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G = D9×C22×C6order 432 = 24·33

Direct product of C22×C6 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — D9×C22×C6
 Chief series C1 — C3 — C9 — C3×C9 — C3×D9 — C6×D9 — C2×C6×D9 — D9×C22×C6
 Lower central C9 — D9×C22×C6
 Upper central C1 — C22×C6

Generators and relations for D9×C22×C6
G = < a,b,c,d,e | a2=b2=c6=d9=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1486 in 434 conjugacy classes, 198 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, C6, C6, C23, C23, C9, C9, C32, D6, C2×C6, C2×C6, C24, D9, C18, C18, C3×S3, C3×C6, C22×S3, C22×C6, C22×C6, C3×C9, D18, C2×C18, C2×C18, S3×C6, C62, S3×C23, C23×C6, C3×D9, C3×C18, C22×D9, C22×C18, C22×C18, S3×C2×C6, C2×C62, C6×D9, C6×C18, C23×D9, S3×C22×C6, C2×C6×D9, C2×C6×C18, D9×C22×C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C24, D9, C3×S3, C22×S3, C22×C6, D18, S3×C6, S3×C23, C23×C6, C3×D9, C22×D9, S3×C2×C6, C6×D9, C23×D9, S3×C22×C6, C2×C6×D9, D9×C22×C6

Smallest permutation representation of D9×C22×C6
On 144 points
Generators in S144
(1 68)(2 69)(3 70)(4 71)(5 72)(6 64)(7 65)(8 66)(9 67)(10 55)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(73 136)(74 137)(75 138)(76 139)(77 140)(78 141)(79 142)(80 143)(81 144)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)(106 115)(107 116)(108 117)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 28)(7 29)(8 30)(9 31)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(44 71)(45 72)(46 55)(47 56)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(109 136)(110 137)(111 138)(112 139)(113 140)(114 141)(115 142)(116 143)(117 144)(118 127)(119 128)(120 129)(121 130)(122 131)(123 132)(124 133)(125 134)(126 135)
(1 11 4 14 7 17)(2 12 5 15 8 18)(3 13 6 16 9 10)(19 34 22 28 25 31)(20 35 23 29 26 32)(21 36 24 30 27 33)(37 52 40 46 43 49)(38 53 41 47 44 50)(39 54 42 48 45 51)(55 70 58 64 61 67)(56 71 59 65 62 68)(57 72 60 66 63 69)(73 85 79 82 76 88)(74 86 80 83 77 89)(75 87 81 84 78 90)(91 103 97 100 94 106)(92 104 98 101 95 107)(93 105 99 102 96 108)(109 121 115 118 112 124)(110 122 116 119 113 125)(111 123 117 120 114 126)(127 139 133 136 130 142)(128 140 134 137 131 143)(129 141 135 138 132 144)
(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 84)(2 83)(3 82)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 79)(11 78)(12 77)(13 76)(14 75)(15 74)(16 73)(17 81)(18 80)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 100)(26 108)(27 107)(28 97)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 99)(36 98)(37 124)(38 123)(39 122)(40 121)(41 120)(42 119)(43 118)(44 126)(45 125)(46 115)(47 114)(48 113)(49 112)(50 111)(51 110)(52 109)(53 117)(54 116)(55 142)(56 141)(57 140)(58 139)(59 138)(60 137)(61 136)(62 144)(63 143)(64 133)(65 132)(66 131)(67 130)(68 129)(69 128)(70 127)(71 135)(72 134)

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

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

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

144 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 3C 3D 3E 6A ··· 6N 6O ··· 6AI 6AJ ··· 6AY 9A ··· 9I 18A ··· 18BK order 1 2 ··· 2 2 ··· 2 3 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 ··· 1 9 ··· 9 1 1 2 2 2 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 D9 C3×S3 D18 S3×C6 C3×D9 C6×D9 kernel D9×C22×C6 C2×C6×D9 C2×C6×C18 C23×D9 C22×D9 C22×C18 C2×C62 C62 C22×C6 C22×C6 C2×C6 C2×C6 C23 C22 # reps 1 14 1 2 28 2 1 7 3 2 21 14 6 42

Matrix representation of D9×C22×C6 in GL4(𝔽19) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 11 0 0 0 0 12 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 17
,
 1 0 0 0 0 1 0 0 0 0 0 2 0 0 10 0
G:=sub<GL(4,GF(19))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[11,0,0,0,0,12,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,17],[1,0,0,0,0,1,0,0,0,0,0,10,0,0,2,0] >;

D9×C22×C6 in GAP, Magma, Sage, TeX

D_9\times C_2^2\times C_6
% in TeX

G:=Group("D9xC2^2xC6");
// GroupNames label

G:=SmallGroup(432,556);
// by ID

G=gap.SmallGroup(432,556);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,10085,292,14118]);
// Polycyclic

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

׿
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