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## G = D9×C3⋊C8order 432 = 24·33

### Direct product of D9 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — D9×C3⋊C8
 Chief series C1 — C3 — C9 — C3×C9 — C3×C18 — C3×C36 — C12×D9 — D9×C3⋊C8
 Lower central C3×C9 — D9×C3⋊C8
 Upper central C1 — C4

Generators and relations for D9×C3⋊C8
G = < a,b,c,d | a9=b2=c3=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 300 in 74 conjugacy classes, 33 normal (29 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C2×C12, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, S3×C8, C2×C3⋊C8, C3×D9, C3×C18, C9⋊C8, C72, C4×D9, C3×C3⋊C8, C324C8, S3×C12, C3×Dic9, C3×C36, C6×D9, C8×D9, S3×C3⋊C8, C9×C3⋊C8, C36.S3, C12×D9, D9×C3⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, D9, C3⋊C8, C4×S3, C2×Dic3, D18, S32, S3×C8, C2×C3⋊C8, C4×D9, S3×Dic3, S3×D9, C8×D9, S3×C3⋊C8, Dic3×D9, D9×C3⋊C8

Smallest permutation representation of D9×C3⋊C8
On 144 points
Generators in S144
(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 75)(2 74)(3 73)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 90)(18 89)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 99)(27 98)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 108)(36 107)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 117)(45 116)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 126)(54 125)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 135)(63 134)(64 142)(65 141)(66 140)(67 139)(68 138)(69 137)(70 136)(71 144)(72 143)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)
(1 68 32 50 14 59 23 41)(2 69 33 51 15 60 24 42)(3 70 34 52 16 61 25 43)(4 71 35 53 17 62 26 44)(5 72 36 54 18 63 27 45)(6 64 28 46 10 55 19 37)(7 65 29 47 11 56 20 38)(8 66 30 48 12 57 21 39)(9 67 31 49 13 58 22 40)(73 136 100 118 82 127 91 109)(74 137 101 119 83 128 92 110)(75 138 102 120 84 129 93 111)(76 139 103 121 85 130 94 112)(77 140 104 122 86 131 95 113)(78 141 105 123 87 132 96 114)(79 142 106 124 88 133 97 115)(80 143 107 125 89 134 98 116)(81 144 108 126 90 135 99 117)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E 8F 8G 8H 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A 18B 18C 18D 18E 18F 24A 24B 24C 24D 36A ··· 36F 36G ··· 36L 72A ··· 72L order 1 2 2 2 3 3 3 4 4 4 4 6 6 6 6 6 8 8 8 8 8 8 8 8 9 9 9 9 9 9 12 12 12 12 12 12 12 12 18 18 18 18 18 18 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 9 9 2 2 4 1 1 9 9 2 2 4 18 18 3 3 3 3 27 27 27 27 2 2 2 4 4 4 2 2 2 2 4 4 18 18 2 2 2 4 4 4 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + - + + + + - + - image C1 C2 C2 C2 C4 C4 C8 S3 S3 Dic3 D6 Dic3 D6 D9 C3⋊C8 C4×S3 D18 S3×C8 C4×D9 C8×D9 S32 S3×Dic3 S3×D9 S3×C3⋊C8 Dic3×D9 D9×C3⋊C8 kernel D9×C3⋊C8 C9×C3⋊C8 C36.S3 C12×D9 C3×Dic9 C6×D9 C3×D9 C4×D9 C3×C3⋊C8 Dic9 C36 D18 C3×C12 C3⋊C8 D9 C3×C6 C12 C32 C6 C3 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 8 1 1 1 1 1 1 3 4 2 3 4 6 12 1 1 3 2 3 6

Matrix representation of D9×C3⋊C8 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 31 70 0 0 3 28
,
 72 0 0 0 0 72 0 0 0 0 70 45 0 0 42 3
,
 0 1 0 0 72 72 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 72 72 0 0 0 0 51 0 0 0 0 51
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,31,3,0,0,70,28],[72,0,0,0,0,72,0,0,0,0,70,42,0,0,45,3],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[1,72,0,0,0,72,0,0,0,0,51,0,0,0,0,51] >;

D9×C3⋊C8 in GAP, Magma, Sage, TeX

D_9\times C_3\rtimes C_8
% in TeX

G:=Group("D9xC3:C8");
// GroupNames label

G:=SmallGroup(432,58);
// by ID

G=gap.SmallGroup(432,58);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,36,58,571,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^3=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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