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G = C33×C6order 162 = 2·34

Abelian group of type [3,3,3,6]

direct product, abelian, monomial, 3-elementary

Aliases: C33×C6, SmallGroup(162,55)

Series: Derived Chief Lower central Upper central

C1 — C33×C6
C1C3C32C33C34 — C33×C6
C1 — C33×C6
C1 — C33×C6

Generators and relations for C33×C6
 G = < a,b,c,d | a3=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 424, all normal (4 characteristic)
C1, C2, C3 [×40], C6 [×40], C32 [×130], C3×C6 [×130], C33 [×40], C32×C6 [×40], C34, C33×C6
Quotients: C1, C2, C3 [×40], C6 [×40], C32 [×130], C3×C6 [×130], C33 [×40], C32×C6 [×40], C34, C33×C6

Smallest permutation representation of C33×C6
Regular action on 162 points
Generators in S162
(1 145 94)(2 146 95)(3 147 96)(4 148 91)(5 149 92)(6 150 93)(7 114 64)(8 109 65)(9 110 66)(10 111 61)(11 112 62)(12 113 63)(13 38 142)(14 39 143)(15 40 144)(16 41 139)(17 42 140)(18 37 141)(19 123 76)(20 124 77)(21 125 78)(22 126 73)(23 121 74)(24 122 75)(25 47 69)(26 48 70)(27 43 71)(28 44 72)(29 45 67)(30 46 68)(31 135 87)(32 136 88)(33 137 89)(34 138 90)(35 133 85)(36 134 86)(49 156 99)(50 151 100)(51 152 101)(52 153 102)(53 154 97)(54 155 98)(55 83 118)(56 84 119)(57 79 120)(58 80 115)(59 81 116)(60 82 117)(103 131 161)(104 132 162)(105 127 157)(106 128 158)(107 129 159)(108 130 160)
(1 127 11)(2 128 12)(3 129 7)(4 130 8)(5 131 9)(6 132 10)(13 124 59)(14 125 60)(15 126 55)(16 121 56)(17 122 57)(18 123 58)(19 115 141)(20 116 142)(21 117 143)(22 118 144)(23 119 139)(24 120 140)(25 51 86)(26 52 87)(27 53 88)(28 54 89)(29 49 90)(30 50 85)(31 48 153)(32 43 154)(33 44 155)(34 45 156)(35 46 151)(36 47 152)(37 76 80)(38 77 81)(39 78 82)(40 73 83)(41 74 84)(42 75 79)(61 93 104)(62 94 105)(63 95 106)(64 96 107)(65 91 108)(66 92 103)(67 99 138)(68 100 133)(69 101 134)(70 102 135)(71 97 136)(72 98 137)(109 148 160)(110 149 161)(111 150 162)(112 145 157)(113 146 158)(114 147 159)
(1 121 85)(2 122 86)(3 123 87)(4 124 88)(5 125 89)(6 126 90)(7 18 52)(8 13 53)(9 14 54)(10 15 49)(11 16 50)(12 17 51)(19 135 96)(20 136 91)(21 137 92)(22 138 93)(23 133 94)(24 134 95)(25 128 57)(26 129 58)(27 130 59)(28 131 60)(29 132 55)(30 127 56)(31 147 76)(32 148 77)(33 149 78)(34 150 73)(35 145 74)(36 146 75)(37 153 114)(38 154 109)(39 155 110)(40 156 111)(41 151 112)(42 152 113)(43 160 81)(44 161 82)(45 162 83)(46 157 84)(47 158 79)(48 159 80)(61 144 99)(62 139 100)(63 140 101)(64 141 102)(65 142 97)(66 143 98)(67 104 118)(68 105 119)(69 106 120)(70 107 115)(71 108 116)(72 103 117)
(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)

