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## G = C9×GL2(𝔽3)  order 432 = 24·33

### Direct product of C9 and GL2(𝔽3)

Aliases: C9×GL2(𝔽3), C18.9S4, SL2(𝔽3)⋊C18, Q8⋊(S3×C9), C2.3(C9×S4), (Q8×C9)⋊2S3, C6.17(C3×S4), (C3×GL2(𝔽3)).C3, (C9×SL2(𝔽3))⋊4C2, C3.4(C3×GL2(𝔽3)), (C3×SL2(𝔽3)).6C6, (C3×Q8).9(C3×S3), SmallGroup(432,241)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C9×GL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — C3×SL2(𝔽3) — C9×SL2(𝔽3) — C9×GL2(𝔽3)
 Lower central SL2(𝔽3) — C9×GL2(𝔽3)
 Upper central C1 — C18

Generators and relations for C9×GL2(𝔽3)
G = < a,b,c,d,e | a9=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece=b-1, dbd-1=bc, ebe=b2c, dcd-1=b, ede=d-1 >

Smallest permutation representation of C9×GL2(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 51 30 15)(2 52 31 16)(3 53 32 17)(4 54 33 18)(5 46 34 10)(6 47 35 11)(7 48 36 12)(8 49 28 13)(9 50 29 14)(19 45 57 72)(20 37 58 64)(21 38 59 65)(22 39 60 66)(23 40 61 67)(24 41 62 68)(25 42 63 69)(26 43 55 70)(27 44 56 71)
(1 58 30 20)(2 59 31 21)(3 60 32 22)(4 61 33 23)(5 62 34 24)(6 63 35 25)(7 55 36 26)(8 56 28 27)(9 57 29 19)(10 68 46 41)(11 69 47 42)(12 70 48 43)(13 71 49 44)(14 72 50 45)(15 64 51 37)(16 65 52 38)(17 66 53 39)(18 67 54 40)
(10 24 68)(11 25 69)(12 26 70)(13 27 71)(14 19 72)(15 20 64)(16 21 65)(17 22 66)(18 23 67)(37 51 58)(38 52 59)(39 53 60)(40 54 61)(41 46 62)(42 47 63)(43 48 55)(44 49 56)(45 50 57)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 28)(9 29)(10 24)(11 25)(12 26)(13 27)(14 19)(15 20)(16 21)(17 22)(18 23)(46 62)(47 63)(48 55)(49 56)(50 57)(51 58)(52 59)(53 60)(54 61)

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

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

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

72 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F 6G 8A 8B 9A ··· 9F 9G ··· 9L 12A 12B 18A ··· 18F 18G ··· 18L 18M ··· 18R 24A 24B 24C 24D 36A ··· 36F 72A ··· 72L order 1 2 2 3 3 3 3 3 4 6 6 6 6 6 6 6 8 8 9 ··· 9 9 ··· 9 12 12 18 ··· 18 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 72 ··· 72 size 1 1 12 1 1 8 8 8 6 1 1 8 8 8 12 12 6 6 1 ··· 1 8 ··· 8 6 6 1 ··· 1 8 ··· 8 12 ··· 12 6 6 6 6 6 ··· 6 6 ··· 6

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 4 4 type + + + + + image C1 C2 C3 C6 C9 C18 S3 C3×S3 GL2(𝔽3) S3×C9 C3×GL2(𝔽3) C9×GL2(𝔽3) S4 C3×S4 C9×S4 GL2(𝔽3) C3×GL2(𝔽3) C9×GL2(𝔽3) kernel C9×GL2(𝔽3) C9×SL2(𝔽3) C3×GL2(𝔽3) C3×SL2(𝔽3) GL2(𝔽3) SL2(𝔽3) Q8×C9 C3×Q8 C9 Q8 C3 C1 C18 C6 C2 C9 C3 C1 # reps 1 1 2 2 6 6 1 2 2 6 4 12 2 4 12 1 2 6

Matrix representation of C9×GL2(𝔽3) in GL2(𝔽19) generated by

 16 0 0 16
,
 7 2 13 12
,
 0 5 15 0
,
 7 5 0 11
,
 17 9 6 2
G:=sub<GL(2,GF(19))| [16,0,0,16],[7,13,2,12],[0,15,5,0],[7,0,5,11],[17,6,9,2] >;

C9×GL2(𝔽3) in GAP, Magma, Sage, TeX

C_9\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C9xGL(2,3)");
// GroupNames label

G:=SmallGroup(432,241);
// by ID

G=gap.SmallGroup(432,241);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,50,1011,3784,655,172,2273,404,285,124]);
// Polycyclic

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

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