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## G = C32.GL2(𝔽3)  order 432 = 24·33

### The non-split extension by C32 of GL2(𝔽3) acting via GL2(𝔽3)/Q8=S3

Aliases: C32.GL2(𝔽3), Q8⋊C9⋊C6, Q8⋊D9⋊C3, Q8⋊(C9⋊C6), C6.3(C3×S4), (C3×C6).2S4, (Q8×C32).7S3, C2.3(C32.S4), Q8⋊3- 1+2⋊C2, C3.1(C3×GL2(𝔽3)), (C3×Q8).3(C3×S3), SmallGroup(432,245)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — Q8⋊C9 — C32.GL2(𝔽3)
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — Q8⋊3- 1+2 — C32.GL2(𝔽3)
 Lower central Q8⋊C9 — C32.GL2(𝔽3)
 Upper central C1 — C2

Generators and relations for C32.GL2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c4=f2=1, d2=c2, e3=fbf=b-1, eae-1=ab=ba, ac=ca, ad=da, af=fa, bc=cb, bd=db, be=eb, dcd-1=fdf=c-1, ece-1=cd, fcf=c2d, ede-1=c, fef=be2 >

Character table of C32.GL2(𝔽3)

 class 1 2A 2B 3A 3B 3C 4 6A 6B 6C 6D 6E 8A 8B 9A 9B 9C 12A 12B 12C 12D 12E 18A 18B 18C 24A 24B 24C 24D size 1 1 36 2 3 3 6 2 3 3 36 36 18 18 24 24 24 6 6 12 12 12 24 24 24 18 18 18 18 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 -1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ65 ζ6 -1 -1 1 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 ζ32 1 ζ65 ζ6 ζ65 ζ6 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ5 1 1 1 1 ζ3 ζ32 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ6 1 1 -1 1 ζ32 ζ3 1 1 ζ3 ζ32 ζ6 ζ65 -1 -1 1 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 ζ3 1 ζ6 ζ65 ζ6 ζ65 linear of order 6 ρ7 2 2 0 2 2 2 2 2 2 2 0 0 0 0 -1 -1 -1 2 2 2 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ8 2 2 0 2 -1-√-3 -1+√-3 2 2 -1+√-3 -1-√-3 0 0 0 0 -1 ζ6 ζ65 -1-√-3 -1+√-3 -1-√-3 2 -1+√-3 ζ6 ζ65 -1 0 0 0 0 complex lifted from C3×S3 ρ9 2 2 0 2 -1+√-3 -1-√-3 2 2 -1-√-3 -1+√-3 0 0 0 0 -1 ζ65 ζ6 -1+√-3 -1-√-3 -1+√-3 2 -1-√-3 ζ65 ζ6 -1 0 0 0 0 complex lifted from C3×S3 ρ10 2 -2 0 2 2 2 0 -2 -2 -2 0 0 √-2 -√-2 -1 -1 -1 0 0 0 0 0 1 1 1 -√-2 √-2 √-2 -√-2 complex lifted from GL2(𝔽3) ρ11 2 -2 0 2 2 2 0 -2 -2 -2 0 0 -√-2 √-2 -1 -1 -1 0 0 0 0 0 1 1 1 √-2 -√-2 -√-2 √-2 complex lifted from GL2(𝔽3) ρ12 2 -2 0 2 -1-√-3 -1+√-3 0 -2 1-√-3 1+√-3 0 0 -√-2 √-2 -1 ζ6 ζ65 0 0 0 0 0 ζ32 ζ3 1 ζ83ζ32+ζ8ζ32 ζ87ζ3+ζ85ζ3 