Copied to
clipboard

G = C9×D20order 360 = 23·32·5

Direct product of C9 and D20

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

Aliases: C9×D20, C456D4, C363D5, C201C18, C1804C2, C60.4C6, D101C18, C18.15D10, C90.20C22, C4⋊(C9×D5), C51(D4×C9), C3.(C3×D20), (C3×D20).C3, (D5×C18)⋊4C2, (C6×D5).2C6, C12.4(C3×D5), C15.1(C3×D4), C2.4(D5×C18), C6.15(C6×D5), C10.3(C2×C18), C30.15(C2×C6), SmallGroup(360,17)

Series: Derived Chief Lower central Upper central

C1C10 — C9×D20
C1C5C15C30C90D5×C18 — C9×D20
C5C10 — C9×D20
C1C18C36

Generators and relations for C9×D20
 G = < a,b,c | a9=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

10C2
10C2
5C22
5C22
10C6
10C6
2D5
2D5
5D4
5C2×C6
5C2×C6
10C18
10C18
2C3×D5
2C3×D5
5C3×D4
5C2×C18
5C2×C18
2C9×D5
2C9×D5
5D4×C9

Smallest permutation representation of C9×D20
On 180 points
Generators in S180
(1 63 95 34 112 165 147 134 57)(2 64 96 35 113 166 148 135 58)(3 65 97 36 114 167 149 136 59)(4 66 98 37 115 168 150 137 60)(5 67 99 38 116 169 151 138 41)(6 68 100 39 117 170 152 139 42)(7 69 81 40 118 171 153 140 43)(8 70 82 21 119 172 154 121 44)(9 71 83 22 120 173 155 122 45)(10 72 84 23 101 174 156 123 46)(11 73 85 24 102 175 157 124 47)(12 74 86 25 103 176 158 125 48)(13 75 87 26 104 177 159 126 49)(14 76 88 27 105 178 160 127 50)(15 77 89 28 106 179 141 128 51)(16 78 90 29 107 180 142 129 52)(17 79 91 30 108 161 143 130 53)(18 80 92 31 109 162 144 131 54)(19 61 93 32 110 163 145 132 55)(20 62 94 33 111 164 146 133 56)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 64)(62 63)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)(81 88)(82 87)(83 86)(84 85)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(121 126)(122 125)(123 124)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 152)(142 151)(143 150)(144 149)(145 148)(146 147)(153 160)(154 159)(155 158)(156 157)(161 168)(162 167)(163 166)(164 165)(169 180)(170 179)(171 178)(172 177)(173 176)(174 175)

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

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

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

117 conjugacy classes

class 1 2A2B2C3A3B 4 5A5B6A6B6C6D6E6F9A···9F10A10B12A12B15A15B15C15D18A···18F18G···18R20A20B20C20D30A30B30C30D36A···36F45A···45L60A···60H90A···90L180A···180X
order1222334556666669···9101012121515151518···1818···18202020203030303036···3645···4560···6090···90180···180
size1110101122211101010101···1222222221···110···10222222222···22···22···22···22···2

117 irreducible representations

dim111111111222222222222
type+++++++
imageC1C2C2C3C6C6C9C18C18D4D5D10C3×D4C3×D5D20C6×D5D4×C9C9×D5C3×D20D5×C18C9×D20
kernelC9×D20C180D5×C18C3×D20C60C6×D5D20C20D10C45C36C18C15C12C9C6C5C4C3C2C1
# reps1122246612122244461281224

Matrix representation of C9×D20 in GL2(𝔽19) generated by

90
09
,
91
1016
,
156
74
G:=sub<GL(2,GF(19))| [9,0,0,9],[9,10,1,16],[15,7,6,4] >;

C9×D20 in GAP, Magma, Sage, TeX

C_9\times D_{20}
% in TeX

G:=Group("C9xD20");
// GroupNames label

G:=SmallGroup(360,17);
// by ID

G=gap.SmallGroup(360,17);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,169,79,122,10373]);
// Polycyclic

G:=Group<a,b,c|a^9=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C9×D20 in TeX

׿
×
𝔽