Copied to
clipboard

## G = S3×C3×C18order 324 = 22·34

### Direct product of C3×C18 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C3×C18
 Chief series C1 — C3 — C32 — C3×C9 — C32×C9 — S3×C3×C9 — S3×C3×C18
 Lower central C3 — S3×C3×C18
 Upper central C1 — C3×C18

Generators and relations for S3×C3×C18
G = < a,b,c,d | a3=b18=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 220 in 130 conjugacy classes, 70 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C22, S3, C6, C6, C6, C9, C9, C32, C32, C32, D6, C2×C6, C18, C18, C3×S3, C3×C6, C3×C6, C3×C6, C3×C9, C3×C9, C3×C9, C33, C2×C18, S3×C6, S3×C6, C62, S3×C9, C3×C18, C3×C18, C3×C18, S3×C32, C32×C6, C32×C9, S3×C18, C6×C18, S3×C3×C6, S3×C3×C9, C32×C18, S3×C3×C18
Quotients: C1, C2, C3, C22, S3, C6, C9, C32, D6, C2×C6, C18, C3×S3, C3×C6, C3×C9, C2×C18, S3×C6, C62, S3×C9, C3×C18, S3×C32, S3×C18, C6×C18, S3×C3×C6, S3×C3×C9, S3×C3×C18

Smallest permutation representation of S3×C3×C18
On 108 points
Generators in S108
(1 55 36)(2 56 19)(3 57 20)(4 58 21)(5 59 22)(6 60 23)(7 61 24)(8 62 25)(9 63 26)(10 64 27)(11 65 28)(12 66 29)(13 67 30)(14 68 31)(15 69 32)(16 70 33)(17 71 34)(18 72 35)(37 79 98)(38 80 99)(39 81 100)(40 82 101)(41 83 102)(42 84 103)(43 85 104)(44 86 105)(45 87 106)(46 88 107)(47 89 108)(48 90 91)(49 73 92)(50 74 93)(51 75 94)(52 76 95)(53 77 96)(54 78 97)
(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)
(1 67 24)(2 68 25)(3 69 26)(4 70 27)(5 71 28)(6 72 29)(7 55 30)(8 56 31)(9 57 32)(10 58 33)(11 59 34)(12 60 35)(13 61 36)(14 62 19)(15 63 20)(16 64 21)(17 65 22)(18 66 23)(37 104 73)(38 105 74)(39 106 75)(40 107 76)(41 108 77)(42 91 78)(43 92 79)(44 93 80)(45 94 81)(46 95 82)(47 96 83)(48 97 84)(49 98 85)(50 99 86)(51 100 87)(52 101 88)(53 102 89)(54 103 90)
(1 51)(2 52)(3 53)(4 54)(5 37)(6 38)(7 39)(8 40)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 91)(34 92)(35 93)(36 94)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 85)(66 86)(67 87)(68 88)(69 89)(70 90)(71 73)(72 74)

G:=sub<Sym(108)| (1,55,36)(2,56,19)(3,57,20)(4,58,21)(5,59,22)(6,60,23)(7,61,24)(8,62,25)(9,63,26)(10,64,27)(11,65,28)(12,66,29)(13,67,30)(14,68,31)(15,69,32)(16,70,33)(17,71,34)(18,72,35)(37,79,98)(38,80,99)(39,81,100)(40,82,101)(41,83,102)(42,84,103)(43,85,104)(44,86,105)(45,87,106)(46,88,107)(47,89,108)(48,90,91)(49,73,92)(50,74,93)(51,75,94)(52,76,95)(53,77,96)(54,78,97), (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), (1,67,24)(2,68,25)(3,69,26)(4,70,27)(5,71,28)(6,72,29)(7,55,30)(8,56,31)(9,57,32)(10,58,33)(11,59,34)(12,60,35)(13,61,36)(14,62,19)(15,63,20)(16,64,21)(17,65,22)(18,66,23)(37,104,73)(38,105,74)(39,106,75)(40,107,76)(41,108,77)(42,91,78)(43,92,79)(44,93,80)(45,94,81)(46,95,82)(47,96,83)(48,97,84)(49,98,85)(50,99,86)(51,100,87)(52,101,88)(53,102,89)(54,103,90), (1,51)(2,52)(3,53)(4,54)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,91)(34,92)(35,93)(36,94)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,73)(72,74)>;