ζ87ζ32+ζ85ζ32 ζ83ζ3+ζ8ζ3 complex lifted from C3×GL2(𝔽3) ρ13 2 -2 0 2 -1+√-3 -1-√-3 0 -2 1+√-3 1-√-3 0 0 -√-2 √-2 -1 ζ65 ζ6 0 0 0 0 0 ζ3 ζ32 1 ζ83ζ3+ζ8ζ3 ζ87ζ32+ζ85ζ32 ζ87ζ3+ζ85ζ3 ζ83ζ32+ζ8ζ32 complex lifted from C3×GL2(𝔽3) ρ14 2 -2 0 2 -1+√-3 -1-√-3 0 -2 1+√-3 1-√-3 0 0 √-2 -√-2 -1 ζ65 ζ6 0 0 0 0 0 ζ3 ζ32 1 ζ87ζ3+ζ85ζ3 ζ83ζ32+ζ8ζ32 ζ83ζ3+ζ8ζ3 ζ87ζ32+ζ85ζ32 complex lifted from C3×GL2(𝔽3) ρ15 2 -2 0 2 -1-√-3 -1+√-3 0 -2 1-√-3 1+√-3 0 0 √-2 -√-2 -1 ζ6 ζ65 0 0 0 0 0 ζ32 ζ3 1 ζ87ζ32+ζ85ζ32 ζ83ζ3+ζ8ζ3 ζ83ζ32+ζ8ζ32 ζ87ζ3+ζ85ζ3 complex lifted from C3×GL2(𝔽3) ρ16 3 3 -1 3 3 3 -1 3 3 3 -1 -1 1 1 0 0 0 -1 -1 -1 -1 -1 0 0 0 1 1 1 1 orthogonal lifted from S4 ρ17 3 3 1 3 3 3 -1 3 3 3 1 1 -1 -1 0 0 0 -1 -1 -1 -1 -1 0 0 0 -1 -1 -1 -1 orthogonal lifted from S4 ρ18 3 3 1 3 -3-3√-3/2 -3+3√-3/2 -1 3 -3+3√-3/2 -3-3√-3/2 ζ32 ζ3 -1 -1 0 0 0 ζ6 ζ65 ζ6 -1 ζ65 0 0 0 ζ6 ζ65 ζ6 ζ65 complex lifted from C3×S4 ρ19 3 3 -1 3 -3+3√-3/2 -3-3√-3/2 -1 3 -3-3√-3/2 -3+3√-3/2 ζ65 ζ6 1 1 0 0 0 ζ65 ζ6 ζ65 -1 ζ6 0 0 0 ζ3 ζ32 ζ3 ζ32 complex lifted from C3×S4 ρ20 3 3 -1 3 -3-3√-3/2 -3+3√-3/2 -1 3 -3+3√-3/2 -3-3√-3/2 ζ6 ζ65 1 1 0 0 0 ζ6 ζ65 ζ6 -1 ζ65 0 0 0 ζ32 ζ3 ζ32 ζ3 complex lifted from C3×S4 ρ21 3 3 1 3 -3+3√-3/2 -3-3√-3/2 -1 3 -3-3√-3/2 -3+3√-3/2 ζ3 ζ32 -1 -1 0 0 0 ζ65 ζ6 ζ65 -1 ζ6 0 0 0 ζ65 ζ6 ζ65 ζ6 complex lifted from C3×S4 ρ22 4 -4 0 4 4 4 0 -4 -4 -4 0 0 0 0 1 1 1 0 0 0 0 0 -1 -1 -1 0 0 0 0 orthogonal lifted from GL2(𝔽3) ρ23 4 -4 0 4 -2+2√-3 -2-2√-3 0 -4 2+2√-3 2-2√-3 0 0 0 0 1 ζ3 ζ32 0 0 0 0 0 ζ65 ζ6 -1 0 0 0 0 complex lifted from C3×GL2(𝔽3) ρ24 4 -4 0 4 -2-2√-3 -2+2√-3 0 -4 2-2√-3 2+2√-3 0 0 0 0 1 ζ32 ζ3 0 0 0 0 0 ζ6 ζ65 -1 0 0 0 0 complex lifted from C3×GL2(𝔽3) ρ25 6 6 0 -3 0 0 6 -3 0 0 0 0 0 0 0 0 0 0 0 0 -3 0 0 0 0 0 0 0 0 orthogonal lifted from C9⋊C6 ρ26 6 6 0 -3 0 0 -2 -3 0 0 0 0 0 0 0 0 0 4 4 -2 1 -2 0 0 0 0 0 0 0 orthogonal lifted from C32.S4 ρ27 6 6 0 -3 0 0 -2 -3 0 0 0 0 0 0 0 0 0 -2-2√-3 -2+2√-3 1+√-3 1 1-√-3 0 0 0 0 0 0 0 complex lifted from C32.S4 ρ28 6 6 0 -3 0 0 -2 -3 0 0 0 0 0 0 0 0 0 -2+2√-3 -2-2√-3 1-√-3 1 1+√-3 0 0 0 0 0 0 0 complex lifted from C32.S4 ρ29 12 -12 0 -6 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful

