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## G = D4×C18order 144 = 24·32

### Direct product of C18 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×C18
G = < a,b,c | a18=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 105 in 81 conjugacy classes, 57 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×4], C22 [×4], C6, C6 [×2], C6 [×4], C2×C4, D4 [×4], C23 [×2], C9, C12 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C2×D4, C18, C18 [×2], C18 [×4], C2×C12, C3×D4 [×4], C22×C6 [×2], C36 [×2], C2×C18, C2×C18 [×4], C2×C18 [×4], C6×D4, C2×C36, D4×C9 [×4], C22×C18 [×2], D4×C18
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, C9, C2×C6 [×7], C2×D4, C18 [×7], C3×D4 [×2], C22×C6, C2×C18 [×7], C6×D4, D4×C9 [×2], C22×C18, D4×C18

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

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

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

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

D4×C18 is a maximal subgroup of
C36.D4  D4⋊Dic9  C232Dic9  D366C22  C23.23D18  C36.17D4  C232D18  C362D4  Dic9⋊D4  C36⋊D4  D46D18

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A ··· 6F 6G ··· 6N 9A ··· 9F 12A 12B 12C 12D 18A ··· 18R 18S ··· 18AP 36A ··· 36L order 1 2 2 2 2 2 2 2 3 3 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 2 1 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 D4 C3×D4 D4×C9 kernel D4×C18 C2×C36 D4×C9 C22×C18 C6×D4 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C18 C6 C2 # reps 1 1 4 2 2 2 8 4 6 6 24 12 2 4 12

Matrix representation of D4×C18 in GL3(𝔽37) generated by

 11 0 0 0 12 0 0 0 12
,
 1 0 0 0 1 14 0 21 36
,
 1 0 0 0 1 0 0 21 36
G:=sub<GL(3,GF(37))| [11,0,0,0,12,0,0,0,12],[1,0,0,0,1,21,0,14,36],[1,0,0,0,1,21,0,0,36] >;

D4×C18 in GAP, Magma, Sage, TeX

D_4\times C_{18}
% in TeX

G:=Group("D4xC18");
// GroupNames label

G:=SmallGroup(144,48);
// by ID

G=gap.SmallGroup(144,48);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,313,165]);
// Polycyclic

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

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