Copied to
clipboard

## G = D4×C3×C15order 360 = 23·32·5

### Direct product of C3×C15 and D4

direct product, metacyclic, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C3×C15
 Chief series C1 — C2 — C10 — C30 — C3×C30 — C6×C30 — D4×C3×C15
 Lower central C1 — C2 — D4×C3×C15
 Upper central C1 — C3×C30 — D4×C3×C15

Generators and relations for D4×C3×C15
G = < a,b,c,d | a3=b15=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 120 in 96 conjugacy classes, 72 normal (16 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, D4, C32, C10, C10, C12, C2×C6, C15, C3×C6, C3×C6, C20, C2×C10, C3×D4, C30, C30, C3×C12, C62, C5×D4, C3×C15, C60, C2×C30, D4×C32, C3×C30, C3×C30, D4×C15, C3×C60, C6×C30, D4×C3×C15
Quotients: C1, C2, C3, C22, C5, C6, D4, C32, C10, C2×C6, C15, C3×C6, C2×C10, C3×D4, C30, C62, C5×D4, C3×C15, C2×C30, D4×C32, C3×C30, D4×C15, C6×C30, D4×C3×C15

Smallest permutation representation of D4×C3×C15
On 180 points
Generators in S180
(1 148 41)(2 149 42)(3 150 43)(4 136 44)(5 137 45)(6 138 31)(7 139 32)(8 140 33)(9 141 34)(10 142 35)(11 143 36)(12 144 37)(13 145 38)(14 146 39)(15 147 40)(16 177 102)(17 178 103)(18 179 104)(19 180 105)(20 166 91)(21 167 92)(22 168 93)(23 169 94)(24 170 95)(25 171 96)(26 172 97)(27 173 98)(28 174 99)(29 175 100)(30 176 101)(46 62 155)(47 63 156)(48 64 157)(49 65 158)(50 66 159)(51 67 160)(52 68 161)(53 69 162)(54 70 163)(55 71 164)(56 72 165)(57 73 151)(58 74 152)(59 75 153)(60 61 154)(76 113 131)(77 114 132)(78 115 133)(79 116 134)(80 117 135)(81 118 121)(82 119 122)(83 120 123)(84 106 124)(85 107 125)(86 108 126)(87 109 127)(88 110 128)(89 111 129)(90 112 130)
(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)
(1 101 77 51)(2 102 78 52)(3 103 79 53)(4 104 80 54)(5 105 81 55)(6 91 82 56)(7 92 83 57)(8 93 84 58)(9 94 85 59)(10 95 86 60)(11 96 87 46)(12 97 88 47)(13 98 89 48)(14 99 90 49)(15 100 76 50)(16 115 68 149)(17 116 69 150)(18 117 70 136)(19 118 71 137)(20 119 72 138)(21 120 73 139)(22 106 74 140)(23 107 75 141)(24 108 61 142)(25 109 62 143)(26 110 63 144)(27 111 64 145)(28 112 65 146)(29 113 66 147)(30 114 67 148)(31 166 122 165)(32 167 123 151)(33 168 124 152)(34 169 125 153)(35 170 126 154)(36 171 127 155)(37 172 128 156)(38 173 129 157)(39 174 130 158)(40 175 131 159)(41 176 132 160)(42 177 133 161)(43 178 134 162)(44 179 135 163)(45 180 121 164)
(16 68)(17 69)(18 70)(19 71)(20 72)(21 73)(22 74)(23 75)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(46 96)(47 97)(48 98)(49 99)(50 100)(51 101)(52 102)(53 103)(54 104)(55 105)(56 91)(57 92)(58 93)(59 94)(60 95)(151 167)(152 168)(153 169)(154 170)(155 171)(156 172)(157 173)(158 174)(159 175)(160 176)(161 177)(162 178)(163 179)(164 180)(165 166)

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

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

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

225 conjugacy classes

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

225 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 D4 C3×D4 C5×D4 D4×C15 kernel D4×C3×C15 C3×C60 C6×C30 D4×C15 D4×C32 C60 C2×C30 C3×C12 C62 C3×D4 C12 C2×C6 C3×C15 C15 C32 C3 # reps 1 1 2 8 4 8 16 4 8 32 32 64 1 8 4 32

Matrix representation of D4×C3×C15 in GL4(𝔽61) generated by

 13 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 47 0 0 0 0 58 0 0 0 0 13 0 0 0 0 13
,
 60 0 0 0 0 1 0 0 0 0 0 60 0 0 1 0
,
 60 0 0 0 0 60 0 0 0 0 1 0 0 0 0 60
G:=sub<GL(4,GF(61))| [13,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[47,0,0,0,0,58,0,0,0,0,13,0,0,0,0,13],[60,0,0,0,0,1,0,0,0,0,0,1,0,0,60,0],[60,0,0,0,0,60,0,0,0,0,1,0,0,0,0,60] >;

D4×C3×C15 in GAP, Magma, Sage, TeX

D_4\times C_3\times C_{15}
% in TeX

G:=Group("D4xC3xC15");
// GroupNames label

G:=SmallGroup(360,116);
// by ID

G=gap.SmallGroup(360,116);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-5,-2,2185]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^15=c^4=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

׿
×
𝔽