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## G = Dic3×Dic9order 432 = 24·33

### Direct product of Dic3 and Dic9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — Dic3×Dic9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C6×C18 — Dic3×C18 — Dic3×Dic9
 Lower central C3×C9 — Dic3×Dic9
 Upper central C1 — C22

Generators and relations for Dic3×Dic9
G = < a,b,c,d | a6=c18=1, b2=a3, d2=c9, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 444 in 106 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C9, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C18, C18, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, C4×Dic3, C3×C18, C2×Dic9, C2×Dic9, C2×C36, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×C18, C4×Dic9, Dic32, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, Dic3×Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, D9, C4×S3, C2×Dic3, Dic9, D18, S32, C4×Dic3, C4×D9, C2×Dic9, S3×Dic3, C6.D6, S3×D9, C4×Dic9, Dic32, Dic3×D9, C18.D6, S3×Dic9, Dic3×Dic9

Smallest permutation representation of Dic3×Dic9
On 144 points
Generators in S144
(1 70 7 58 13 64)(2 71 8 59 14 65)(3 72 9 60 15 66)(4 55 10 61 16 67)(5 56 11 62 17 68)(6 57 12 63 18 69)(19 100 31 94 25 106)(20 101 32 95 26 107)(21 102 33 96 27 108)(22 103 34 97 28 91)(23 104 35 98 29 92)(24 105 36 99 30 93)(37 89 43 77 49 83)(38 90 44 78 50 84)(39 73 45 79 51 85)(40 74 46 80 52 86)(41 75 47 81 53 87)(42 76 48 82 54 88)(109 132 121 144 115 138)(110 133 122 127 116 139)(111 134 123 128 117 140)(112 135 124 129 118 141)(113 136 125 130 119 142)(114 137 126 131 120 143)
(1 97 58 22)(2 98 59 23)(3 99 60 24)(4 100 61 25)(5 101 62 26)(6 102 63 27)(7 103 64 28)(8 104 65 29)(9 105 66 30)(10 106 67 31)(11 107 68 32)(12 108 69 33)(13 91 70 34)(14 92 71 35)(15 93 72 36)(16 94 55 19)(17 95 56 20)(18 96 57 21)(37 121 77 138)(38 122 78 139)(39 123 79 140)(40 124 80 141)(41 125 81 142)(42 126 82 143)(43 109 83 144)(44 110 84 127)(45 111 85 128)(46 112 86 129)(47 113 87 130)(48 114 88 131)(49 115 89 132)(50 116 90 133)(51 117 73 134)(52 118 74 135)(53 119 75 136)(54 120 76 137)
(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)
(1 123 10 114)(2 122 11 113)(3 121 12 112)(4 120 13 111)(5 119 14 110)(6 118 15 109)(7 117 16 126)(8 116 17 125)(9 115 18 124)(19 42 28 51)(20 41 29 50)(21 40 30 49)(22 39 31 48)(23 38 32 47)(24 37 33 46)(25 54 34 45)(26 53 35 44)(27 52 36 43)(55 143 64 134)(56 142 65 133)(57 141 66 132)(58 140 67 131)(59 139 68 130)(60 138 69 129)(61 137 70 128)(62 136 71 127)(63 135 72 144)(73 94 82 103)(74 93 83 102)(75 92 84 101)(76 91 85 100)(77 108 86 99)(78 107 87 98)(79 106 88 97)(80 105 89 96)(81 104 90 95)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6F 6G 6H 6I 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18I 18J ··· 18R 36A ··· 36L order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 4 3 3 3 3 9 9 9 9 27 27 27 27 2 ··· 2 4 4 4 2 2 2 4 4 4 6 6 6 6 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + - + - + + - + + - + + - + - image C1 C2 C2 C2 C4 C4 C4 S3 S3 Dic3 D6 Dic3 D6 D9 C4×S3 C4×S3 Dic9 D18 C4×D9 S32 S3×Dic3 C6.D6 S3×D9 Dic3×D9 C18.D6 S3×Dic9 kernel Dic3×Dic9 C6×Dic9 Dic3×C18 C2×C9⋊Dic3 C3×Dic9 C9×Dic3 C9⋊Dic3 C2×Dic9 C6×Dic3 Dic9 C2×C18 C3×Dic3 C62 C2×Dic3 C18 C3×C6 Dic3 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 4 4 1 1 2 1 2 1 3 4 4 6 3 12 1 2 1 3 3 3 3

Matrix representation of Dic3×Dic9 in GL6(𝔽37)

 0 36 0 0 0 0 1 36 0 0 0 0 0 0 1 1 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 31 0 0 0 0 31 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 26 3 0 0 0 0 25 20
,
 6 0 0 0 0 0 0 6 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 35 36

G:=sub<GL(6,GF(37))| [0,1,0,0,0,0,36,36,0,0,0,0,0,0,1,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,31,0,0,0,0,31,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,25,0,0,0,0,3,20],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,35,0,0,0,0,0,36] >;

Dic3×Dic9 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times {\rm Dic}_9
% in TeX

G:=Group("Dic3xDic9");
// GroupNames label

G:=SmallGroup(432,87);
// by ID

G=gap.SmallGroup(432,87);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,36,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^18=1,b^2=a^3,d^2=c^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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