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## G = D4×C9order 72 = 23·32

### Direct product of C9 and D4

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

Aliases: D4×C9, C4⋊C18, C363C2, C12.3C6, C222C18, C18.6C22, C3.(C3×D4), C18(C3×D4), (C3×D4).C3, (C2×C18)⋊1C2, (C2×C6).2C6, C6.6(C2×C6), C2.1(C2×C18), SmallGroup(72,10)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D4×C9
G = < a,b,c | a9=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D4×C9
On 36 points
Generators in S36
(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)
(1 34 22 14)(2 35 23 15)(3 36 24 16)(4 28 25 17)(5 29 26 18)(6 30 27 10)(7 31 19 11)(8 32 20 12)(9 33 21 13)
(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 28)(18 29)

G:=sub<Sym(36)| (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), (1,34,22,14)(2,35,23,15)(3,36,24,16)(4,28,25,17)(5,29,26,18)(6,30,27,10)(7,31,19,11)(8,32,20,12)(9,33,21,13), (10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29)>;

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), (1,34,22,14)(2,35,23,15)(3,36,24,16)(4,28,25,17)(5,29,26,18)(6,30,27,10)(7,31,19,11)(8,32,20,12)(9,33,21,13), (10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,28)(18,29) );

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)], [(1,34,22,14),(2,35,23,15),(3,36,24,16),(4,28,25,17),(5,29,26,18),(6,30,27,10),(7,31,19,11),(8,32,20,12),(9,33,21,13)], [(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,28),(18,29)])

D4×C9 is a maximal subgroup of   D4.D9  D4⋊D9  D42D9  2- 1+4⋊C9

45 conjugacy classes

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

45 irreducible representations

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

Matrix representation of D4×C9 in GL2(𝔽19) generated by

 6 0 0 6
,
 0 18 1 0
,
 0 1 1 0
G:=sub<GL(2,GF(19))| [6,0,0,6],[0,1,18,0],[0,1,1,0] >;

D4×C9 in GAP, Magma, Sage, TeX

D_4\times C_9
% in TeX

G:=Group("D4xC9");
// GroupNames label

G:=SmallGroup(72,10);
// by ID

G=gap.SmallGroup(72,10);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-3,141,102]);
// Polycyclic

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

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