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

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

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

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

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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