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## G = C3×D72order 432 = 24·33

### Direct product of C3 and D72

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C3×D72
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C3×D36 — C3×D72
 Lower central C9 — C18 — C36 — C3×D72
 Upper central C1 — C6 — C12 — C24

Generators and relations for C3×D72
G = < a,b,c | a3=b72=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 430 in 74 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C9, C9, C32, C12, C12, D6, C2×C6, D8, D9, C18, C18, C3×S3, C3×C6, C24, C24, D12, C3×D4, C3×C9, C36, C36, D18, C3×C12, S3×C6, D24, C3×D8, C3×D9, C3×C18, C72, C72, D36, C3×C24, C3×D12, C3×C36, C6×D9, D72, C3×D24, C3×C72, C3×D36, C3×D72
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, D9, C3×S3, D12, C3×D4, D18, S3×C6, D24, C3×D8, C3×D9, D36, C3×D12, C6×D9, D72, C3×D24, C3×D36, C3×D72

Smallest permutation representation of C3×D72
On 144 points
Generators in S144
(1 49 25)(2 50 26)(3 51 27)(4 52 28)(5 53 29)(6 54 30)(7 55 31)(8 56 32)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)(73 97 121)(74 98 122)(75 99 123)(76 100 124)(77 101 125)(78 102 126)(79 103 127)(80 104 128)(81 105 129)(82 106 130)(83 107 131)(84 108 132)(85 109 133)(86 110 134)(87 111 135)(88 112 136)(89 113 137)(90 114 138)(91 115 139)(92 116 140)(93 117 141)(94 118 142)(95 119 143)(96 120 144)
(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)
(1 141)(2 140)(3 139)(4 138)(5 137)(6 136)(7 135)(8 134)(9 133)(10 132)(11 131)(12 130)(13 129)(14 128)(15 127)(16 126)(17 125)(18 124)(19 123)(20 122)(21 121)(22 120)(23 119)(24 118)(25 117)(26 116)(27 115)(28 114)(29 113)(30 112)(31 111)(32 110)(33 109)(34 108)(35 107)(36 106)(37 105)(38 104)(39 103)(40 102)(41 101)(42 100)(43 99)(44 98)(45 97)(46 96)(47 95)(48 94)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 85)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 144)(71 143)(72 142)

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

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

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

117 conjugacy classes

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

117 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D4 D6 D8 D9 C3×S3 C3×D4 D12 D18 S3×C6 C3×D8 D24 C3×D9 D36 C3×D12 C6×D9 D72 C3×D24 C3×D36 C3×D72 kernel C3×D72 C3×C72 C3×D36 D72 C72 D36 C3×C24 C3×C18 C3×C12 C3×C9 C24 C24 C18 C3×C6 C12 C12 C9 C32 C8 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 1 2 3 2 2 2 3 2 4 4 6 6 4 6 12 8 12 24

Matrix representation of C3×D72 in GL4(𝔽73) generated by

 64 0 0 0 0 64 0 0 0 0 1 0 0 0 0 1
,
 36 0 0 0 37 71 0 0 0 0 0 25 0 0 35 32
,
 29 59 0 0 60 44 0 0 0 0 27 9 0 0 57 46
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,1,0,0,0,0,1],[36,37,0,0,0,71,0,0,0,0,0,35,0,0,25,32],[29,60,0,0,59,44,0,0,0,0,27,57,0,0,9,46] >;

C3×D72 in GAP, Magma, Sage, TeX

C_3\times D_{72}
% in TeX

G:=Group("C3xD72");
// GroupNames label

G:=SmallGroup(432,108);
// by ID

G=gap.SmallGroup(432,108);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,260,1011,80,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^3=b^72=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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