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## G = D4×C9⋊C6order 432 = 24·33

### Direct product of D4 and C9⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — D4×C9⋊C6
 Chief series C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C22×C9⋊C6 — D4×C9⋊C6
 Lower central C9 — C18 — D4×C9⋊C6
 Upper central C1 — C2 — D4

Generators and relations for D4×C9⋊C6
G = < a,b,c,d | a4=b2=c9=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c2 >

Subgroups: 750 in 170 conjugacy classes, 56 normal (30 characteristic)
C1, C2, C2 [×6], C3, C3, C4, C4, C22 [×2], C22 [×7], S3 [×4], C6, C6 [×9], C2×C4, D4, D4 [×3], C23 [×2], C9, C9, C32, Dic3, C12, C12 [×2], D6 [×7], C2×C6 [×2], C2×C6 [×9], C2×D4, D9 [×2], D9 [×2], C18, C18 [×5], C3×S3 [×4], C3×C6, C3×C6 [×2], C4×S3, D12, C3⋊D4 [×2], C2×C12, C3×D4, C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], 3- 1+2, Dic9, C36, C36, D18, D18 [×2], D18 [×4], C2×C18 [×2], C2×C18 [×2], C3×Dic3, C3×C12, S3×C6 [×7], C62 [×2], S3×D4, C6×D4, C9⋊C6 [×2], C9⋊C6 [×2], C2×3- 1+2, C2×3- 1+2 [×2], C4×D9, D36, C9⋊D4 [×2], D4×C9, D4×C9, C22×D9 [×2], S3×C12, C3×D12, C3×C3⋊D4 [×2], D4×C32, S3×C2×C6 [×2], C9⋊C12, C4×3- 1+2, C2×C9⋊C6, C2×C9⋊C6 [×2], C2×C9⋊C6 [×4], C22×3- 1+2 [×2], D4×D9, C3×S3×D4, C4×C9⋊C6, D36⋊C3, Dic9⋊C6 [×2], D4×3- 1+2, C22×C9⋊C6 [×2], D4×C9⋊C6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, S3×C6 [×3], S3×D4, C6×D4, C9⋊C6, S3×C2×C6, C2×C9⋊C6 [×3], C3×S3×D4, C22×C9⋊C6, D4×C9⋊C6

Smallest permutation representation of D4×C9⋊C6
On 36 points
Generators in S36
(1 29 11 20)(2 30 12 21)(3 31 13 22)(4 32 14 23)(5 33 15 24)(6 34 16 25)(7 35 17 26)(8 36 18 27)(9 28 10 19)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 19)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 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 11)(2 16 8 10 5 13)(3 12 6 18 9 15)(4 17)(7 14)(19 33 22 30 25 36)(20 29)(21 34 27 28 24 31)(23 35)(26 32)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 6P 6Q 9A 9B 9C 12A 12B 12C 12D 12E 18A 18B 18C 18D ··· 18I 36A 36B 36C order 1 2 2 2 2 2 2 2 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 9 9 9 12 12 12 12 12 18 18 18 18 ··· 18 36 36 36 size 1 1 2 2 9 9 18 18 2 3 3 2 18 2 3 3 4 4 6 6 6 6 9 9 9 9 18 18 18 18 6 6 6 4 6 6 18 18 6 6 6 12 ··· 12 12 12 12

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 12 2 2 2 2 2 2 2 2 4 4 6 6 6 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4×C9⋊C6 S3 D4 D6 D6 C3×S3 C3×D4 S3×C6 S3×C6 S3×D4 C3×S3×D4 C9⋊C6 C2×C9⋊C6 C2×C9⋊C6 kernel D4×C9⋊C6 C4×C9⋊C6 D36⋊C3 Dic9⋊C6 D4×3- 1+2 C22×C9⋊C6 D4×D9 C4×D9 D36 C9⋊D4 D4×C9 C22×D9 C1 D4×C32 C9⋊C6 C3×C12 C62 C3×D4 D9 C12 C2×C6 C32 C3 D4 C4 C22 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 1 2 1 2 2 4 2 4 1 2 1 1 2

Matrix representation of D4×C9⋊C6 in GL10(ℤ)

 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1
,
 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1
,
 -1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 1 0 0

G:=sub<GL(10,Integers())| [0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,-1],[-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0],[0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0] >;

D4×C9⋊C6 in GAP, Magma, Sage, TeX

D_4\times C_9\rtimes C_6
% in TeX

G:=Group("D4xC9:C6");
// GroupNames label

G:=SmallGroup(432,362);
// by ID

G=gap.SmallGroup(432,362);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,303,10085,1034,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^9=d^6=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^2>;
// generators/relations

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