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G = C5×D31order 310 = 2·5·31

Direct product of C5 and D31

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

Aliases: C5×D31, C313C10, C1552C2, SmallGroup(310,4)

Series: Derived Chief Lower central Upper central

C1C31 — C5×D31
C1C31C155 — C5×D31
C31 — C5×D31
C1C5

Generators and relations for C5×D31
 G = < a,b,c | a5=b31=c2=1, ab=ba, ac=ca, cbc=b-1 >

31C2
31C10

Smallest permutation representation of C5×D31
On 155 points
Generators in S155
(1 136 114 87 35)(2 137 115 88 36)(3 138 116 89 37)(4 139 117 90 38)(5 140 118 91 39)(6 141 119 92 40)(7 142 120 93 41)(8 143 121 63 42)(9 144 122 64 43)(10 145 123 65 44)(11 146 124 66 45)(12 147 94 67 46)(13 148 95 68 47)(14 149 96 69 48)(15 150 97 70 49)(16 151 98 71 50)(17 152 99 72 51)(18 153 100 73 52)(19 154 101 74 53)(20 155 102 75 54)(21 125 103 76 55)(22 126 104 77 56)(23 127 105 78 57)(24 128 106 79 58)(25 129 107 80 59)(26 130 108 81 60)(27 131 109 82 61)(28 132 110 83 62)(29 133 111 84 32)(30 134 112 85 33)(31 135 113 86 34)
(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)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 37)(33 36)(34 35)(38 62)(39 61)(40 60)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(80 93)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(94 102)(95 101)(96 100)(97 99)(103 124)(104 123)(105 122)(106 121)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)(125 146)(126 145)(127 144)(128 143)(129 142)(130 141)(131 140)(132 139)(133 138)(134 137)(135 136)(147 155)(148 154)(149 153)(150 152)

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

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

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

85 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D31A···31O155A···155BH
order1255551010101031···31155···155
size1311111313131312···22···2

85 irreducible representations

dim111122
type+++
imageC1C2C5C10D31C5×D31
kernelC5×D31C155D31C31C5C1
# reps11441560

Matrix representation of C5×D31 in GL2(𝔽311) generated by

60
06
,
146134
31014
,
755
214304
G:=sub<GL(2,GF(311))| [6,0,0,6],[146,310,134,14],[7,214,55,304] >;

C5×D31 in GAP, Magma, Sage, TeX

C_5\times D_{31}
% in TeX

G:=Group("C5xD31");
// GroupNames label

G:=SmallGroup(310,4);
// by ID

G=gap.SmallGroup(310,4);
# by ID

G:=PCGroup([3,-2,-5,-31,2702]);
// Polycyclic

G:=Group<a,b,c|a^5=b^31=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D31 in TeX

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