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G = C3×C72⋊C2order 432 = 24·33

Direct product of C3 and C72⋊C2

direct product, metacyclic, supersoluble, monomial

Aliases: C3×C72⋊C2, C246D9, C7210C6, D36.3C6, C6.20D36, Dic187C6, C12.72D18, C82(C3×D9), (C3×C72)⋊4C2, C4.9(C6×D9), (C3×C9)⋊8SD16, C24.8(C3×S3), C95(C3×SD16), C6.2(C3×D12), C2.4(C3×D36), C12.65(S3×C6), (C3×C24).18S3, C36.32(C2×C6), (C3×D36).2C2, C18.18(C3×D4), (C3×C18).25D4, (C3×C6).50D12, (C3×Dic18)⋊2C2, (C3×C12).203D6, (C3×C36).56C22, C32.4(C24⋊C2), C3.1(C3×C24⋊C2), SmallGroup(432,107)

Series: Derived Chief Lower central Upper central

C1C36 — C3×C72⋊C2
C1C3C9C18C36C3×C36C3×D36 — C3×C72⋊C2
C9C18C36 — C3×C72⋊C2
C1C6C12C24

Generators and relations for C3×C72⋊C2
 G = < a,b,c | a3=b72=c2=1, ab=ba, ac=ca, cbc=b35 >

Subgroups: 326 in 68 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, D4, Q8, C9, C9, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, SD16, D9, C18, C18, C3×S3, C3×C6, C24 [×2], C24, Dic6, D12, C3×D4, C3×Q8, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, C24⋊C2, C3×SD16, C3×D9, C3×C18, C72, C72, Dic18, D36, C3×C24, C3×Dic6, C3×D12, C3×Dic9, C3×C36, C6×D9, C72⋊C2, C3×C24⋊C2, C3×C72, C3×Dic18, C3×D36, C3×C72⋊C2
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D4, D6, C2×C6, SD16, D9, C3×S3, D12, C3×D4, D18, S3×C6, C24⋊C2, C3×SD16, C3×D9, D36, C3×D12, C6×D9, C72⋊C2, C3×C24⋊C2, C3×D36, C3×C72⋊C2

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

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

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

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

117 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B6A6B6C6D6E6F6G8A8B9A···9I12A···12H12I12J18A···18I24A···24P36A···36R72A···72AJ
order12233333446666666889···912···12121218···1824···2436···3672···72
size113611222236112223636222···22···236362···22···22···22···2

117 irreducible representations

dim1111111122222222222222222222
type+++++++++++
imageC1C2C2C2C3C6C6C6S3D4D6SD16D9C3×S3C3×D4D12D18S3×C6C3×SD16C24⋊C2C3×D9D36C3×D12C6×D9C72⋊C2C3×C24⋊C2C3×D36C3×C72⋊C2
kernelC3×C72⋊C2C3×C72C3×Dic18C3×D36C72⋊C2C72Dic18D36C3×C24C3×C18C3×C12C3×C9C24C24C18C3×C6C12C12C9C32C8C6C6C4C3C3C2C1
# reps1111222211123222324466461281224

Matrix representation of C3×C72⋊C2 in GL2(𝔽73) generated by

640
064
,
50
529
,
417
5232
G:=sub<GL(2,GF(73))| [64,0,0,64],[5,5,0,29],[41,52,7,32] >;

C3×C72⋊C2 in GAP, Magma, Sage, TeX

C_3\times C_{72}\rtimes C_2
% in TeX

G:=Group("C3xC72:C2");
// GroupNames label

G:=SmallGroup(432,107);
// by ID

G=gap.SmallGroup(432,107);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,92,1011,80,10085,292,14118]);
// 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^35>;
// generators/relations

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