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G = D5×C39order 390 = 2·3·5·13

Direct product of C39 and D5

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

Aliases: D5×C39, C5⋊C78, C657C6, C152C26, C1956C2, SmallGroup(390,6)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C39
C1C5C65C195 — D5×C39
C5 — D5×C39
C1C39

Generators and relations for D5×C39
 G = < a,b,c | a39=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C6
5C26
5C78

Smallest permutation representation of D5×C39
On 195 points
Generators in S195
(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)(157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195)
(1 65 176 139 87)(2 66 177 140 88)(3 67 178 141 89)(4 68 179 142 90)(5 69 180 143 91)(6 70 181 144 92)(7 71 182 145 93)(8 72 183 146 94)(9 73 184 147 95)(10 74 185 148 96)(11 75 186 149 97)(12 76 187 150 98)(13 77 188 151 99)(14 78 189 152 100)(15 40 190 153 101)(16 41 191 154 102)(17 42 192 155 103)(18 43 193 156 104)(19 44 194 118 105)(20 45 195 119 106)(21 46 157 120 107)(22 47 158 121 108)(23 48 159 122 109)(24 49 160 123 110)(25 50 161 124 111)(26 51 162 125 112)(27 52 163 126 113)(28 53 164 127 114)(29 54 165 128 115)(30 55 166 129 116)(31 56 167 130 117)(32 57 168 131 79)(33 58 169 132 80)(34 59 170 133 81)(35 60 171 134 82)(36 61 172 135 83)(37 62 173 136 84)(38 63 174 137 85)(39 64 175 138 86)
(1 87)(2 88)(3 89)(4 90)(5 91)(6 92)(7 93)(8 94)(9 95)(10 96)(11 97)(12 98)(13 99)(14 100)(15 101)(16 102)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 113)(28 114)(29 115)(30 116)(31 117)(32 79)(33 80)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 153)(41 154)(42 155)(43 156)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 137)(64 138)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)

G:=sub<Sym(195)| (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)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (1,65,176,139,87)(2,66,177,140,88)(3,67,178,141,89)(4,68,179,142,90)(5,69,180,143,91)(6,70,181,144,92)(7,71,182,145,93)(8,72,183,146,94)(9,73,184,147,95)(10,74,185,148,96)(11,75,186,149,97)(12,76,187,150,98)(13,77,188,151,99)(14,78,189,152,100)(15,40,190,153,101)(16,41,191,154,102)(17,42,192,155,103)(18,43,193,156,104)(19,44,194,118,105)(20,45,195,119,106)(21,46,157,120,107)(22,47,158,121,108)(23,48,159,122,109)(24,49,160,123,110)(25,50,161,124,111)(26,51,162,125,112)(27,52,163,126,113)(28,53,164,127,114)(29,54,165,128,115)(30,55,166,129,116)(31,56,167,130,117)(32,57,168,131,79)(33,58,169,132,80)(34,59,170,133,81)(35,60,171,134,82)(36,61,172,135,83)(37,62,173,136,84)(38,63,174,137,85)(39,64,175,138,86), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,153)(41,154)(42,155)(43,156)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)>;

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)(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156)(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (1,65,176,139,87)(2,66,177,140,88)(3,67,178,141,89)(4,68,179,142,90)(5,69,180,143,91)(6,70,181,144,92)(7,71,182,145,93)(8,72,183,146,94)(9,73,184,147,95)(10,74,185,148,96)(11,75,186,149,97)(12,76,187,150,98)(13,77,188,151,99)(14,78,189,152,100)(15,40,190,153,101)(16,41,191,154,102)(17,42,192,155,103)(18,43,193,156,104)(19,44,194,118,105)(20,45,195,119,106)(21,46,157,120,107)(22,47,158,121,108)(23,48,159,122,109)(24,49,160,123,110)(25,50,161,124,111)(26,51,162,125,112)(27,52,163,126,113)(28,53,164,127,114)(29,54,165,128,115)(30,55,166,129,116)(31,56,167,130,117)(32,57,168,131,79)(33,58,169,132,80)(34,59,170,133,81)(35,60,171,134,82)(36,61,172,135,83)(37,62,173,136,84)(38,63,174,137,85)(39,64,175,138,86), (1,87)(2,88)(3,89)(4,90)(5,91)(6,92)(7,93)(8,94)(9,95)(10,96)(11,97)(12,98)(13,99)(14,100)(15,101)(16,102)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,113)(28,114)(29,115)(30,116)(31,117)(32,79)(33,80)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,153)(41,154)(42,155)(43,156)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,137)(64,138)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152) );

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),(118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156),(157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)], [(1,65,176,139,87),(2,66,177,140,88),(3,67,178,141,89),(4,68,179,142,90),(5,69,180,143,91),(6,70,181,144,92),(7,71,182,145,93),(8,72,183,146,94),(9,73,184,147,95),(10,74,185,148,96),(11,75,186,149,97),(12,76,187,150,98),(13,77,188,151,99),(14,78,189,152,100),(15,40,190,153,101),(16,41,191,154,102),(17,42,192,155,103),(18,43,193,156,104),(19,44,194,118,105),(20,45,195,119,106),(21,46,157,120,107),(22,47,158,121,108),(23,48,159,122,109),(24,49,160,123,110),(25,50,161,124,111),(26,51,162,125,112),(27,52,163,126,113),(28,53,164,127,114),(29,54,165,128,115),(30,55,166,129,116),(31,56,167,130,117),(32,57,168,131,79),(33,58,169,132,80),(34,59,170,133,81),(35,60,171,134,82),(36,61,172,135,83),(37,62,173,136,84),(38,63,174,137,85),(39,64,175,138,86)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,92),(7,93),(8,94),(9,95),(10,96),(11,97),(12,98),(13,99),(14,100),(15,101),(16,102),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,113),(28,114),(29,115),(30,116),(31,117),(32,79),(33,80),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,153),(41,154),(42,155),(43,156),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,129),(56,130),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,137),(64,138),(65,139),(66,140),(67,141),(68,142),(69,143),(70,144),(71,145),(72,146),(73,147),(74,148),(75,149),(76,150),(77,151),(78,152)])

156 conjugacy classes

class 1  2 3A3B5A5B6A6B13A···13L15A15B15C15D26A···26L39A···39X65A···65X78A···78X195A···195AV
order1233556613···131515151526···2639···3965···6578···78195···195
size151122551···122225···51···12···25···52···2

156 irreducible representations

dim111111112222
type+++
imageC1C2C3C6C13C26C39C78D5C3×D5D5×C13D5×C39
kernelD5×C39C195D5×C13C65C3×D5C15D5C5C39C13C3C1
# reps112212122424242448

Matrix representation of D5×C39 in GL3(𝔽1171) generated by

42000
03700
00370
,
100
01131
011700
,
117000
01113
001170
G:=sub<GL(3,GF(1171))| [420,0,0,0,370,0,0,0,370],[1,0,0,0,113,1170,0,1,0],[1170,0,0,0,1,0,0,113,1170] >;

D5×C39 in GAP, Magma, Sage, TeX

D_5\times C_{39}
% in TeX

G:=Group("D5xC39");
// GroupNames label

G:=SmallGroup(390,6);
// by ID

G=gap.SmallGroup(390,6);
# by ID

G:=PCGroup([4,-2,-3,-13,-5,4995]);
// Polycyclic

G:=Group<a,b,c|a^39=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×C39 in TeX

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