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## G = C42×D9order 288 = 25·32

### Direct product of C42 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C42×D9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×C4×D9 — C42×D9
 Lower central C9 — C42×D9
 Upper central C1 — C42

Generators and relations for C42×D9
G = < a,b,c,d | a4=b4=c9=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 552 in 162 conjugacy classes, 84 normal (11 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C4 [×6], C22, C22 [×6], S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×15], C23, C9, Dic3 [×6], C12 [×6], D6 [×6], C2×C6, C42, C42 [×3], C22×C4 [×3], D9 [×4], C18 [×3], C4×S3 [×12], C2×Dic3 [×3], C2×C12 [×3], C22×S3, C2×C42, Dic9 [×6], C36 [×6], D18 [×6], C2×C18, C4×Dic3 [×3], C4×C12, S3×C2×C4 [×3], C4×D9 [×12], C2×Dic9 [×3], C2×C36 [×3], C22×D9, S3×C42, C4×Dic9 [×3], C4×C36, C2×C4×D9 [×3], C42×D9
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3, C2×C4 [×18], C23, D6 [×3], C42 [×4], C22×C4 [×3], D9, C4×S3 [×6], C22×S3, C2×C42, D18 [×3], S3×C2×C4 [×3], C4×D9 [×6], C22×D9, S3×C42, C2×C4×D9 [×3], C42×D9

Smallest permutation representation of C42×D9
On 144 points
Generators in S144
(1 86 14 77)(2 87 15 78)(3 88 16 79)(4 89 17 80)(5 90 18 81)(6 82 10 73)(7 83 11 74)(8 84 12 75)(9 85 13 76)(19 100 28 91)(20 101 29 92)(21 102 30 93)(22 103 31 94)(23 104 32 95)(24 105 33 96)(25 106 34 97)(26 107 35 98)(27 108 36 99)(37 118 46 109)(38 119 47 110)(39 120 48 111)(40 121 49 112)(41 122 50 113)(42 123 51 114)(43 124 52 115)(44 125 53 116)(45 126 54 117)(55 136 64 127)(56 137 65 128)(57 138 66 129)(58 139 67 130)(59 140 68 131)(60 141 69 132)(61 142 70 133)(62 143 71 134)(63 144 72 135)
(1 68 23 50)(2 69 24 51)(3 70 25 52)(4 71 26 53)(5 72 27 54)(6 64 19 46)(7 65 20 47)(8 66 21 48)(9 67 22 49)(10 55 28 37)(11 56 29 38)(12 57 30 39)(13 58 31 40)(14 59 32 41)(15 60 33 42)(16 61 34 43)(17 62 35 44)(18 63 36 45)(73 136 91 118)(74 137 92 119)(75 138 93 120)(76 139 94 121)(77 140 95 122)(78 141 96 123)(79 142 97 124)(80 143 98 125)(81 144 99 126)(82 127 100 109)(83 128 101 110)(84 129 102 111)(85 130 103 112)(86 131 104 113)(87 132 105 114)(88 133 106 115)(89 134 107 116)(90 135 108 117)
(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 22)(2 21)(3 20)(4 19)(5 27)(6 26)(7 25)(8 24)(9 23)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 36)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 63)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 72)(73 98)(74 97)(75 96)(76 95)(77 94)(78 93)(79 92)(80 91)(81 99)(82 107)(83 106)(84 105)(85 104)(86 103)(87 102)(88 101)(89 100)(90 108)(109 134)(110 133)(111 132)(112 131)(113 130)(114 129)(115 128)(116 127)(117 135)(118 143)(119 142)(120 141)(121 140)(122 139)(123 138)(124 137)(125 136)(126 144)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4L 4M ··· 4X 6A 6B 6C 9A 9B 9C 12A ··· 12L 18A ··· 18I 36A ··· 36AJ order 1 2 2 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 6 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 9 9 9 9 2 1 ··· 1 9 ··· 9 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C4 S3 D6 D9 C4×S3 D18 C4×D9 kernel C42×D9 C4×Dic9 C4×C36 C2×C4×D9 C4×D9 C4×C12 C2×C12 C42 C12 C2×C4 C4 # reps 1 3 1 3 24 1 3 3 12 9 36

Matrix representation of C42×D9 in GL3(𝔽37) generated by

 31 0 0 0 31 0 0 0 31
,
 31 0 0 0 36 0 0 0 36
,
 1 0 0 0 31 20 0 17 11
,
 1 0 0 0 31 20 0 26 6
G:=sub<GL(3,GF(37))| [31,0,0,0,31,0,0,0,31],[31,0,0,0,36,0,0,0,36],[1,0,0,0,31,17,0,20,11],[1,0,0,0,31,26,0,20,6] >;

C42×D9 in GAP, Magma, Sage, TeX

C_4^2\times D_9
% in TeX

G:=Group("C4^2xD9");
// GroupNames label

G:=SmallGroup(288,81);
// by ID

G=gap.SmallGroup(288,81);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,58,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^9=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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