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G = C3×D61order 366 = 2·3·61

Direct product of C3 and D61

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C3×D61, C613C6, C1832C2, SmallGroup(366,4)

Series: Derived Chief Lower central Upper central

C1C61 — C3×D61
C1C61C183 — C3×D61
C61 — C3×D61
C1C3

Generators and relations for C3×D61
 G = < a,b,c | a3=b61=c2=1, ab=ba, ac=ca, cbc=b-1 >

61C2
61C6

Smallest permutation representation of C3×D61
On 183 points
Generators in S183
(1 142 115)(2 143 116)(3 144 117)(4 145 118)(5 146 119)(6 147 120)(7 148 121)(8 149 122)(9 150 62)(10 151 63)(11 152 64)(12 153 65)(13 154 66)(14 155 67)(15 156 68)(16 157 69)(17 158 70)(18 159 71)(19 160 72)(20 161 73)(21 162 74)(22 163 75)(23 164 76)(24 165 77)(25 166 78)(26 167 79)(27 168 80)(28 169 81)(29 170 82)(30 171 83)(31 172 84)(32 173 85)(33 174 86)(34 175 87)(35 176 88)(36 177 89)(37 178 90)(38 179 91)(39 180 92)(40 181 93)(41 182 94)(42 183 95)(43 123 96)(44 124 97)(45 125 98)(46 126 99)(47 127 100)(48 128 101)(49 129 102)(50 130 103)(51 131 104)(52 132 105)(53 133 106)(54 134 107)(55 135 108)(56 136 109)(57 137 110)(58 138 111)(59 139 112)(60 140 113)(61 141 114)
(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)
(1 61)(2 60)(3 59)(4 58)(5 57)(6 56)(7 55)(8 54)(9 53)(10 52)(11 51)(12 50)(13 49)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(62 106)(63 105)(64 104)(65 103)(66 102)(67 101)(68 100)(69 99)(70 98)(71 97)(72 96)(73 95)(74 94)(75 93)(76 92)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(107 122)(108 121)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(123 160)(124 159)(125 158)(126 157)(127 156)(128 155)(129 154)(130 153)(131 152)(132 151)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(141 142)(161 183)(162 182)(163 181)(164 180)(165 179)(166 178)(167 177)(168 176)(169 175)(170 174)(171 173)

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

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

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

96 conjugacy classes

class 1  2 3A3B6A6B61A···61AD183A···183BH
order12336661···61183···183
size1611161612···22···2

96 irreducible representations

dim111122
type+++
imageC1C2C3C6D61C3×D61
kernelC3×D61C183D61C61C3C1
# reps11223060

Matrix representation of C3×D61 in GL2(𝔽367) generated by

2830
0283
,
81
126337
,
337366
16530
G:=sub<GL(2,GF(367))| [283,0,0,283],[8,126,1,337],[337,165,366,30] >;

C3×D61 in GAP, Magma, Sage, TeX

C_3\times D_{61}
% in TeX

G:=Group("C3xD61");
// GroupNames label

G:=SmallGroup(366,4);
// by ID

G=gap.SmallGroup(366,4);
# by ID

G:=PCGroup([3,-2,-3,-61,3242]);
// Polycyclic

G:=Group<a,b,c|a^3=b^61=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D61 in TeX

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