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## G = C12×C3⋊Dic3order 432 = 24·33

### Direct product of C12 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C12×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C3×C62 — C6×C3⋊Dic3 — C12×C3⋊Dic3
 Lower central C32 — C12×C3⋊Dic3
 Upper central C1 — C2×C12

Generators and relations for C12×C3⋊Dic3
G = < a,b,c,d | a12=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 532 in 244 conjugacy classes, 110 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C62, C4×Dic3, C4×C12, C32×C6, C32×C6, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, C3×C3⋊Dic3, C32×C12, C3×C62, Dic3×C12, C4×C3⋊Dic3, C6×C3⋊Dic3, C3×C6×C12, C12×C3⋊Dic3
Quotients:

Smallest permutation representation of C12×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 40 56)(2 41 57)(3 42 58)(4 43 59)(5 44 60)(6 45 49)(7 46 50)(8 47 51)(9 48 52)(10 37 53)(11 38 54)(12 39 55)(13 87 35)(14 88 36)(15 89 25)(16 90 26)(17 91 27)(18 92 28)(19 93 29)(20 94 30)(21 95 31)(22 96 32)(23 85 33)(24 86 34)(61 76 123)(62 77 124)(63 78 125)(64 79 126)(65 80 127)(66 81 128)(67 82 129)(68 83 130)(69 84 131)(70 73 132)(71 74 121)(72 75 122)(97 142 114)(98 143 115)(99 144 116)(100 133 117)(101 134 118)(102 135 119)(103 136 120)(104 137 109)(105 138 110)(106 139 111)(107 140 112)(108 141 113)
(1 144 60 103 48 112)(2 133 49 104 37 113)(3 134 50 105 38 114)(4 135 51 106 39 115)(5 136 52 107 40 116)(6 137 53 108 41 117)(7 138 54 97 42 118)(8 139 55 98 43 119)(9 140 56 99 44 120)(10 141 57 100 45 109)(11 142 58 101 46 110)(12 143 59 102 47 111)(13 71 31 129 91 78)(14 72 32 130 92 79)(15 61 33 131 93 80)(16 62 34 132 94 81)(17 63 35 121 95 82)(18 64 36 122 96 83)(19 65 25 123 85 84)(20 66 26 124 86 73)(21 67 27 125 87 74)(22 68 28 126 88 75)(23 69 29 127 89 76)(24 70 30 128 90 77)
(1 131 103 15)(2 132 104 16)(3 121 105 17)(4 122 106 18)(5 123 107 19)(6 124 108 20)(7 125 97 21)(8 126 98 22)(9 127 99 23)(10 128 100 24)(11 129 101 13)(12 130 102 14)(25 40 84 136)(26 41 73 137)(27 42 74 138)(28 43 75 139)(29 44 76 140)(30 45 77 141)(31 46 78 142)(32 47 79 143)(33 48 80 144)(34 37 81 133)(35 38 82 134)(36 39 83 135)(49 62 113 94)(50 63 114 95)(51 64 115 96)(52 65 116 85)(53 66 117 86)(54 67 118 87)(55 68 119 88)(56 69 120 89)(57 70 109 90)(58 71 110 91)(59 72 111 92)(60 61 112 93)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 6G ··· 6AP 12A ··· 12H 12I ··· 12BD 12BE ··· 12BT order 1 2 2 2 3 3 3 ··· 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 ··· 2 1 1 1 1 9 ··· 9 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 9 ··· 9

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 C3×S3 C4×S3 C3×Dic3 S3×C6 S3×C12 kernel C12×C3⋊Dic3 C6×C3⋊Dic3 C3×C6×C12 C4×C3⋊Dic3 C3×C3⋊Dic3 C32×C12 C2×C3⋊Dic3 C6×C12 C3⋊Dic3 C3×C12 C6×C12 C3×C12 C62 C2×C12 C3×C6 C12 C2×C6 C6 # reps 1 2 1 2 8 4 4 2 16 8 4 8 4 8 16 16 8 32

Matrix representation of C12×C3⋊Dic3 in GL5(𝔽13)

 7 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 11 0 0 0 0 0 11
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 3 0 0 0 0 0 9
,
 1 0 0 0 0 0 4 0 0 0 0 0 10 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 8 0 0 0 8 0 0 0 0 0 0 0 3 0 0 0 4 0

G:=sub<GL(5,GF(13))| [7,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,3,0,0,0,0,0,9],[1,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,0,4,0,0,0,3,0] >;

C12×C3⋊Dic3 in GAP, Magma, Sage, TeX

C_{12}\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("C12xC3:Dic3");
// GroupNames label

G:=SmallGroup(432,487);
// by ID

G=gap.SmallGroup(432,487);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^3=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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