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## G = C22×C4×D9order 288 = 25·32

### Direct product of C22×C4 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C22×C4×D9
 Chief series C1 — C3 — C9 — C18 — D18 — C22×D9 — C23×D9 — C22×C4×D9
 Lower central C9 — C22×C4×D9
 Upper central C1 — C22×C4

Generators and relations for C22×C4×D9
G = < a,b,c,d,e | a2=b2=c4=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: 1264 in 354 conjugacy classes, 172 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C4 [×4], C22 [×7], C22 [×28], S3 [×8], C6, C6 [×6], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], C9, Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×7], C22×C4, C22×C4 [×13], C24, D9 [×8], C18, C18 [×6], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×14], C22×C6, C23×C4, Dic9 [×4], C36 [×4], D18 [×28], C2×C18 [×7], S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, C4×D9 [×16], C2×Dic9 [×6], C2×C36 [×6], C22×D9 [×14], C22×C18, S3×C22×C4, C2×C4×D9 [×12], C22×Dic9, C22×C36, C23×D9, C22×C4×D9
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, D9, C4×S3 [×4], C22×S3 [×7], C23×C4, D18 [×7], S3×C2×C4 [×6], S3×C23, C4×D9 [×4], C22×D9 [×7], S3×C22×C4, C2×C4×D9 [×6], C23×D9, C22×C4×D9

Smallest permutation representation of C22×C4×D9
On 144 points
Generators in S144
(1 86)(2 87)(3 88)(4 89)(5 90)(6 82)(7 83)(8 84)(9 85)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 91)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 118)(38 119)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 115)(53 116)(54 117)(55 136)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 46)(7 47)(8 48)(9 49)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 64)(20 65)(21 66)(22 67)(23 68)(24 69)(25 70)(26 71)(27 72)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 118)(74 119)(75 120)(76 121)(77 122)(78 123)(79 124)(80 125)(81 126)(82 109)(83 110)(84 111)(85 112)(86 113)(87 114)(88 115)(89 116)(90 117)(91 136)(92 137)(93 138)(94 139)(95 140)(96 141)(97 142)(98 143)(99 144)(100 127)(101 128)(102 129)(103 130)(104 131)(105 132)(106 133)(107 134)(108 135)
(1 23 14 32)(2 24 15 33)(3 25 16 34)(4 26 17 35)(5 27 18 36)(6 19 10 28)(7 20 11 29)(8 21 12 30)(9 22 13 31)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)(73 91 82 100)(74 92 83 101)(75 93 84 102)(76 94 85 103)(77 95 86 104)(78 96 87 105)(79 97 88 106)(80 98 89 107)(81 99 90 108)(109 127 118 136)(110 128 119 137)(111 129 120 138)(112 130 121 139)(113 131 122 140)(114 132 123 141)(115 133 124 142)(116 134 125 143)(117 135 126 144)
(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 76)(2 75)(3 74)(4 73)(5 81)(6 80)(7 79)(8 78)(9 77)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 90)(19 98)(20 97)(21 96)(22 95)(23 94)(24 93)(25 92)(26 91)(27 99)(28 107)(29 106)(30 105)(31 104)(32 103)(33 102)(34 101)(35 100)(36 108)(37 116)(38 115)(39 114)(40 113)(41 112)(42 111)(43 110)(44 109)(45 117)(46 125)(47 124)(48 123)(49 122)(50 121)(51 120)(52 119)(53 118)(54 126)(55 134)(56 133)(57 132)(58 131)(59 130)(60 129)(61 128)(62 127)(63 135)(64 143)(65 142)(66 141)(67 140)(68 139)(69 138)(70 137)(71 136)(72 144)

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

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

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

96 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4H 4I ··· 4P 6A ··· 6G 9A 9B 9C 12A ··· 12H 18A ··· 18U 36A ··· 36X order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 ··· 1 9 ··· 9 2 1 ··· 1 9 ··· 9 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D6 D6 D9 C4×S3 D18 D18 C4×D9 kernel C22×C4×D9 C2×C4×D9 C22×Dic9 C22×C36 C23×D9 C22×D9 C22×C12 C2×C12 C22×C6 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 1 1 1 16 1 6 1 3 8 18 3 24

Matrix representation of C22×C4×D9 in GL4(𝔽37) generated by

 36 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 36 0 0 0 0 1 0 0 0 0 6 0 0 0 0 6
,
 1 0 0 0 0 1 0 0 0 0 20 6 0 0 31 26
,
 1 0 0 0 0 36 0 0 0 0 31 26 0 0 20 6
G:=sub<GL(4,GF(37))| [36,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,1,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,1,0,0,0,0,20,31,0,0,6,26],[1,0,0,0,0,36,0,0,0,0,31,20,0,0,26,6] >;

C22×C4×D9 in GAP, Magma, Sage, TeX

C_2^2\times C_4\times D_9
% in TeX

G:=Group("C2^2xC4xD9");
// GroupNames label

G:=SmallGroup(288,353);
// by ID

G=gap.SmallGroup(288,353);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,80,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=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

׿
×
𝔽