Copied to
clipboard

## G = C33×C6order 162 = 2·34

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

Aliases: C33×C6, SmallGroup(162,55)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33×C6
 Chief series C1 — C3 — C32 — C33 — C34 — C33×C6
 Lower central C1 — C33×C6
 Upper central 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, C6, C32, C3×C6, C33, C32×C6, C34, C33×C6
Quotients: C1, C2, C3, C6, C32, C3×C6, C33, C32×C6, C34, C33×C6

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

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

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

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

C33×C6 is a maximal subgroup of   C348C4

162 conjugacy classes

 class 1 2 3A ··· 3CB 6A ··· 6CB order 1 2 3 ··· 3 6 ··· 6 size 1 1 1 ··· 1 1 ··· 1

162 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C3 C6 kernel C33×C6 C34 C32×C6 C33 # reps 1 1 80 80

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

 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
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

׿
×
𝔽