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G = C3×D72order 432 = 24·33

Direct product of C3 and D72

direct product, metacyclic, supersoluble, monomial

Aliases: C3×D72, C729C6, C243D9, D367C6, C6.21D36, C12.73D18, C32.3D24, (C3×C9)⋊5D8, C94(C3×D8), C81(C3×D9), (C3×C72)⋊2C2, (C3×D36)⋊2C2, C24.2(C3×S3), C4.10(C6×D9), C3.1(C3×D24), C6.3(C3×D12), C2.5(C3×D36), C12.66(S3×C6), (C3×C24).15S3, C36.33(C2×C6), (C3×C18).26D4, (C3×C6).51D12, C18.19(C3×D4), (C3×C12).204D6, (C3×C36).57C22, SmallGroup(432,108)

Series: Derived Chief Lower central Upper central

C1C36 — C3×D72
C1C3C9C18C36C3×C36C3×D36 — C3×D72
C9C18C36 — C3×D72
C1C6C12C24

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 [×2], C3 [×2], C3, C4, C22 [×2], S3 [×2], C6 [×2], C6 [×3], C8, D4 [×2], C9, C9, C32, C12 [×2], C12, D6 [×2], C2×C6 [×2], D8, D9 [×2], C18, C18, C3×S3 [×2], C3×C6, C24 [×2], C24, D12 [×2], C3×D4 [×2], C3×C9, C36, C36, D18 [×2], C3×C12, S3×C6 [×2], D24, C3×D8, C3×D9 [×2], C3×C18, C72, C72, D36 [×2], C3×C24, C3×D12 [×2], C3×C36, C6×D9 [×2], D72, C3×D24, C3×C72, C3×D36 [×2], C3×D72
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], 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)
(1 107)(2 106)(3 105)(4 104)(5 103)(6 102)(7 101)(8 100)(9 99)(10 98)(11 97)(12 96)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 84)(25 83)(26 82)(27 81)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 144)(37 143)(38 142)(39 141)(40 140)(41 139)(42 138)(43 137)(44 136)(45 135)(46 134)(47 133)(48 132)(49 131)(50 130)(51 129)(52 128)(53 127)(54 126)(55 125)(56 124)(57 123)(58 122)(59 121)(60 120)(61 119)(62 118)(63 117)(64 116)(65 115)(66 114)(67 113)(68 112)(69 111)(70 110)(71 109)(72 108)

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

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

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

117 conjugacy classes

class 1 2A2B2C3A3B3C3D3E 4 6A6B6C6D6E6F6G6H6I8A8B9A···9I12A···12H18A···18I24A···24P36A···36R72A···72AJ
order1222333334666666666889···912···1218···1824···2436···3672···72
size1136361122221122236363636222···22···22···22···22···22···2

117 irreducible representations

dim11111122222222222222222222
type+++++++++++++
imageC1C2C2C3C6C6S3D4D6D8D9C3×S3C3×D4D12D18S3×C6C3×D8D24C3×D9D36C3×D12C6×D9D72C3×D24C3×D36C3×D72
kernelC3×D72C3×C72C3×D36D72C72D36C3×C24C3×C18C3×C12C3×C9C24C24C18C3×C6C12C12C9C32C8C6C6C4C3C3C2C1
# reps11222411123222324466461281224

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

64000
06400
0010
0001
,
36000
377100
00025
003532
,
295900
604400
00279
005746
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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