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## G = C18×C3⋊D4order 432 = 24·33

### Direct product of C18 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C18×C3⋊D4
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — S3×C2×C18 — C18×C3⋊D4
 Lower central C3 — C6 — C18×C3⋊D4
 Upper central C1 — C2×C18 — C22×C18

Generators and relations for C18×C3⋊D4
G = < a,b,c,d | a18=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 388 in 194 conjugacy classes, 81 normal (33 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C9, C9, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×D4, C18, C18, C18, C3×S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C3×C9, C36, C2×C18, C2×C18, C2×C18, C3×Dic3, S3×C6, S3×C6, C62, C62, C62, C2×C3⋊D4, C6×D4, S3×C9, C3×C18, C3×C18, C3×C18, C2×C36, D4×C9, C22×C18, C22×C18, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, C9×Dic3, S3×C18, S3×C18, C6×C18, C6×C18, C6×C18, D4×C18, C6×C3⋊D4, Dic3×C18, C9×C3⋊D4, S3×C2×C18, C2×C6×C18, C18×C3⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, C9, D6, C2×C6, C2×D4, C18, C3×S3, C3⋊D4, C3×D4, C22×S3, C22×C6, C2×C18, S3×C6, C2×C3⋊D4, C6×D4, S3×C9, D4×C9, C22×C18, C3×C3⋊D4, S3×C2×C6, S3×C18, D4×C18, C6×C3⋊D4, C9×C3⋊D4, S3×C2×C18, C18×C3⋊D4

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

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

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

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

162 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A ··· 6F 6G ··· 6AE 6AF 6AG 6AH 6AI 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 18A ··· 18R 18S ··· 18BT 18BU ··· 18CF 36A ··· 36L order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 6 6 1 1 2 2 2 6 6 1 ··· 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 6 ··· 6

162 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 C9 C18 C18 C18 C18 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 S3×C9 D4×C9 C3×C3⋊D4 S3×C18 C9×C3⋊D4 kernel C18×C3⋊D4 Dic3×C18 C9×C3⋊D4 S3×C2×C18 C2×C6×C18 C6×C3⋊D4 C6×Dic3 C3×C3⋊D4 S3×C2×C6 C2×C62 C2×C3⋊D4 C2×Dic3 C3⋊D4 C22×S3 C22×C6 C22×C18 C3×C18 C2×C18 C22×C6 C18 C3×C6 C2×C6 C23 C6 C6 C22 C2 # reps 1 1 4 1 1 2 2 8 2 2 6 6 24 6 6 1 2 3 2 4 4 6 6 12 8 18 24

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

 11 0 0 0 0 11 0 0 0 0 34 0 0 0 0 34
,
 1 0 0 0 0 1 0 0 0 0 26 0 0 0 0 10
,
 18 4 0 0 2 19 0 0 0 0 0 1 0 0 36 0
,
 19 32 0 0 35 18 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(37))| [11,0,0,0,0,11,0,0,0,0,34,0,0,0,0,34],[1,0,0,0,0,1,0,0,0,0,26,0,0,0,0,10],[18,2,0,0,4,19,0,0,0,0,0,36,0,0,1,0],[19,35,0,0,32,18,0,0,0,0,0,1,0,0,1,0] >;

C18×C3⋊D4 in GAP, Magma, Sage, TeX

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

G:=Group("C18xC3:D4");
// GroupNames label

G:=SmallGroup(432,375);
// by ID

G=gap.SmallGroup(432,375);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,192,14118]);
// Polycyclic

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

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