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## G = D9×SL2(𝔽3)  order 432 = 24·33

### Direct product of D9 and SL2(𝔽3)

Aliases: D9×SL2(𝔽3), D18.2A4, C6.4(S3×A4), (Q8×D9)⋊3C3, (Q8×C9)⋊5C6, Q82(C3×D9), C2.3(A4×D9), C18.8(C2×A4), C93(C2×SL2(𝔽3)), (C9×SL2(𝔽3))⋊3C2, C3.2(S3×SL2(𝔽3)), (C3×SL2(𝔽3)).7S3, (C3×Q8).14(C3×S3), SmallGroup(432,264)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C9 — D9×SL2(𝔽3)
 Chief series C1 — C3 — C6 — C18 — Q8×C9 — C9×SL2(𝔽3) — D9×SL2(𝔽3)
 Lower central Q8×C9 — D9×SL2(𝔽3)
 Upper central C1 — C2

Generators and relations for D9×SL2(𝔽3)
G = < a,b,c,d,e | a9=b2=c4=e3=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 419 in 60 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×Q8, D9, C18, C18, C3×S3, C3×C6, SL2(𝔽3), SL2(𝔽3), Dic6, C4×S3, C3×Q8, C3×C9, Dic9, C36, D18, S3×C6, C2×SL2(𝔽3), S3×Q8, C3×D9, C3×C18, Q8⋊C9, Dic18, C4×D9, Q8×C9, C3×SL2(𝔽3), C6×D9, Q8×D9, S3×SL2(𝔽3), C9×SL2(𝔽3), D9×SL2(𝔽3)
Quotients: C1, C2, C3, S3, C6, A4, D9, C3×S3, SL2(𝔽3), C2×A4, C2×SL2(𝔽3), C3×D9, S3×A4, S3×SL2(𝔽3), A4×D9, D9×SL2(𝔽3)

Smallest permutation representation of D9×SL2(𝔽3)
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 19)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 66)(11 65)(12 64)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(28 40)(29 39)(30 38)(31 37)(32 45)(33 44)(34 43)(35 42)(36 41)(46 58)(47 57)(48 56)(49 55)(50 63)(51 62)(52 61)(53 60)(54 59)
(1 38 20 31)(2 39 21 32)(3 40 22 33)(4 41 23 34)(5 42 24 35)(6 43 25 36)(7 44 26 28)(8 45 27 29)(9 37 19 30)(10 49 67 56)(11 50 68 57)(12 51 69 58)(13 52 70 59)(14 53 71 60)(15 54 72 61)(16 46 64 62)(17 47 65 63)(18 48 66 55)
(1 10 20 67)(2 11 21 68)(3 12 22 69)(4 13 23 70)(5 14 24 71)(6 15 25 72)(7 16 26 64)(8 17 27 65)(9 18 19 66)(28 46 44 62)(29 47 45 63)(30 48 37 55)(31 49 38 56)(32 50 39 57)(33 51 40 58)(34 52 41 59)(35 53 42 60)(36 54 43 61)
(10 38 56)(11 39 57)(12 40 58)(13 41 59)(14 42 60)(15 43 61)(16 44 62)(17 45 63)(18 37 55)(28 46 64)(29 47 65)(30 48 66)(31 49 67)(32 50 68)(33 51 69)(34 52 70)(35 53 71)(36 54 72)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A 9B 9C 9D ··· 9I 12 18A 18B 18C 18D ··· 18I 36A 36B 36C order 1 2 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 9 9 9 9 ··· 9 12 18 18 18 18 ··· 18 36 36 36 size 1 1 9 9 2 4 4 8 8 6 54 2 4 4 8 8 36 36 36 36 2 2 2 8 ··· 8 12 2 2 2 8 ··· 8 12 12 12

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 3 4 4 4 4 6 6 type + + + + - + + - - + + image C1 C2 C3 C6 S3 D9 C3×S3 SL2(𝔽3) SL2(𝔽3) C3×D9 A4 C2×A4 S3×SL2(𝔽3) S3×SL2(𝔽3) D9×SL2(𝔽3) D9×SL2(𝔽3) S3×A4 A4×D9 kernel D9×SL2(𝔽3) C9×SL2(𝔽3) Q8×D9 Q8×C9 C3×SL2(𝔽3) SL2(𝔽3) C3×Q8 D9 D9 Q8 D18 C18 C3 C3 C1 C1 C6 C2 # reps 1 1 2 2 1 3 2 2 4 6 1 1 1 2 3 6 1 3

Matrix representation of D9×SL2(𝔽3) in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 31 20 0 0 17 11
,
 36 0 0 0 0 36 0 0 0 0 17 6 0 0 26 20
,
 0 1 0 0 36 0 0 0 0 0 1 0 0 0 0 1
,
 10 26 0 0 26 27 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 10 26 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,31,17,0,0,20,11],[36,0,0,0,0,36,0,0,0,0,17,26,0,0,6,20],[0,36,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[10,26,0,0,26,27,0,0,0,0,1,0,0,0,0,1],[1,10,0,0,0,26,0,0,0,0,1,0,0,0,0,1] >;

D9×SL2(𝔽3) in GAP, Magma, Sage, TeX

D_9\times {\rm SL}_2({\mathbb F}_3)
% in TeX

G:=Group("D9xSL(2,3)");
// GroupNames label

G:=SmallGroup(432,264);
// by ID

G=gap.SmallGroup(432,264);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-3,-2,-3,198,268,94,409,192,6724,452,14118]);
// Polycyclic

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

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