C21C9C32C3
# reps1626183612108

Matrix representation of C3×C63 in GL2(𝔽127) generated by

1070
0107
,
1070
044
G:=sub<GL(2,GF(127))| [107,0,0,107],[107,0,0,44] >;

C3×C63 in GAP, Magma, Sage, TeX

C_3\times C_{63}
% in TeX

G:=Group("C3xC63");
// GroupNames label

G:=SmallGroup(189,9);
// by ID

G=gap.SmallGroup(189,9);
# by ID

G:=PCGroup([4,-3,-3,-7,-3,252]);
// Polycyclic

G:=Group<a,b|a^3=b^63=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C63 in TeX

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