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## G = C3×C18.D4order 432 = 24·33

### Direct product of C3 and C18.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×C18.D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C6×C18 — C6×Dic9 — C3×C18.D4
 Lower central C9 — C18 — C3×C18.D4
 Upper central C1 — C2×C6 — C22×C6

Generators and relations for C3×C18.D4
G = < a,b,c,d | a3=b18=c4=1, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b9c-1 >

Subgroups: 342 in 134 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C18, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, Dic9, C2×C18, C2×C18, C2×C18, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C3×C18, C3×C18, C3×C18, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C3×Dic9, C6×C18, C6×C18, C6×C18, C18.D4, C3×C6.D4, C6×Dic9, C2×C6×C18, C3×C18.D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, D9, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, Dic9, D18, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C3×D9, C2×Dic9, C9⋊D4, C6×Dic3, C3×C3⋊D4, C3×Dic9, C6×D9, C18.D4, C3×C6.D4, C6×Dic9, C3×C9⋊D4, C3×C18.D4

Smallest permutation representation of C3×C18.D4
On 72 points
Generators in S72
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)
(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)
(1 21 53 64)(2 20 54 63)(3 19 37 62)(4 36 38 61)(5 35 39 60)(6 34 40 59)(7 33 41 58)(8 32 42 57)(9 31 43 56)(10 30 44 55)(11 29 45 72)(12 28 46 71)(13 27 47 70)(14 26 48 69)(15 25 49 68)(16 24 50 67)(17 23 51 66)(18 22 52 65)
(1 55 10 64)(2 72 11 63)(3 71 12 62)(4 70 13 61)(5 69 14 60)(6 68 15 59)(7 67 16 58)(8 66 17 57)(9 65 18 56)(19 37 28 46)(20 54 29 45)(21 53 30 44)(22 52 31 43)(23 51 32 42)(24 50 33 41)(25 49 34 40)(26 48 35 39)(27 47 36 38)

G:=sub<Sym(72)| (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (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), (1,21,53,64)(2,20,54,63)(3,19,37,62)(4,36,38,61)(5,35,39,60)(6,34,40,59)(7,33,41,58)(8,32,42,57)(9,31,43,56)(10,30,44,55)(11,29,45,72)(12,28,46,71)(13,27,47,70)(14,26,48,69)(15,25,49,68)(16,24,50,67)(17,23,51,66)(18,22,52,65), (1,55,10,64)(2,72,11,63)(3,71,12,62)(4,70,13,61)(5,69,14,60)(6,68,15,59)(7,67,16,58)(8,66,17,57)(9,65,18,56)(19,37,28,46)(20,54,29,45)(21,53,30,44)(22,52,31,43)(23,51,32,42)(24,50,33,41)(25,49,34,40)(26,48,35,39)(27,47,36,38)>;

G:=Group( (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (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), (1,21,53,64)(2,20,54,63)(3,19,37,62)(4,36,38,61)(5,35,39,60)(6,34,40,59)(7,33,41,58)(8,32,42,57)(9,31,43,56)(10,30,44,55)(11,29,45,72)(12,28,46,71)(13,27,47,70)(14,26,48,69)(15,25,49,68)(16,24,50,67)(17,23,51,66)(18,22,52,65), (1,55,10,64)(2,72,11,63)(3,71,12,62)(4,70,13,61)(5,69,14,60)(6,68,15,59)(7,67,16,58)(8,66,17,57)(9,65,18,56)(19,37,28,46)(20,54,29,45)(21,53,30,44)(22,52,31,43)(23,51,32,42)(24,50,33,41)(25,49,34,40)(26,48,35,39)(27,47,36,38) );

G=PermutationGroup([[(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,61,67),(56,62,68),(57,63,69),(58,64,70),(59,65,71),(60,66,72)], [(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)], [(1,21,53,64),(2,20,54,63),(3,19,37,62),(4,36,38,61),(5,35,39,60),(6,34,40,59),(7,33,41,58),(8,32,42,57),(9,31,43,56),(10,30,44,55),(11,29,45,72),(12,28,46,71),(13,27,47,70),(14,26,48,69),(15,25,49,68),(16,24,50,67),(17,23,51,66),(18,22,52,65)], [(1,55,10,64),(2,72,11,63),(3,71,12,62),(4,70,13,61),(5,69,14,60),(6,68,15,59),(7,67,16,58),(8,66,17,57),(9,65,18,56),(19,37,28,46),(20,54,29,45),(21,53,30,44),(22,52,31,43),(23,51,32,42),(24,50,33,41),(25,49,34,40),(26,48,35,39),(27,47,36,38)]])

126 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6AE 9A ··· 9I 12A ··· 12H 18A ··· 18BK order 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 1 1 2 2 1 1 2 2 2 18 18 18 18 1 ··· 1 2 ··· 2 2 ··· 2 18 ··· 18 2 ··· 2

126 irreducible representations

 dim 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 type + + + + + - + + - + image C1 C2 C2 C3 C4 C6 C6 C12 S3 D4 Dic3 D6 D9 C3×S3 C3×D4 C3⋊D4 Dic9 D18 C3×Dic3 S3×C6 C3×D9 C9⋊D4 C3×C3⋊D4 C3×Dic9 C6×D9 C3×C9⋊D4 kernel C3×C18.D4 C6×Dic9 C2×C6×C18 C18.D4 C6×C18 C2×Dic9 C22×C18 C2×C18 C2×C62 C3×C18 C62 C62 C22×C6 C22×C6 C18 C3×C6 C2×C6 C2×C6 C2×C6 C2×C6 C23 C6 C6 C22 C22 C2 # reps 1 2 1 2 4 4 2 8 1 2 2 1 3 2 4 4 6 3 4 2 6 12 8 12 6 24

Matrix representation of C3×C18.D4 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 26 0 0 0 0 26
,
 10 0 0 0 5 26 0 0 0 0 28 0 0 0 0 4
,
 31 3 0 0 0 6 0 0 0 0 0 1 0 0 36 0
,
 31 3 0 0 13 6 0 0 0 0 0 1 0 0 36 0
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,26,0,0,0,0,26],[10,5,0,0,0,26,0,0,0,0,28,0,0,0,0,4],[31,0,0,0,3,6,0,0,0,0,0,36,0,0,1,0],[31,13,0,0,3,6,0,0,0,0,0,36,0,0,1,0] >;

C3×C18.D4 in GAP, Magma, Sage, TeX

C_3\times C_{18}.D_4
% in TeX

G:=Group("C3xC18.D4");
// GroupNames label

G:=SmallGroup(432,164);
// by ID

G=gap.SmallGroup(432,164);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^18=c^4=1,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^9*c^-1>;
// generators/relations

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