Smallest permutation representation of C32.GL2(𝔽3)
On 72 points
Generators in S72
(2 8 5)(3 6 9)(10 16 13)(11 14 17)(19 25 22)(20 23 26)(28 34 31)(29 32 35)(38 44 41)(39 42 45)(46 52 49)(47 50 53)(56 62 59)(57 60 63)(64 70 67)(65 68 71)
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)
(1 40 36 15)(2 22 28 52)(3 63 29 68)(4 43 30 18)(5 25 31 46)(6 57 32 71)(7 37 33 12)(8 19 34 49)(9 60 35 65)(10 56 44 70)(11 47 45 26)(13 59 38 64)(14 50 39 20)(16 62 41 67)(17 53 42 23)(21 61 51 66)(24 55 54 69)(27 58 48 72)
(1 61 36 66)(2 41 28 16)(3 23 29 53)(4 55 30 69)(5 44 31 10)(6 26 32 47)(7 58 33 72)(8 38 34 13)(9 20 35 50)(11 57 45 71)(12 48 37 27)(14 60 39 65)(15 51 40 21)(17 63 42 68)(18 54 43 24)(19 59 49 64)(22 62 52 67)(25 56 46 70)
(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)
(2 9)(3 8)(4 7)(5 6)(10 57)(11 56)(12 55)(13 63)(14 62)(15 61)(16 60)(17 59)(18 58)(19 53)(20 52)(21 51)(22 50)(23 49)(24 48)(25 47)(26 46)(27 54)(28 35)(29 34)(30 33)(31 32)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 72)(44 71)(45 70)

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

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

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

Matrix representation of C32.GL2(𝔽3) in GL8(𝔽73)

 64 0 0 0 0 0 0 0 0 64 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 36 65 36 45 0 0 0 0 27 48 45 36 0 0 0 0 20 72 0 0 36 28 0 0 46 55 0 0 28 36
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 36 28 0 0 0 0 0 0 28 36 0 0 0 0 0 0 24 5 36 28 0 0 0 0 68 49 28 36 0 0 0 0 45 64 0 0 36 28 0 0 9 28 0 0 28 36
,
 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 8 38 0 1 0 0 0 0 38 8 1 0 0 0 0 0 36 37 0 0 72 0 0 0 9 64 0 0 0 72
,
 61 72 0 0 0 0 0 0 72 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 44 44 72 0 0 0 0 0 56 56 0 72 0 0 0 0 29 56 0 0 0 72 0 0 56 29 0 0 72 0
,
 66 5 0 0 0 0 0 0 6 6 0 0 0 0 0 0 0 0 14 33 72 56 0 0 0 0 43 31 56 72 0 0 0 0 34 5 34 50 36 28 0 0 2 45 20 67 28 36 0 0 25 40 4 12 0 0 0 0 30 61 29 31 0 0
,
 1 0 0 0 0 0 0 0 61 72 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 28 72 0 0 36 28 0 0 48 47 0 0 45 37 0 0 57 39 36 28 0 0 0 0 30 15 45 37 0 0

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,36,27,20,46,0,0,0,1,65,48,72,55,0,0,0,0,36,45,0,0,0,0,0,0,45,36,0,0,0,0,0,0,0,0,36,28,0,0,0,0,0,0,28,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,28,24,68,45,9,0,0,28,36,5,49,64,28,0,0,0,0,36,28,0,0,0,0,0,0,28,36,0,0,0,0,0,0,0,0,36,28,0,0,0,0,0,0,28,36],[0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,8,38,36,9,0,0,72,0,38,8,37,64,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[61,72,0,0,0,0,0,0,72,12,0,0,0,0,0,0,0,0,0,1,44,56,29,56,0,0,1,0,44,56,56,29,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0],[66,6,0,0,0,0,0,0,5,6,0,0,0,0,0,0,0,0,14,43,34,2,25,30,0,0,33,31,5,45,40,61,0,0,72,56,34,20,4,29,0,0,56,72,50,67,12,31,0,0,0,0,36,28,0,0,0,0,0,0,28,36,0,0],[1,61,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,28,48,57,30,0,0,0,72,72,47,39,15,0,0,0,0,0,0,36,45,0,0,0,0,0,0,28,37,0,0,0,0,36,45,0,0,0,0,0,0,28,37,0,0] >;

C32.GL2(𝔽3) in GAP, Magma, Sage, TeX

C_3^2.{\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C3^2.GL(2,3)");
// GroupNames label

G:=SmallGroup(432,245);
// by ID

G=gap.SmallGroup(432,245);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,632,261,142,1011,3784,1908,172,2273,1153,285,124]);
// Polycyclic

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

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