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## G = Dic9⋊Dic3order 432 = 24·33

### The semidirect product of Dic9 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — Dic9⋊Dic3
 Chief series C1 — C3 — C9 — C3×C9 — C3×C18 — C6×C18 — C6×Dic9 — Dic9⋊Dic3
 Lower central C3×C9 — C3×C18 — Dic9⋊Dic3
 Upper central C1 — C22

Generators and relations for Dic9⋊Dic3
G = < a,b,c,d | a18=c6=1, b2=a9, d2=c3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a9b, dcd-1=c-1 >

Subgroups: 444 in 94 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C4⋊Dic3, C3×C18, C2×Dic9, C2×Dic9, C2×C36, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×C18, Dic9⋊C4, Dic3⋊Dic3, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, Dic9⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, D9, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, D18, S32, Dic3⋊C4, C4⋊Dic3, Dic18, C4×D9, C9⋊D4, S3×Dic3, C3⋊D12, C322Q8, S3×D9, Dic9⋊C4, Dic3⋊Dic3, C9⋊Dic6, Dic3×D9, C9⋊D12, Dic9⋊Dic3

Smallest permutation representation of Dic9⋊Dic3
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)
(1 100 10 91)(2 99 11 108)(3 98 12 107)(4 97 13 106)(5 96 14 105)(6 95 15 104)(7 94 16 103)(8 93 17 102)(9 92 18 101)(19 88 28 79)(20 87 29 78)(21 86 30 77)(22 85 31 76)(23 84 32 75)(24 83 33 74)(25 82 34 73)(26 81 35 90)(27 80 36 89)(37 130 46 139)(38 129 47 138)(39 128 48 137)(40 127 49 136)(41 144 50 135)(42 143 51 134)(43 142 52 133)(44 141 53 132)(45 140 54 131)(55 114 64 123)(56 113 65 122)(57 112 66 121)(58 111 67 120)(59 110 68 119)(60 109 69 118)(61 126 70 117)(62 125 71 116)(63 124 72 115)
(1 29 7 35 13 23)(2 30 8 36 14 24)(3 31 9 19 15 25)(4 32 10 20 16 26)(5 33 11 21 17 27)(6 34 12 22 18 28)(37 68 49 62 43 56)(38 69 50 63 44 57)(39 70 51 64 45 58)(40 71 52 65 46 59)(41 72 53 66 47 60)(42 55 54 67 48 61)(73 107 85 101 79 95)(74 108 86 102 80 96)(75 91 87 103 81 97)(76 92 88 104 82 98)(77 93 89 105 83 99)(78 94 90 106 84 100)(109 144 115 132 121 138)(110 127 116 133 122 139)(111 128 117 134 123 140)(112 129 118 135 124 141)(113 130 119 136 125 142)(114 131 120 137 126 143)
(1 69 35 44)(2 70 36 45)(3 71 19 46)(4 72 20 47)(5 55 21 48)(6 56 22 49)(7 57 23 50)(8 58 24 51)(9 59 25 52)(10 60 26 53)(11 61 27 54)(12 62 28 37)(13 63 29 38)(14 64 30 39)(15 65 31 40)(16 66 32 41)(17 67 33 42)(18 68 34 43)(73 133 101 110)(74 134 102 111)(75 135 103 112)(76 136 104 113)(77 137 105 114)(78 138 106 115)(79 139 107 116)(80 140 108 117)(81 141 91 118)(82 142 92 119)(83 143 93 120)(84 144 94 121)(85 127 95 122)(86 128 96 123)(87 129 97 124)(88 130 98 125)(89 131 99 126)(90 132 100 109)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A ··· 18I 18J ··· 18R 36A ··· 36L order 1 2 2 2 3 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 9 9 9 9 9 9 12 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 4 6 6 18 18 54 54 2 ··· 2 4 4 4 2 2 2 4 4 4 6 6 6 6 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + - - + + + - + - + - + - + - + - - + image C1 C2 C2 C2 C4 S3 S3 D4 Q8 Dic3 D6 D6 D9 Dic6 D12 Dic6 C4×S3 C3⋊D4 D18 Dic18 C4×D9 C9⋊D4 S32 S3×Dic3 C3⋊D12 C32⋊2Q8 S3×D9 C9⋊Dic6 Dic3×D9 C9⋊D12 kernel Dic9⋊Dic3 C6×Dic9 Dic3×C18 C2×C9⋊Dic3 C3×Dic9 C2×Dic9 C6×Dic3 C3×C18 C3×C18 Dic9 C2×C18 C62 C2×Dic3 C18 C18 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 1 1 1 2 1 1 3 2 2 2 2 2 3 6 6 6 1 1 1 1 3 3 3 3

Matrix representation of Dic9⋊Dic3 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 6 11 0 0 0 0 26 17
,
 31 24 0 0 0 0 0 6 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 1 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 31 0 0 0 0 0 34 6 0 0 0 0 0 0 31 0 0 0 0 0 6 6 0 0 0 0 0 0 31 0 0 0 0 0 0 31

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,26,0,0,0,0,11,17],[31,0,0,0,0,0,24,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[31,34,0,0,0,0,0,6,0,0,0,0,0,0,31,6,0,0,0,0,0,6,0,0,0,0,0,0,31,0,0,0,0,0,0,31] >;

Dic9⋊Dic3 in GAP, Magma, Sage, TeX

{\rm Dic}_9\rtimes {\rm Dic}_3
% in TeX

G:=Group("Dic9:Dic3");
// GroupNames label

G:=SmallGroup(432,88);
// by ID

G=gap.SmallGroup(432,88);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,571,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^18=c^6=1,b^2=a^9,d^2=c^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=c^-1>;
// generators/relations

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