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## G = C9×CSU2(𝔽3)  order 432 = 24·33

### Direct product of C9 and CSU2(𝔽3)

Aliases: C9×CSU2(𝔽3), C18.8S4, SL2(𝔽3).C18, Q8.(S3×C9), C2.2(C9×S4), C6.16(C3×S4), (Q8×C9).4S3, (C3×CSU2(𝔽3)).C3, (C9×SL2(𝔽3)).2C2, (C3×SL2(𝔽3)).5C6, C3.4(C3×CSU2(𝔽3)), (C3×Q8).8(C3×S3), SmallGroup(432,240)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C9×CSU2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C9×SL2(𝔽3) — C9×CSU2(𝔽3)
 Lower central SL2(𝔽3) — C9×CSU2(𝔽3)
 Upper central C1 — C18

Generators and relations for C9×CSU2(𝔽3)
G = < a,b,c,d,e | a9=b4=d3=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=b-1, dbd-1=bc, ebe-1=b2c, dcd-1=b, ede-1=d-1 >

Smallest permutation representation of C9×CSU2(𝔽3)
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 98 134 78)(2 99 135 79)(3 91 127 80)(4 92 128 81)(5 93 129 73)(6 94 130 74)(7 95 131 75)(8 96 132 76)(9 97 133 77)(10 64 39 54)(11 65 40 46)(12 66 41 47)(13 67 42 48)(14 68 43 49)(15 69 44 50)(16 70 45 51)(17 71 37 52)(18 72 38 53)(19 29 57 137)(20 30 58 138)(21 31 59 139)(22 32 60 140)(23 33 61 141)(24 34 62 142)(25 35 63 143)(26 36 55 144)(27 28 56 136)(82 121 109 102)(83 122 110 103)(84 123 111 104)(85 124 112 105)(86 125 113 106)(87 126 114 107)(88 118 115 108)(89 119 116 100)(90 120 117 101)
(1 89 134 116)(2 90 135 117)(3 82 127 109)(4 83 128 110)(5 84 129 111)(6 85 130 112)(7 86 131 113)(8 87 132 114)(9 88 133 115)(10 138 39 30)(11 139 40 31)(12 140 41 32)(13 141 42 33)(14 142 43 34)(15 143 44 35)(16 144 45 36)(17 136 37 28)(18 137 38 29)(19 72 57 53)(20 64 58 54)(21 65 59 46)(22 66 60 47)(23 67 61 48)(24 68 62 49)(25 69 63 50)(26 70 55 51)(27 71 56 52)(73 123 93 104)(74 124 94 105)(75 125 95 106)(76 126 96 107)(77 118 97 108)(78 119 98 100)(79 120 99 101)(80 121 91 102)(81 122 92 103)
(10 54 58)(11 46 59)(12 47 60)(13 48 61)(14 49 62)(15 50 63)(16 51 55)(17 52 56)(18 53 57)(19 38 72)(20 39 64)(21 40 65)(22 41 66)(23 42 67)(24 43 68)(25 44 69)(26 45 70)(27 37 71)(73 111 123)(74 112 124)(75 113 125)(76 114 126)(77 115 118)(78 116 119)(79 117 120)(80 109 121)(81 110 122)(82 102 91)(83 103 92)(84 104 93)(85 105 94)(86 106 95)(87 107 96)(88 108 97)(89 100 98)(90 101 99)
(1 30 134 138)(2 31 135 139)(3 32 127 140)(4 33 128 141)(5 34 129 142)(6 35 130 143)(7 36 131 144)(8 28 132 136)(9 29 133 137)(10 98 39 78)(11 99 40 79)(12 91 41 80)(13 92 42 81)(14 93 43 73)(15 94 44 74)(16 95 45 75)(17 96 37 76)(18 97 38 77)(19 115 57 88)(20 116 58 89)(21 117 59 90)(22 109 60 82)(23 110 61 83)(24 111 62 84)(25 112 63 85)(26 113 55 86)(27 114 56 87)(46 101 65 120)(47 102 66 121)(48 103 67 122)(49 104 68 123)(50 105 69 124)(51 106 70 125)(52 107 71 126)(53 108 72 118)(54 100 64 119)

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

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

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

72 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 8A 8B 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 18A ··· 18F 18G ··· 18L 24A 24B 24C 24D 36A ··· 36F 36G ··· 36L 72A ··· 72L order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 8 8 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 1 1 8 8 8 6 12 1 1 8 8 8 6 6 1 ··· 1 8 ··· 8 6 6 12 12 1 ··· 1 8 ··· 8 6 6 6 6 6 ··· 6 12 ··· 12 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 4 4 type + + + - + - image C1 C2 C3 C6 C9 C18 S3 C3×S3 CSU2(𝔽3) S3×C9 C3×CSU2(𝔽3) C9×CSU2(𝔽3) S4 C3×S4 C9×S4 CSU2(𝔽3) C3×CSU2(𝔽3) C9×CSU2(𝔽3) kernel C9×CSU2(𝔽3) C9×SL2(𝔽3) C3×CSU2(𝔽3) C3×SL2(𝔽3) CSU2(𝔽3) SL2(𝔽3) Q8×C9 C3×Q8 C9 Q8 C3 C1 C18 C6 C2 C9 C3 C1 # reps 1 1 2 2 6 6 1 2 2 6 4 12 2 4 12 1 2 6

Matrix representation of C9×CSU2(𝔽3) in GL2(𝔽73) generated by

 2 0 0 2
,
 6 9 4 67
,
 63 69 7 10
,
 69 6 10 3
,
 37 62 25 36
G:=sub<GL(2,GF(73))| [2,0,0,2],[6,4,9,67],[63,7,69,10],[69,10,6,3],[37,25,62,36] >;

C9×CSU2(𝔽3) in GAP, Magma, Sage, TeX

C_9\times {\rm CSU}_2({\mathbb F}_3)
% in TeX

G:=Group("C9xCSU(2,3)");
// GroupNames label

G:=SmallGroup(432,240);
// by ID

G=gap.SmallGroup(432,240);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,50,1011,3784,655,172,2273,404,285,124]);
// Polycyclic

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

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