G:=sub<Sym(162)| (1,145,94)(2,146,95)(3,147,96)(4,148,91)(5,149,92)(6,150,93)(7,114,64)(8,109,65)(9,110,66)(10,111,61)(11,112,62)(12,113,63)(13,38,142)(14,39,143)(15,40,144)(16,41,139)(17,42,140)(18,37,141)(19,123,76)(20,124,77)(21,125,78)(22,126,73)(23,121,74)(24,122,75)(25,47,69)(26,48,70)(27,43,71)(28,44,72)(29,45,67)(30,46,68)(31,135,87)(32,136,88)(33,137,89)(34,138,90)(35,133,85)(36,134,86)(49,156,99)(50,151,100)(51,152,101)(52,153,102)(53,154,97)(54,155,98)(55,83,118)(56,84,119)(57,79,120)(58,80,115)(59,81,116)(60,82,117)(103,131,161)(104,132,162)(105,127,157)(106,128,158)(107,129,159)(108,130,160), (1,127,11)(2,128,12)(3,129,7)(4,130,8)(5,131,9)(6,132,10)(13,124,59)(14,125,60)(15,126,55)(16,121,56)(17,122,57)(18,123,58)(19,115,141)(20,116,142)(21,117,143)(22,118,144)(23,119,139)(24,120,140)(25,51,86)(26,52,87)(27,53,88)(28,54,89)(29,49,90)(30,50,85)(31,48,153)(32,43,154)(33,44,155)(34,45,156)(35,46,151)(36,47,152)(37,76,80)(38,77,81)(39,78,82)(40,73,83)(41,74,84)(42,75,79)(61,93,104)(62,94,105)(63,95,106)(64,96,107)(65,91,108)(66,92,103)(67,99,138)(68,100,133)(69,101,134)(70,102,135)(71,97,136)(72,98,137)(109,148,160)(110,149,161)(111,150,162)(112,145,157)(113,146,158)(114,147,159), (1,121,85)(2,122,86)(3,123,87)(4,124,88)(5,125,89)(6,126,90)(7,18,52)(8,13,53)(9,14,54)(10,15,49)(11,16,50)(12,17,51)(19,135,96)(20,136,91)(21,137,92)(22,138,93)(23,133,94)(24,134,95)(25,128,57)(26,129,58)(27,130,59)(28,131,60)(29,132,55)(30,127,56)(31,147,76)(32,148,77)(33,149,78)(34,150,73)(35,145,74)(36,146,75)(37,153,114)(38,154,109)(39,155,110)(40,156,111)(41,151,112)(42,152,113)(43,160,81)(44,161,82)(45,162,83)(46,157,84)(47,158,79)(48,159,80)(61,144,99)(62,139,100)(63,140,101)(64,141,102)(65,142,97)(66,143,98)(67,104,118)(68,105,119)(69,106,120)(70,107,115)(71,108,116)(72,103,117), (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)>;

G:=Group( (1,145,94)(2,146,95)(3,147,96)(4,148,91)(5,149,92)(6,150,93)(7,114,64)(8,109,65)(9,110,66)(10,111,61)(11,112,62)(12,113,63)(13,38,142)(14,39,143)(15,40,144)(16,41,139)(17,42,140)(18,37,141)(19,123,76)(20,124,77)(21,125,78)(22,126,73)(23,121,74)(24,122,75)(25,47,69)(26,48,70)(27,43,71)(28,44,72)(29,45,67)(30,46,68)(31,135,87)(32,136,88)(33,137,89)(34,138,90)(35,133,85)(36,134,86)(49,156,99)(50,151,100)(51,152,101)(52,153,102)(53,154,97)(54,155,98)(55,83,118)(56,84,119)(57,79,120)(58,80,115)(59,81,116)(60,82,117)(103,131,161)(104,132,162)(105,127,157)(106,128,158)(107,129,159)(108,130,160), (1,127,11)(2,128,12)(3,129,7)(4,130,8)(5,131,9)(6,132,10)(13,124,59)(14,125,60)(15,126,55)(16,121,56)(17,122,57)(18,123,58)(19,115,141)(20,116,142)(21,117,143)(22,118,144)(23,119,139)(24,120,140)(25,51,86)(26,52,87)(27,53,88)(28,54,89)(29,49,90)(30,50,85)(31,48,153)(32,43,154)(33,44,155)(34,45,156)(35,46,151)(36,47,152)(37,76,80)(38,77,81)(39,78,82)(40,73,83)(41,74,84)(42,75,79)(61,93,104)(62,94,105)(63,95,106)(64,96,107)(65,91,108)(66,92,103)(67,99,138)(68,100,133)(69,101,134)(70,102,135)(71,97,136)(72,98,137)(109,148,160)(110,149,161)(111,150,162)(112,145,157)(113,146,158)(114,147,159), (1,121,85)(2,122,86)(3,123,87)(4,124,88)(5,125,89)(6,126,90)(7,18,52)(8,13,53)(9,14,54)(10,15,49)(11,16,50)(12,17,51)(19,135,96)(20,136,91)(21,137,92)(22,138,93)(23,133,94)(24,134,95)(25,128,57)(26,129,58)(27,130,59)(28,131,60)(29,132,55)(30,127,56)(31,147,76)(32,148,77)(33,149,78)(34,150,73)(35,145,74)(36,146,75)(37,153,114)(38,154,109)(39,155,110)(40,156,111)(41,151,112)(42,152,113)(43,160,81)(44,161,82)(45,162,83)(46,157,84)(47,158,79)(48,159,80)(61,144,99)(62,139,100)(63,140,101)(64,141,102)(65,142,97)(66,143,98)(67,104,118)(68,105,119)(69,106,120)(70,107,115)(71,108,116)(72,103,117), (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) );

