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G = D5×C31order 310 = 2·5·31

Direct product of C31 and D5

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

Aliases: D5×C31, C5⋊C62, C1553C2, SmallGroup(310,3)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C31
C1C5C155 — D5×C31
C5 — D5×C31
C1C31

Generators and relations for D5×C31
 G = < a,b,c | a31=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C62

Smallest permutation representation of D5×C31
On 155 points
Generators in S155
(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 41 76 135 104)(2 42 77 136 105)(3 43 78 137 106)(4 44 79 138 107)(5 45 80 139 108)(6 46 81 140 109)(7 47 82 141 110)(8 48 83 142 111)(9 49 84 143 112)(10 50 85 144 113)(11 51 86 145 114)(12 52 87 146 115)(13 53 88 147 116)(14 54 89 148 117)(15 55 90 149 118)(16 56 91 150 119)(17 57 92 151 120)(18 58 93 152 121)(19 59 63 153 122)(20 60 64 154 123)(21 61 65 155 124)(22 62 66 125 94)(23 32 67 126 95)(24 33 68 127 96)(25 34 69 128 97)(26 35 70 129 98)(27 36 71 130 99)(28 37 72 131 100)(29 38 73 132 101)(30 39 74 133 102)(31 40 75 134 103)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 121)(19 122)(20 123)(21 124)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 102)(31 103)(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 151)(58 152)(59 153)(60 154)(61 155)(62 125)

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

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

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

124 conjugacy classes

class 1  2 5A5B31A···31AD62A···62AD155A···155BH
order125531···3162···62155···155
size15221···15···52···2

124 irreducible representations

dim111122
type+++
imageC1C2C31C62D5D5×C31
kernelD5×C31C155D5C5C31C1
# reps113030260

Matrix representation of D5×C31 in GL2(𝔽311) generated by

2700
0270
,
31060
31059
,
3100
3101
G:=sub<GL(2,GF(311))| [270,0,0,270],[310,310,60,59],[310,310,0,1] >;

D5×C31 in GAP, Magma, Sage, TeX

D_5\times C_{31}
% in TeX

G:=Group("D5xC31");
// GroupNames label

G:=SmallGroup(310,3);
// by ID

G=gap.SmallGroup(310,3);
# by ID

G:=PCGroup([3,-2,-31,-5,2234]);
// Polycyclic

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

Export

Subgroup lattice of D5×C31 in TeX

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