G:=Group( (1,55,36)(2,56,19)(3,57,20)(4,58,21)(5,59,22)(6,60,23)(7,61,24)(8,62,25)(9,63,26)(10,64,27)(11,65,28)(12,66,29)(13,67,30)(14,68,31)(15,69,32)(16,70,33)(17,71,34)(18,72,35)(37,79,98)(38,80,99)(39,81,100)(40,82,101)(41,83,102)(42,84,103)(43,85,104)(44,86,105)(45,87,106)(46,88,107)(47,89,108)(48,90,91)(49,73,92)(50,74,93)(51,75,94)(52,76,95)(53,77,96)(54,78,97), (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), (1,67,24)(2,68,25)(3,69,26)(4,70,27)(5,71,28)(6,72,29)(7,55,30)(8,56,31)(9,57,32)(10,58,33)(11,59,34)(12,60,35)(13,61,36)(14,62,19)(15,63,20)(16,64,21)(17,65,22)(18,66,23)(37,104,73)(38,105,74)(39,106,75)(40,107,76)(41,108,77)(42,91,78)(43,92,79)(44,93,80)(45,94,81)(46,95,82)(47,96,83)(48,97,84)(49,98,85)(50,99,86)(51,100,87)(52,101,88)(53,102,89)(54,103,90), (1,51)(2,52)(3,53)(4,54)(5,37)(6,38)(7,39)(8,40)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,91)(34,92)(35,93)(36,94)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)(63,83)(64,84)(65,85)(66,86)(67,87)(68,88)(69,89)(70,90)(71,73)(72,74) );

G=PermutationGroup([[(1,55,36),(2,56,19),(3,57,20),(4,58,21),(5,59,22),(6,60,23),(7,61,24),(8,62,25),(9,63,26),(10,64,27),(11,65,28),(12,66,29),(13,67,30),(14,68,31),(15,69,32),(16,70,33),(17,71,34),(18,72,35),(37,79,98),(38,80,99),(39,81,100),(40,82,101),(41,83,102),(42,84,103),(43,85,104),(44,86,105),(45,87,106),(46,88,107),(47,89,108),(48,90,91),(49,73,92),(50,74,93),(51,75,94),(52,76,95),(53,77,96),(54,78,97)], [(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)], [(1,67,24),(2,68,25),(3,69,26),(4,70,27),(5,71,28),(6,72,29),(7,55,30),(8,56,31),(9,57,32),(10,58,33),(11,59,34),(12,60,35),(13,61,36),(14,62,19),(15,63,20),(16,64,21),(17,65,22),(18,66,23),(37,104,73),(38,105,74),(39,106,75),(40,107,76),(41,108,77),(42,91,78),(43,92,79),(44,93,80),(45,94,81),(46,95,82),(47,96,83),(48,97,84),(49,98,85),(50,99,86),(51,100,87),(52,101,88),(53,102,89),(54,103,90)], [(1,51),(2,52),(3,53),(4,54),(5,37),(6,38),(7,39),(8,40),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,91),(34,92),(35,93),(36,94),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82),(63,83),(64,84),(65,85),(66,86),(67,87),(68,88),(69,89),(70,90),(71,73),(72,74)]])

162 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 6A ··· 6H 6I ··· 6Q 6R ··· 6AG 9A ··· 9R 9S ··· 9AJ 18A ··· 18R 18S ··· 18AJ 18AK ··· 18BT order 1 2 2 2 3 ··· 3 3 ··· 3 6 ··· 6 6 ··· 6 6 ··· 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 3 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 C9 C18 C18 S3 D6 C3×S3 C3×S3 S3×C6 S3×C6 S3×C9 S3×C18 kernel S3×C3×C18 S3×C3×C9 C32×C18 S3×C18 S3×C3×C6 S3×C9 C3×C18 S3×C32 C32×C6 S3×C6 C3×S3 C3×C6 C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C3 # reps 1 2 1 6 2 12 6 4 2 18 36 18 1 1 6 2 6 2 18 18

Matrix representation of S3×C3×C18 in GL3(𝔽19) generated by

 7 0 0 0 1 0 0 0 1
,
 8 0 0 0 14 0 0 0 14
,
 1 0 0 0 11 0 0 18 7
,
 18 0 0 0 11 6 0 18 8
G:=sub<GL(3,GF(19))| [7,0,0,0,1,0,0,0,1],[8,0,0,0,14,0,0,0,14],[1,0,0,0,11,18,0,0,7],[18,0,0,0,11,18,0,6,8] >;

S3×C3×C18 in GAP, Magma, Sage, TeX

S_3\times C_3\times C_{18}
% in TeX

G:=Group("S3xC3xC18");
// GroupNames label

G:=SmallGroup(324,137);
// by ID

G=gap.SmallGroup(324,137);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,93,7781]);
// Polycyclic

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

׿
×
𝔽