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## G = D5×C36order 360 = 23·32·5

### Direct product of C36 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C36
 Chief series C1 — C5 — C15 — C30 — C90 — D5×C18 — D5×C36
 Lower central C5 — D5×C36
 Upper central C1 — C36

Generators and relations for D5×C36
G = < a,b,c | a36=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C36
On 180 points
Generators in S180
(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)
(1 134 47 152 85)(2 135 48 153 86)(3 136 49 154 87)(4 137 50 155 88)(5 138 51 156 89)(6 139 52 157 90)(7 140 53 158 91)(8 141 54 159 92)(9 142 55 160 93)(10 143 56 161 94)(11 144 57 162 95)(12 109 58 163 96)(13 110 59 164 97)(14 111 60 165 98)(15 112 61 166 99)(16 113 62 167 100)(17 114 63 168 101)(18 115 64 169 102)(19 116 65 170 103)(20 117 66 171 104)(21 118 67 172 105)(22 119 68 173 106)(23 120 69 174 107)(24 121 70 175 108)(25 122 71 176 73)(26 123 72 177 74)(27 124 37 178 75)(28 125 38 179 76)(29 126 39 180 77)(30 127 40 145 78)(31 128 41 146 79)(32 129 42 147 80)(33 130 43 148 81)(34 131 44 149 82)(35 132 45 150 83)(36 133 46 151 84)
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 145)(128 146)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 153)(136 154)(137 155)(138 156)(139 157)(140 158)(141 159)(142 160)(143 161)(144 162)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 9A ··· 9F 10A 10B 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 18A ··· 18F 18G ··· 18R 20A 20B 20C 20D 30A 30B 30C 30D 36A ··· 36L 36M ··· 36X 45A ··· 45L 60A ··· 60H 90A ··· 90L 180A ··· 180X order 1 2 2 2 3 3 4 4 4 4 5 5 6 6 6 6 6 6 9 ··· 9 10 10 12 12 12 12 12 12 12 12 15 15 15 15 18 ··· 18 18 ··· 18 20 20 20 20 30 30 30 30 36 ··· 36 36 ··· 36 45 ··· 45 60 ··· 60 90 ··· 90 180 ··· 180 size 1 1 5 5 1 1 1 1 5 5 2 2 1 1 5 5 5 5 1 ··· 1 2 2 1 1 1 1 5 5 5 5 2 2 2 2 1 ··· 1 5 ··· 5 2 2 2 2 2 2 2 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C9 C12 C18 C18 C18 C36 D5 D10 C3×D5 C4×D5 C6×D5 C9×D5 D5×C12 D5×C18 D5×C36 kernel D5×C36 C9×Dic5 C180 D5×C18 D5×C12 C9×D5 C3×Dic5 C60 C6×D5 C4×D5 C3×D5 Dic5 C20 D10 D5 C36 C18 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 6 8 6 6 6 24 2 2 4 4 4 12 8 12 24

Matrix representation of D5×C36 in GL3(𝔽181) generated by

 142 0 0 0 7 0 0 0 7
,
 1 0 0 0 167 1 0 180 0
,
 180 0 0 0 1 167 0 0 180
G:=sub<GL(3,GF(181))| [142,0,0,0,7,0,0,0,7],[1,0,0,0,167,180,0,1,0],[180,0,0,0,1,0,0,167,180] >;

D5×C36 in GAP, Magma, Sage, TeX

D_5\times C_{36}
% in TeX

G:=Group("D5xC36");
// GroupNames label

G:=SmallGroup(360,16);
// by ID

G=gap.SmallGroup(360,16);
# by ID

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

G:=Group<a,b,c|a^36=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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