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G = C6×D31order 372 = 22·3·31

Direct product of C6 and D31

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

Aliases: C6×D31, C623C6, C1862C2, C933C22, C313(C2×C6), SmallGroup(372,12)

Series: Derived Chief Lower central Upper central

C1C31 — C6×D31
C1C31C93C3×D31 — C6×D31
C31 — C6×D31
C1C6

Generators and relations for C6×D31
 G = < a,b,c | a6=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >

31C2
31C2
31C22
31C6
31C6
31C2×C6

Smallest permutation representation of C6×D31
On 186 points
Generators in S186
(1 149 63 101 52 156)(2 150 64 102 53 157)(3 151 65 103 54 158)(4 152 66 104 55 159)(5 153 67 105 56 160)(6 154 68 106 57 161)(7 155 69 107 58 162)(8 125 70 108 59 163)(9 126 71 109 60 164)(10 127 72 110 61 165)(11 128 73 111 62 166)(12 129 74 112 32 167)(13 130 75 113 33 168)(14 131 76 114 34 169)(15 132 77 115 35 170)(16 133 78 116 36 171)(17 134 79 117 37 172)(18 135 80 118 38 173)(19 136 81 119 39 174)(20 137 82 120 40 175)(21 138 83 121 41 176)(22 139 84 122 42 177)(23 140 85 123 43 178)(24 141 86 124 44 179)(25 142 87 94 45 180)(26 143 88 95 46 181)(27 144 89 96 47 182)(28 145 90 97 48 183)(29 146 91 98 49 184)(30 147 92 99 50 185)(31 148 93 100 51 186)
(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)
(1 100)(2 99)(3 98)(4 97)(5 96)(6 95)(7 94)(8 124)(9 123)(10 122)(11 121)(12 120)(13 119)(14 118)(15 117)(16 116)(17 115)(18 114)(19 113)(20 112)(21 111)(22 110)(23 109)(24 108)(25 107)(26 106)(27 105)(28 104)(29 103)(30 102)(31 101)(32 137)(33 136)(34 135)(35 134)(36 133)(37 132)(38 131)(39 130)(40 129)(41 128)(42 127)(43 126)(44 125)(45 155)(46 154)(47 153)(48 152)(49 151)(50 150)(51 149)(52 148)(53 147)(54 146)(55 145)(56 144)(57 143)(58 142)(59 141)(60 140)(61 139)(62 138)(63 186)(64 185)(65 184)(66 183)(67 182)(68 181)(69 180)(70 179)(71 178)(72 177)(73 176)(74 175)(75 174)(76 173)(77 172)(78 171)(79 170)(80 169)(81 168)(82 167)(83 166)(84 165)(85 164)(86 163)(87 162)(88 161)(89 160)(90 159)(91 158)(92 157)(93 156)

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

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

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

102 conjugacy classes

class 1 2A2B2C3A3B6A6B6C6D6E6F31A···31O62A···62O93A···93AD186A···186AD
order12223366666631···3162···6293···93186···186
size1131311111313131312···22···22···22···2

102 irreducible representations

dim1111112222
type+++++
imageC1C2C2C3C6C6D31D62C3×D31C6×D31
kernelC6×D31C3×D31C186D62D31C62C6C3C2C1
# reps12124215153030

Matrix representation of C6×D31 in GL2(𝔽373) generated by

890
089
,
3461
2290
,
25383
213120
G:=sub<GL(2,GF(373))| [89,0,0,89],[346,2,1,290],[253,213,83,120] >;

C6×D31 in GAP, Magma, Sage, TeX

C_6\times D_{31}
% in TeX

G:=Group("C6xD31");
// GroupNames label

G:=SmallGroup(372,12);
// by ID

G=gap.SmallGroup(372,12);
# by ID

G:=PCGroup([4,-2,-2,-3,-31,5763]);
// Polycyclic

G:=Group<a,b,c|a^6=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D31 in TeX

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