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## G = Dic3×C3⋊Dic3order 432 = 24·33

### Direct product of Dic3 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — Dic3×C3⋊Dic3
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C3×C62 — Dic3×C3×C6 — Dic3×C3⋊Dic3
 Lower central C33 — Dic3×C3⋊Dic3
 Upper central C1 — C22

Generators and relations for Dic3×C3⋊Dic3
G = < a,b,c,d,e | a6=c3=d6=1, b2=a3, e2=d3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 984 in 244 conjugacy classes, 88 normal (26 characteristic)
C1, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C32, C32, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C62, C62, C4×Dic3, C32×C6, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C2×C3⋊Dic3, C6×C12, C32×Dic3, C3×C3⋊Dic3, C335C4, C3×C62, Dic32, C4×C3⋊Dic3, Dic3×C3×C6, C6×C3⋊Dic3, C2×C335C4, Dic3×C3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C3⋊S3, C4×S3, C2×Dic3, C3⋊Dic3, S32, C2×C3⋊S3, C4×Dic3, S3×Dic3, C6.D6, C4×C3⋊S3, C2×C3⋊Dic3, S3×C3⋊S3, Dic32, C4×C3⋊Dic3, S3×C3⋊Dic3, Dic3×C3⋊S3, C338(C2×C4), Dic3×C3⋊Dic3

Smallest permutation representation of Dic3×C3⋊Dic3
On 144 points
Generators in S144
(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 98 4 101)(2 97 5 100)(3 102 6 99)(7 91 10 94)(8 96 11 93)(9 95 12 92)(13 89 16 86)(14 88 17 85)(15 87 18 90)(19 78 22 75)(20 77 23 74)(21 76 24 73)(25 79 28 82)(26 84 29 81)(27 83 30 80)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 113 40 110)(38 112 41 109)(39 111 42 114)(43 117 46 120)(44 116 47 119)(45 115 48 118)(49 125 52 122)(50 124 53 121)(51 123 54 126)(55 127 58 130)(56 132 59 129)(57 131 60 128)(61 138 64 135)(62 137 65 134)(63 136 66 133)(67 140 70 143)(68 139 71 142)(69 144 72 141)
(1 9 16)(2 10 17)(3 11 18)(4 12 13)(5 7 14)(6 8 15)(19 142 117)(20 143 118)(21 144 119)(22 139 120)(23 140 115)(24 141 116)(25 31 42)(26 32 37)(27 33 38)(28 34 39)(29 35 40)(30 36 41)(43 75 71)(44 76 72)(45 77 67)(46 78 68)(47 73 69)(48 74 70)(49 60 62)(50 55 63)(51 56 64)(52 57 65)(53 58 66)(54 59 61)(79 104 114)(80 105 109)(81 106 110)(82 107 111)(83 108 112)(84 103 113)(85 97 94)(86 98 95)(87 99 96)(88 100 91)(89 101 92)(90 102 93)(121 130 133)(122 131 134)(123 132 135)(124 127 136)(125 128 137)(126 129 138)
(1 42 14 35 11 27)(2 37 15 36 12 28)(3 38 16 31 7 29)(4 39 17 32 8 30)(5 40 18 33 9 25)(6 41 13 34 10 26)(19 136 115 131 144 126)(20 137 116 132 139 121)(21 138 117 127 140 122)(22 133 118 128 141 123)(23 134 119 129 142 124)(24 135 120 130 143 125)(43 55 67 52 73 61)(44 56 68 53 74 62)(45 57 69 54 75 63)(46 58 70 49 76 64)(47 59 71 50 77 65)(48 60 72 51 78 66)(79 100 110 90 108 95)(80 101 111 85 103 96)(81 102 112 86 104 91)(82 97 113 87 105 92)(83 98 114 88 106 93)(84 99 109 89 107 94)
(1 71 35 65)(2 72 36 66)(3 67 31 61)(4 68 32 62)(5 69 33 63)(6 70 34 64)(7 73 38 55)(8 74 39 56)(9 75 40 57)(10 76 41 58)(11 77 42 59)(12 78 37 60)(13 46 26 49)(14 47 27 50)(15 48 28 51)(16 43 29 52)(17 44 30 53)(18 45 25 54)(19 110 131 95)(20 111 132 96)(21 112 127 91)(22 113 128 92)(23 114 129 93)(24 109 130 94)(79 126 90 115)(80 121 85 116)(81 122 86 117)(82 123 87 118)(83 124 88 119)(84 125 89 120)(97 141 105 133)(98 142 106 134)(99 143 107 135)(100 144 108 136)(101 139 103 137)(102 140 104 138)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3E 3F 3G 3H 3I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6O 6P ··· 6AA 12A ··· 12P 12Q 12R 12S 12T order 1 2 2 2 3 ··· 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 12 12 12 12 size 1 1 1 1 2 ··· 2 4 4 4 4 3 3 3 3 9 9 9 9 27 27 27 27 2 ··· 2 4 ··· 4 6 ··· 6 18 18 18 18

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + - - + + - + image C1 C2 C2 C2 C4 C4 C4 S3 S3 Dic3 Dic3 D6 C4×S3 S32 S3×Dic3 C6.D6 kernel Dic3×C3⋊Dic3 Dic3×C3×C6 C6×C3⋊Dic3 C2×C33⋊5C4 C32×Dic3 C3×C3⋊Dic3 C33⋊5C4 C6×Dic3 C2×C3⋊Dic3 C3×Dic3 C3⋊Dic3 C62 C3×C6 C2×C6 C6 C6 # reps 1 1 1 1 4 4 4 4 1 8 2 5 20 4 8 4

Matrix representation of Dic3×C3⋊Dic3 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 11 3 0 0 0 0 12 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 10 0 0 0 0 1 12 0 0 0 0 0 0 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 8 5 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,12],[11,12,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[2,1,0,0,0,0,10,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,8,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

Dic3×C3⋊Dic3 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("Dic3xC3:Dic3");
// GroupNames label

G:=SmallGroup(432,448);
// by ID

G=gap.SmallGroup(432,448);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,64,571,2028,14118]);
// Polycyclic

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

׿
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