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G = D365S3order 432 = 24·33

The semidirect product of D36 and S3 acting through Inn(D36)

metabelian, supersoluble, monomial

Aliases: D365S3, D6.6D18, D18.2D6, C36.35D6, C12.25D18, Dic3.10D18, C12.49S32, (C4×S3)⋊1D9, (S3×C36)⋊3C2, (C3×D36)⋊3C2, C4.13(S3×D9), D6⋊D92C2, (S3×C12).5S3, (Dic3×D9)⋊4C2, (S3×C6).27D6, (C3×C12).96D6, C92(D42S3), C12.D94C2, C6.9(C22×D9), (C3×C18).9C23, C18.9(C22×S3), (C3×C36).6C22, C33(D365C2), (C6×D9).2C22, (C3×Dic3).37D6, C9⋊Dic3.4C22, C3.1(D125S3), (S3×C18).10C22, C32.3(C4○D12), (C9×Dic3).11C22, C6.28(C2×S32), C2.13(C2×S3×D9), (C3×C9)⋊6(C4○D4), (C3×C6).77(C22×S3), SmallGroup(432,288)

Series: Derived Chief Lower central Upper central

C1C3×C18 — D365S3
C1C3C32C3×C9C3×C18S3×C18D6⋊D9 — D365S3
C3×C9C3×C18 — D365S3
C1C2C4

Generators and relations for D365S3
 G = < a,b,c,d | a36=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a18b, dcd=c-1 >

Subgroups: 760 in 132 conjugacy classes, 41 normal (31 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×C9, Dic9, C36, C36, D18, C2×C18, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C4○D12, D42S3, C3×D9, S3×C9, C3×C18, Dic18, C4×D9, D36, C9⋊D4, C2×C36, S3×Dic3, D6⋊S3, S3×C12, C3×D12, C324Q8, C9×Dic3, C9⋊Dic3, C3×C36, C6×D9, S3×C18, D365C2, D125S3, Dic3×D9, D6⋊D9, C3×D36, S3×C36, C12.D9, D365S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, D9, C22×S3, D18, S32, C4○D12, D42S3, C22×D9, C2×S32, S3×D9, D365C2, D125S3, C2×S3×D9, D365S3

Smallest permutation representation of D365S3
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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 72)(18 71)(19 70)(20 69)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(33 56)(34 55)(35 54)(36 53)(73 130)(74 129)(75 128)(76 127)(77 126)(78 125)(79 124)(80 123)(81 122)(82 121)(83 120)(84 119)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 112)(92 111)(93 110)(94 109)(95 144)(96 143)(97 142)(98 141)(99 140)(100 139)(101 138)(102 137)(103 136)(104 135)(105 134)(106 133)(107 132)(108 131)
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)(73 97 85)(74 98 86)(75 99 87)(76 100 88)(77 101 89)(78 102 90)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)(109 121 133)(110 122 134)(111 123 135)(112 124 136)(113 125 137)(114 126 138)(115 127 139)(116 128 140)(117 129 141)(118 130 142)(119 131 143)(120 132 144)
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 81)(33 82)(34 83)(35 84)(36 85)(37 120)(38 121)(39 122)(40 123)(41 124)(42 125)(43 126)(44 127)(45 128)(46 129)(47 130)(48 131)(49 132)(50 133)(51 134)(52 135)(53 136)(54 137)(55 138)(56 139)(57 140)(58 141)(59 142)(60 143)(61 144)(62 109)(63 110)(64 111)(65 112)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)

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

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

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

63 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G9A9B9C9D9E9F12A12B12C12D12E12F12G18A18B18C18D18E18F18G···18L36A···36F36G···36L36M···36R
order122223334444466666669999991212121212121218181818181818···1836···3636···3636···36
size1161818224233545422466363622244422444662224446···62···24···46···6

63 irreducible representations

dim111111222222222222224444444
type++++++++++++++++++-++-+-
imageC1C2C2C2C2C2S3S3D6D6D6D6D6C4○D4D9D18D18D18C4○D12D365C2S32D42S3C2×S32S3×D9D125S3C2×S3×D9D365S3
kernelD365S3Dic3×D9D6⋊D9C3×D36S3×C36C12.D9D36S3×C12C36D18C3×Dic3C3×C12S3×C6C3×C9C4×S3Dic3C12D6C32C3C12C9C6C4C3C2C1
# reps1221111112111233334121113236

Matrix representation of D365S3 in GL6(𝔽37)

22350000
2150000
001000
000100
00003110
00003411
,
940000
17280000
0036000
0003600
0000627
00002231
,
100000
010000
0036100
0036000
000010
000001
,
21250000
12160000
000100
001000
000010
000001

G:=sub<GL(6,GF(37))| [22,2,0,0,0,0,35,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,34,0,0,0,0,10,11],[9,17,0,0,0,0,4,28,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,6,22,0,0,0,0,27,31],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[21,12,0,0,0,0,25,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D365S3 in GAP, Magma, Sage, TeX

D_{36}\rtimes_5S_3
% in TeX

G:=Group("D36:5S3");
// GroupNames label

G:=SmallGroup(432,288);
// by ID

G=gap.SmallGroup(432,288);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,135,58,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^36=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations

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