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## G = D4×C3×C12order 288 = 25·32

### Direct product of C3×C12 and D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C3×C12
 Chief series C1 — C2 — C22 — C2×C6 — C62 — C6×C12 — C32×C22⋊C4 — D4×C3×C12
 Lower central C1 — C2 — D4×C3×C12
 Upper central C1 — C6×C12 — D4×C3×C12

Generators and relations for D4×C3×C12
G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 372 in 282 conjugacy classes, 192 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×4], C4 [×3], C22, C22 [×4], C22 [×4], C6 [×12], C6 [×16], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C32, C12 [×16], C12 [×12], C2×C6 [×20], C2×C6 [×16], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×C6 [×3], C3×C6 [×4], C2×C12 [×20], C2×C12 [×16], C3×D4 [×16], C22×C6 [×8], C4×D4, C3×C12 [×4], C3×C12 [×3], C62, C62 [×4], C62 [×4], C4×C12 [×4], C3×C22⋊C4 [×8], C3×C4⋊C4 [×4], C22×C12 [×8], C6×D4 [×4], C6×C12 [×3], C6×C12 [×2], C6×C12 [×4], D4×C32 [×4], C2×C62 [×2], D4×C12 [×4], C122, C32×C22⋊C4 [×2], C32×C4⋊C4, C2×C6×C12 [×2], D4×C3×C6, D4×C3×C12
Quotients: C1, C2 [×7], C3 [×4], C4 [×4], C22 [×7], C6 [×28], C2×C4 [×6], D4 [×2], C23, C32, C12 [×16], C2×C6 [×28], C22×C4, C2×D4, C4○D4, C3×C6 [×7], C2×C12 [×24], C3×D4 [×8], C22×C6 [×4], C4×D4, C3×C12 [×4], C62 [×7], C22×C12 [×4], C6×D4 [×4], C3×C4○D4 [×4], C6×C12 [×6], D4×C32 [×2], C2×C62, D4×C12 [×4], C2×C6×C12, D4×C3×C6, C32×C4○D4, D4×C3×C12

Smallest permutation representation of D4×C3×C12
On 144 points
Generators in S144
(1 139 19)(2 140 20)(3 141 21)(4 142 22)(5 143 23)(6 144 24)(7 133 13)(8 134 14)(9 135 15)(10 136 16)(11 137 17)(12 138 18)(25 67 55)(26 68 56)(27 69 57)(28 70 58)(29 71 59)(30 72 60)(31 61 49)(32 62 50)(33 63 51)(34 64 52)(35 65 53)(36 66 54)(37 129 106)(38 130 107)(39 131 108)(40 132 97)(41 121 98)(42 122 99)(43 123 100)(44 124 101)(45 125 102)(46 126 103)(47 127 104)(48 128 105)(73 94 111)(74 95 112)(75 96 113)(76 85 114)(77 86 115)(78 87 116)(79 88 117)(80 89 118)(81 90 119)(82 91 120)(83 92 109)(84 93 110)
(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 47 78 25)(2 48 79 26)(3 37 80 27)(4 38 81 28)(5 39 82 29)(6 40 83 30)(7 41 84 31)(8 42 73 32)(9 43 74 33)(10 44 75 34)(11 45 76 35)(12 46 77 36)(13 98 110 49)(14 99 111 50)(15 100 112 51)(16 101 113 52)(17 102 114 53)(18 103 115 54)(19 104 116 55)(20 105 117 56)(21 106 118 57)(22 107 119 58)(23 108 120 59)(24 97 109 60)(61 133 121 93)(62 134 122 94)(63 135 123 95)(64 136 124 96)(65 137 125 85)(66 138 126 86)(67 139 127 87)(68 140 128 88)(69 141 129 89)(70 142 130 90)(71 143 131 91)(72 144 132 92)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 80)(44 81)(45 82)(46 83)(47 84)(48 73)(61 139)(62 140)(63 141)(64 142)(65 143)(66 144)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(85 131)(86 132)(87 121)(88 122)(89 123)(90 124)(91 125)(92 126)(93 127)(94 128)(95 129)(96 130)(97 115)(98 116)(99 117)(100 118)(101 119)(102 120)(103 109)(104 110)(105 111)(106 112)(107 113)(108 114)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 4A 4B 4C 4D 4E ··· 4L 6A ··· 6X 6Y ··· 6BD 12A ··· 12AF 12AG ··· 12CR order 1 2 2 2 2 2 2 2 3 ··· 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 2 2 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 D4 C4○D4 C3×D4 C3×C4○D4 kernel D4×C3×C12 C122 C32×C22⋊C4 C32×C4⋊C4 C2×C6×C12 D4×C3×C6 D4×C12 D4×C32 C4×C12 C3×C22⋊C4 C3×C4⋊C4 C22×C12 C6×D4 C3×D4 C3×C12 C3×C6 C12 C6 # reps 1 1 2 1 2 1 8 8 8 16 8 16 8 64 2 2 16 16

Matrix representation of D4×C3×C12 in GL4(𝔽13) generated by

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

D4×C3×C12 in GAP, Magma, Sage, TeX

D_4\times C_3\times C_{12}
% in TeX

G:=Group("D4xC3xC12");
// GroupNames label

G:=SmallGroup(288,815);
// by ID

G=gap.SmallGroup(288,815);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037,772]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=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

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