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## G = C9×C4⋊Dic3order 432 = 24·33

### Direct product of C9 and C4⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C9×C4⋊Dic3
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×C18 — Dic3×C18 — C9×C4⋊Dic3
 Lower central C3 — C6 — C9×C4⋊Dic3
 Upper central C1 — C2×C18 — C2×C36

Generators and relations for C9×C4⋊Dic3
G = < a,b,c,d | a9=b4=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 164 in 94 conjugacy classes, 57 normal (39 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C9, C9, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C2×C18, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C3×C18, C2×C36, C2×C36, C6×Dic3, C6×C12, C9×Dic3, C3×C36, C6×C18, C9×C4⋊C4, C3×C4⋊Dic3, Dic3×C18, C6×C36, C9×C4⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C9, Dic3, C12, D6, C2×C6, C4⋊C4, C18, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C36, C2×C18, C3×Dic3, S3×C6, C4⋊Dic3, C3×C4⋊C4, S3×C9, C2×C36, D4×C9, Q8×C9, C3×Dic6, C3×D12, C6×Dic3, C9×Dic3, S3×C18, C9×C4⋊C4, C3×C4⋊Dic3, C9×Dic6, C9×D12, Dic3×C18, C9×C4⋊Dic3

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

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G ··· 6O 9A ··· 9F 9G ··· 9L 12A ··· 12P 12Q ··· 12X 18A ··· 18R 18S ··· 18AJ 36A ··· 36AJ 36AK ··· 36BH order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 1 1 2 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 2 ··· 2 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + - + image C1 C2 C2 C3 C4 C6 C6 C9 C12 C18 C18 C36 S3 D4 Q8 Dic3 D6 C3×S3 Dic6 D12 C3×D4 C3×Q8 C3×Dic3 S3×C6 S3×C9 D4×C9 Q8×C9 C3×Dic6 C3×D12 C9×Dic3 S3×C18 C9×Dic6 C9×D12 kernel C9×C4⋊Dic3 Dic3×C18 C6×C36 C3×C4⋊Dic3 C3×C36 C6×Dic3 C6×C12 C4⋊Dic3 C3×C12 C2×Dic3 C2×C12 C12 C2×C36 C3×C18 C3×C18 C36 C2×C18 C2×C12 C18 C18 C3×C6 C3×C6 C12 C2×C6 C2×C4 C6 C6 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 4 4 2 6 8 12 6 24 1 1 1 2 1 2 2 2 2 2 4 2 6 6 6 4 4 12 6 12 12

Matrix representation of C9×C4⋊Dic3 in GL3(𝔽37) generated by

 12 0 0 0 12 0 0 0 12
,
 36 0 0 0 6 0 0 0 31
,
 36 0 0 0 11 0 0 0 27
,
 6 0 0 0 0 1 0 36 0
G:=sub<GL(3,GF(37))| [12,0,0,0,12,0,0,0,12],[36,0,0,0,6,0,0,0,31],[36,0,0,0,11,0,0,0,27],[6,0,0,0,0,36,0,1,0] >;

C9×C4⋊Dic3 in GAP, Magma, Sage, TeX

C_9\times C_4\rtimes {\rm Dic}_3
% in TeX

G:=Group("C9xC4:Dic3");
// GroupNames label

G:=SmallGroup(432,133);
// by ID

G=gap.SmallGroup(432,133);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,176,268,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^4=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

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