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## G = D5×C3×C12order 360 = 23·32·5

### Direct product of C3×C12 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C3×C12
 Chief series C1 — C5 — C10 — C30 — C3×C30 — D5×C3×C6 — D5×C3×C12
 Lower central C5 — D5×C3×C12
 Upper central C1 — C3×C12

Generators and relations for D5×C3×C12
G = < a,b,c,d | a3=b12=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 216 in 96 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C32, D5, C10, C12, C12, C2×C6, C15, C3×C6, C3×C6, Dic5, C20, D10, C2×C12, C3×D5, C30, C3×C12, C3×C12, C62, C4×D5, C3×C15, C3×Dic5, C60, C6×D5, C6×C12, C32×D5, C3×C30, D5×C12, C32×Dic5, C3×C60, D5×C3×C6, D5×C3×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C32, D5, C12, C2×C6, C3×C6, D10, C2×C12, C3×D5, C3×C12, C62, C4×D5, C6×D5, C6×C12, C32×D5, D5×C12, D5×C3×C6, D5×C3×C12

Smallest permutation representation of D5×C3×C12
On 180 points
Generators in S180
(1 148 84)(2 149 73)(3 150 74)(4 151 75)(5 152 76)(6 153 77)(7 154 78)(8 155 79)(9 156 80)(10 145 81)(11 146 82)(12 147 83)(13 170 26)(14 171 27)(15 172 28)(16 173 29)(17 174 30)(18 175 31)(19 176 32)(20 177 33)(21 178 34)(22 179 35)(23 180 36)(24 169 25)(37 101 94)(38 102 95)(39 103 96)(40 104 85)(41 105 86)(42 106 87)(43 107 88)(44 108 89)(45 97 90)(46 98 91)(47 99 92)(48 100 93)(49 133 159)(50 134 160)(51 135 161)(52 136 162)(53 137 163)(54 138 164)(55 139 165)(56 140 166)(57 141 167)(58 142 168)(59 143 157)(60 144 158)(61 122 115)(62 123 116)(63 124 117)(64 125 118)(65 126 119)(66 127 120)(67 128 109)(68 129 110)(69 130 111)(70 131 112)(71 132 113)(72 121 114)
(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 57 94 119 22)(2 58 95 120 23)(3 59 96 109 24)(4 60 85 110 13)(5 49 86 111 14)(6 50 87 112 15)(7 51 88 113 16)(8 52 89 114 17)(9 53 90 115 18)(10 54 91 116 19)(11 55 92 117 20)(12 56 93 118 21)(25 74 157 103 128)(26 75 158 104 129)(27 76 159 105 130)(28 77 160 106 131)(29 78 161 107 132)(30 79 162 108 121)(31 80 163 97 122)(32 81 164 98 123)(33 82 165 99 124)(34 83 166 100 125)(35 84 167 101 126)(36 73 168 102 127)(37 65 179 148 141)(38 66 180 149 142)(39 67 169 150 143)(40 68 170 151 144)(41 69 171 152 133)(42 70 172 153 134)(43 71 173 154 135)(44 72 174 155 136)(45 61 175 156 137)(46 62 176 145 138)(47 63 177 146 139)(48 64 178 147 140)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 73)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 109)(60 110)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 133)(70 134)(71 135)(72 136)(121 162)(122 163)(123 164)(124 165)(125 166)(126 167)(127 168)(128 157)(129 158)(130 159)(131 160)(132 161)(145 176)(146 177)(147 178)(148 179)(149 180)(150 169)(151 170)(152 171)(153 172)(154 173)(155 174)(156 175)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 5A 5B 6A ··· 6H 6I ··· 6X 10A 10B 12A ··· 12P 12Q ··· 12AF 15A ··· 15P 20A 20B 20C 20D 30A ··· 30P 60A ··· 60AF order 1 2 2 2 3 ··· 3 4 4 4 4 5 5 6 ··· 6 6 ··· 6 10 10 12 ··· 12 12 ··· 12 15 ··· 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 5 5 1 ··· 1 1 1 5 5 2 2 1 ··· 1 5 ··· 5 2 2 1 ··· 1 5 ··· 5 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D5 D10 C3×D5 C4×D5 C6×D5 D5×C12 kernel D5×C3×C12 C32×Dic5 C3×C60 D5×C3×C6 D5×C12 C32×D5 C3×Dic5 C60 C6×D5 C3×D5 C3×C12 C3×C6 C12 C32 C6 C3 # reps 1 1 1 1 8 4 8 8 8 32 2 2 16 4 16 32

Matrix representation of D5×C3×C12 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 47 0 0 0 0 47
,
 48 0 0 0 0 50 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 17 1 0 0 60 0
,
 60 0 0 0 0 60 0 0 0 0 1 17 0 0 0 60
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,47,0,0,0,0,47],[48,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,17,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,17,60] >;

D5×C3×C12 in GAP, Magma, Sage, TeX

D_5\times C_3\times C_{12}
% in TeX

G:=Group("D5xC3xC12");
// GroupNames label

G:=SmallGroup(360,91);
// by ID

G=gap.SmallGroup(360,91);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-5,223,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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