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G = D36.S3order 432 = 24·33

1st non-split extension by D36 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: D36.1S3, C36.30D6, C12.9D18, C6.12D36, C3⋊C81D9, C12.44S32, C4.8(S3×D9), (C3×C9)⋊1SD16, (C3×C18).1D4, C33(C72⋊C2), C91(D4.S3), (C3×D36).1C2, (C3×C6).29D12, (C3×C12).69D6, C12.D91C2, C18.1(C3⋊D4), (C3×C36).1C22, C6.1(C3⋊D12), C2.4(C3⋊D36), C32.2(C24⋊C2), C3.1(D12.S3), (C9×C3⋊C8)⋊2C2, (C3×C3⋊C8).4S3, SmallGroup(432,62)

Series: Derived Chief Lower central Upper central

C1C3×C36 — D36.S3
C1C3C9C3×C9C3×C18C3×C36C3×D36 — D36.S3
C3×C9C3×C18C3×C36 — D36.S3
C1C2C4

Generators and relations for D36.S3
 G = < a,b,c,d | a36=b2=c3=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a27b, dcd-1=c-1 >

Subgroups: 504 in 72 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, SD16, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C3×C9, Dic9, C36, C36, D18, C3⋊Dic3, C3×C12, S3×C6, C24⋊C2, D4.S3, C3×D9, C3×C18, C72, Dic18, D36, C3×C3⋊C8, C3×D12, C324Q8, C9⋊Dic3, C3×C36, C6×D9, C72⋊C2, D12.S3, C9×C3⋊C8, C3×D36, C12.D9, D36.S3
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, D12, C3⋊D4, D18, S32, C24⋊C2, D4.S3, D36, C3⋊D12, S3×D9, C72⋊C2, D12.S3, C3⋊D36, D36.S3

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

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

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

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

60 conjugacy classes

class 1 2A2B3A3B3C4A4B6A6B6C6D6E8A8B9A9B9C9D9E9F12A12B12C12D12E18A18B18C18D18E18F24A24B24C24D36A···36F36G···36L72A···72L
order12233344666668899999912121212121818181818182424242436···3636···3672···72
size113622421082243636662224442244422244466662···24···46···6

60 irreducible representations

dim111122222222222224444444
type++++++++++++++-++-+-
imageC1C2C2C2S3S3D4D6D6SD16D9C3⋊D4D12D18C24⋊C2D36C72⋊C2S32D4.S3C3⋊D12S3×D9D12.S3C3⋊D36D36.S3
kernelD36.S3C9×C3⋊C8C3×D36C12.D9D36C3×C3⋊C8C3×C18C36C3×C12C3×C9C3⋊C8C18C3×C6C12C32C6C3C12C9C6C4C3C2C1
# reps1111111112322346121113236

Matrix representation of D36.S3 in GL6(𝔽73)

010000
7200000
001000
000100
00003128
0000453
,
32270000
27410000
0072000
0007200
00001868
00005055
,
100000
010000
000100
00727200
000010
000001
,
6760000
67670000
0072000
001100
00006659
0000147

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,45,0,0,0,0,28,3],[32,27,0,0,0,0,27,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,18,50,0,0,0,0,68,55],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,66,14,0,0,0,0,59,7] >;

D36.S3 in GAP, Magma, Sage, TeX

D_{36}.S_3
% in TeX

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

G:=SmallGroup(432,62);
// by ID

G=gap.SmallGroup(432,62);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,36,254,58,571,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^36=b^2=c^3=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^27*b,d*c*d^-1=c^-1>;
// generators/relations

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