G=PermutationGroup([(1,145,94),(2,146,95),(3,147,96),(4,148,91),(5,149,92),(6,150,93),(7,114,64),(8,109,65),(9,110,66),(10,111,61),(11,112,62),(12,113,63),(13,38,142),(14,39,143),(15,40,144),(16,41,139),(17,42,140),(18,37,141),(19,123,76),(20,124,77),(21,125,78),(22,126,73),(23,121,74),(24,122,75),(25,47,69),(26,48,70),(27,43,71),(28,44,72),(29,45,67),(30,46,68),(31,135,87),(32,136,88),(33,137,89),(34,138,90),(35,133,85),(36,134,86),(49,156,99),(50,151,100),(51,152,101),(52,153,102),(53,154,97),(54,155,98),(55,83,118),(56,84,119),(57,79,120),(58,80,115),(59,81,116),(60,82,117),(103,131,161),(104,132,162),(105,127,157),(106,128,158),(107,129,159),(108,130,160)], [(1,127,11),(2,128,12),(3,129,7),(4,130,8),(5,131,9),(6,132,10),(13,124,59),(14,125,60),(15,126,55),(16,121,56),(17,122,57),(18,123,58),(19,115,141),(20,116,142),(21,117,143),(22,118,144),(23,119,139),(24,120,140),(25,51,86),(26,52,87),(27,53,88),(28,54,89),(29,49,90),(30,50,85),(31,48,153),(32,43,154),(33,44,155),(34,45,156),(35,46,151),(36,47,152),(37,76,80),(38,77,81),(39,78,82),(40,73,83),(41,74,84),(42,75,79),(61,93,104),(62,94,105),(63,95,106),(64,96,107),(65,91,108),(66,92,103),(67,99,138),(68,100,133),(69,101,134),(70,102,135),(71,97,136),(72,98,137),(109,148,160),(110,149,161),(111,150,162),(112,145,157),(113,146,158),(114,147,159)], [(1,121,85),(2,122,86),(3,123,87),(4,124,88),(5,125,89),(6,126,90),(7,18,52),(8,13,53),(9,14,54),(10,15,49),(11,16,50),(12,17,51),(19,135,96),(20,136,91),(21,137,92),(22,138,93),(23,133,94),(24,134,95),(25,128,57),(26,129,58),(27,130,59),(28,131,60),(29,132,55),(30,127,56),(31,147,76),(32,148,77),(33,149,78),(34,150,73),(35,145,74),(36,146,75),(37,153,114),(38,154,109),(39,155,110),(40,156,111),(41,151,112),(42,152,113),(43,160,81),(44,161,82),(45,162,83),(46,157,84),(47,158,79),(48,159,80),(61,144,99),(62,139,100),(63,140,101),(64,141,102),(65,142,97),(66,143,98),(67,104,118),(68,105,119),(69,106,120),(70,107,115),(71,108,116),(72,103,117)], [(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)])

C33×C6 is a maximal subgroup of   C348C4

162 conjugacy classes

class 1  2 3A···3CB6A···6CB
order123···36···6
size111···11···1

162 irreducible representations

dim1111
type++
imageC1C2C3C6
kernelC33×C6C34C32×C6C33
# reps118080

Matrix representation of C33×C6 in GL4(𝔽7) generated by

4000
0400
0020
0001
,
1000
0100
0020
0002
,
4000
0400
0010
0001
,
6000
0400
0010
0003
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,2,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[6,0,0,0,0,4,0,0,0,0,1,0,0,0,0,3] >;

C33×C6 in GAP, Magma, Sage, TeX

C_3^3\times C_6
% in TeX

G:=Group("C3^3xC6");
// GroupNames label

G:=SmallGroup(162,55);
// by ID

G=gap.SmallGroup(162,55);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
// generators/relations

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