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## G = D4×C3⋊Dic3order 288 = 25·32

### Direct product of D4 and C3⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D4×C3⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C22×C3⋊Dic3 — D4×C3⋊Dic3
 Lower central C32 — C3×C6 — D4×C3⋊Dic3
 Upper central C1 — C22 — C2×D4

Generators and relations for D4×C3⋊Dic3
G = < a,b,c,d,e | a4=b2=c3=d6=1, e2=d3, bab=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: 804 in 282 conjugacy classes, 127 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3 [×4], C4 [×2], C4 [×5], C22, C22 [×4], C22 [×4], C6 [×12], C6 [×16], C2×C4, C2×C4 [×8], D4 [×4], C23 [×2], C32, Dic3 [×20], C12 [×8], C2×C6 [×20], C2×C6 [×16], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×32], C2×C12 [×4], C3×D4 [×16], C22×C6 [×8], C4×D4, C3⋊Dic3 [×2], C3⋊Dic3 [×3], C3×C12 [×2], C62, C62 [×4], C62 [×4], C4×Dic3 [×4], C4⋊Dic3 [×4], C6.D4 [×8], C22×Dic3 [×8], C6×D4 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×4], C6×C12, D4×C32 [×4], C2×C62 [×2], D4×Dic3 [×4], C4×C3⋊Dic3, C12⋊Dic3, C625C4 [×2], C22×C3⋊Dic3 [×2], D4×C3×C6, D4×C3⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×2], C23, Dic3 [×16], D6 [×12], C22×C4, C2×D4, C4○D4, C3⋊S3, C2×Dic3 [×24], C22×S3 [×4], C4×D4, C3⋊Dic3 [×4], C2×C3⋊S3 [×3], S3×D4 [×4], D42S3 [×4], C22×Dic3 [×4], C2×C3⋊Dic3 [×6], C22×C3⋊S3, D4×Dic3 [×4], D4×C3⋊S3, C12.D6, C22×C3⋊Dic3, D4×C3⋊Dic3

Smallest permutation representation of D4×C3⋊Dic3
On 144 points
Generators in S144
(1 77 57 50)(2 78 58 51)(3 73 59 52)(4 74 60 53)(5 75 55 54)(6 76 56 49)(7 131 27 134)(8 132 28 135)(9 127 29 136)(10 128 30 137)(11 129 25 138)(12 130 26 133)(13 70 34 66)(14 71 35 61)(15 72 36 62)(16 67 31 63)(17 68 32 64)(18 69 33 65)(19 110 139 103)(20 111 140 104)(21 112 141 105)(22 113 142 106)(23 114 143 107)(24 109 144 108)(37 98 44 93)(38 99 45 94)(39 100 46 95)(40 101 47 96)(41 102 48 91)(42 97 43 92)(79 125 89 115)(80 126 90 116)(81 121 85 117)(82 122 86 118)(83 123 87 119)(84 124 88 120)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 49)(7 131)(8 132)(9 127)(10 128)(11 129)(12 130)(13 66)(14 61)(15 62)(16 63)(17 64)(18 65)(19 110)(20 111)(21 112)(22 113)(23 114)(24 109)(25 138)(26 133)(27 134)(28 135)(29 136)(30 137)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 93)(38 94)(39 95)(40 96)(41 91)(42 92)(43 97)(44 98)(45 99)(46 100)(47 101)(48 102)(55 75)(56 76)(57 77)(58 78)(59 73)(60 74)(79 115)(80 116)(81 117)(82 118)(83 119)(84 120)(85 121)(86 122)(87 123)(88 124)(89 125)(90 126)(103 139)(104 140)(105 141)(106 142)(107 143)(108 144)
(1 40 31)(2 41 32)(3 42 33)(4 37 34)(5 38 35)(6 39 36)(7 119 21)(8 120 22)(9 115 23)(10 116 24)(11 117 19)(12 118 20)(13 60 44)(14 55 45)(15 56 46)(16 57 47)(17 58 48)(18 59 43)(25 121 139)(26 122 140)(27 123 141)(28 124 142)(29 125 143)(30 126 144)(49 95 72)(50 96 67)(51 91 68)(52 92 69)(53 93 70)(54 94 71)(61 75 99)(62 76 100)(63 77 101)(64 78 102)(65 73 97)(66 74 98)(79 114 127)(80 109 128)(81 110 129)(82 111 130)(83 112 131)(84 113 132)(85 103 138)(86 104 133)(87 105 134)(88 106 135)(89 107 136)(90 108 137)
(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 79 4 82)(2 84 5 81)(3 83 6 80)(7 95 10 92)(8 94 11 91)(9 93 12 96)(13 104 16 107)(14 103 17 106)(15 108 18 105)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 102 28 99)(26 101 29 98)(27 100 30 97)(31 114 34 111)(32 113 35 110)(33 112 36 109)(37 130 40 127)(38 129 41 132)(39 128 42 131)(43 134 46 137)(44 133 47 136)(45 138 48 135)(49 116 52 119)(50 115 53 118)(51 120 54 117)(55 85 58 88)(56 90 59 87)(57 89 60 86)(61 139 64 142)(62 144 65 141)(63 143 66 140)(73 123 76 126)(74 122 77 125)(75 121 78 124)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G ··· 4L 6A ··· 6L 6M ··· 6AB 12A ··· 12H order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 2 2 2 2 9 9 9 9 18 ··· 18 2 ··· 2 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D6 Dic3 D6 C4○D4 S3×D4 D4⋊2S3 kernel D4×C3⋊Dic3 C4×C3⋊Dic3 C12⋊Dic3 C62⋊5C4 C22×C3⋊Dic3 D4×C3×C6 D4×C32 C6×D4 C3⋊Dic3 C2×C12 C3×D4 C22×C6 C3×C6 C6 C6 # reps 1 1 1 2 2 1 8 4 2 4 16 8 2 4 4

Matrix representation of D4×C3⋊Dic3 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 9 0 0 0 0 0 6 3 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
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 12 0 0 0 0 1 0
,
 5 8 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 8 0 0 0 0 8 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[9,6,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[5,0,0,0,0,0,8,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;

D4×C3⋊Dic3 in GAP, Magma, Sage, TeX

D_4\times C_3\rtimes {\rm Dic}_3
% in TeX

G:=Group("D4xC3:Dic3");
// GroupNames label

G:=SmallGroup(288,791);
// by ID

G=gap.SmallGroup(288,791);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,219,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^3=d^6=1,e^2=d^3,b*a*b=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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