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## G = C18×SL2(𝔽3)  order 432 = 24·33

### Direct product of C18 and SL2(𝔽3)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C18×SL2(𝔽3)
G = < a,b,c,d | a18=b4=d3=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=c, dcd-1=bc >

Subgroups: 211 in 82 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, Q8, C9, C9, C32, C12, C2×C6, C2×C6, C2×Q8, C18, C18, C18, C3×C6, SL2(𝔽3), C2×C12, C3×Q8, C3×Q8, C3×C9, C36, C2×C18, C2×C18, C62, C2×SL2(𝔽3), C6×Q8, C3×C18, Q8⋊C9, C2×C36, Q8×C9, Q8×C9, C3×SL2(𝔽3), C6×C18, C2×Q8⋊C9, Q8×C18, C6×SL2(𝔽3), C9×SL2(𝔽3), C18×SL2(𝔽3)
Quotients: C1, C2, C3, C6, C9, C32, A4, C18, C3×C6, SL2(𝔽3), C2×A4, C3×C9, C3×A4, C2×SL2(𝔽3), C3×C18, C3×SL2(𝔽3), C6×A4, C9×A4, C6×SL2(𝔽3), C9×SL2(𝔽3), A4×C18, C18×SL2(𝔽3)

Smallest permutation representation of C18×SL2(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 97 82 65)(2 98 83 66)(3 99 84 67)(4 100 85 68)(5 101 86 69)(6 102 87 70)(7 103 88 71)(8 104 89 72)(9 105 90 55)(10 106 73 56)(11 107 74 57)(12 108 75 58)(13 91 76 59)(14 92 77 60)(15 93 78 61)(16 94 79 62)(17 95 80 63)(18 96 81 64)(19 127 110 47)(20 128 111 48)(21 129 112 49)(22 130 113 50)(23 131 114 51)(24 132 115 52)(25 133 116 53)(26 134 117 54)(27 135 118 37)(28 136 119 38)(29 137 120 39)(30 138 121 40)(31 139 122 41)(32 140 123 42)(33 141 124 43)(34 142 125 44)(35 143 126 45)(36 144 109 46)
(1 48 82 128)(2 49 83 129)(3 50 84 130)(4 51 85 131)(5 52 86 132)(6 53 87 133)(7 54 88 134)(8 37 89 135)(9 38 90 136)(10 39 73 137)(11 40 74 138)(12 41 75 139)(13 42 76 140)(14 43 77 141)(15 44 78 142)(16 45 79 143)(17 46 80 144)(18 47 81 127)(19 96 110 64)(20 97 111 65)(21 98 112 66)(22 99 113 67)(23 100 114 68)(24 101 115 69)(25 102 116 70)(26 103 117 71)(27 104 118 72)(28 105 119 55)(29 106 120 56)(30 107 121 57)(31 108 122 58)(32 91 123 59)(33 92 124 60)(34 93 125 61)(35 94 126 62)(36 95 109 63)
(19 127 64)(20 128 65)(21 129 66)(22 130 67)(23 131 68)(24 132 69)(25 133 70)(26 134 71)(27 135 72)(28 136 55)(29 137 56)(30 138 57)(31 139 58)(32 140 59)(33 141 60)(34 142 61)(35 143 62)(36 144 63)(37 104 118)(38 105 119)(39 106 120)(40 107 121)(41 108 122)(42 91 123)(43 92 124)(44 93 125)(45 94 126)(46 95 109)(47 96 110)(48 97 111)(49 98 112)(50 99 113)(51 100 114)(52 101 115)(53 102 116)(54 103 117)

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

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

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

126 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 4A 4B 6A ··· 6F 6G ··· 6X 9A ··· 9F 9G ··· 9R 12A 12B 12C 12D 18A ··· 18R 18S ··· 18BB 36A ··· 36L order 1 2 2 2 3 3 3 ··· 3 4 4 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 1 1 4 ··· 4 6 6 1 ··· 1 4 ··· 4 1 ··· 1 4 ··· 4 6 6 6 6 1 ··· 1 4 ··· 4 6 ··· 6

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 3 3 3 type + + - + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C18 SL2(𝔽3) SL2(𝔽3) C3×SL2(𝔽3) C9×SL2(𝔽3) A4 C2×A4 C3×A4 C6×A4 C9×A4 A4×C18 kernel C18×SL2(𝔽3) C9×SL2(𝔽3) C2×Q8⋊C9 Q8×C18 C6×SL2(𝔽3) Q8⋊C9 Q8×C9 C3×SL2(𝔽3) C2×SL2(𝔽3) SL2(𝔽3) C18 C18 C6 C2 C2×C18 C18 C2×C6 C6 C22 C2 # reps 1 1 4 2 2 4 2 2 18 18 2 4 12 36 1 1 2 2 6 6

Matrix representation of C18×SL2(𝔽3) in GL3(𝔽37) generated by

 36 0 0 0 3 0 0 0 3
,
 1 0 0 0 0 36 0 1 0
,
 1 0 0 0 26 27 0 27 11
,
 26 0 0 0 1 0 0 11 10
G:=sub<GL(3,GF(37))| [36,0,0,0,3,0,0,0,3],[1,0,0,0,0,1,0,36,0],[1,0,0,0,26,27,0,27,11],[26,0,0,0,1,11,0,0,10] >;

C18×SL2(𝔽3) in GAP, Magma, Sage, TeX

C_{18}\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("C18xSL(2,3)");
// GroupNames label

G:=SmallGroup(432,327);
// by ID

G=gap.SmallGroup(432,327);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,79,1901,172,3414,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^18=b^4=d^3=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=c,d*c*d^-1=b*c>;
// generators/relations

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