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## G = C32⋊3GL2(𝔽3)  order 432 = 24·33

### 2nd semidirect product of C32 and GL2(𝔽3) acting via GL2(𝔽3)/Q8=S3

Aliases: C323GL2(𝔽3), (C3×C6).8S4, Q8⋊He32C2, C6.18(C3⋊S4), Q8⋊(He3⋊C2), (Q8×C32)⋊4S3, C3.3(C6.6S4), (C3×SL2(𝔽3))⋊3S3, C2.3(C32⋊S4), (C3×Q8).7(C3⋊S3), SmallGroup(432,258)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — Q8⋊He3 — C32⋊3GL2(𝔽3)
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C32 — Q8⋊He3 — C32⋊3GL2(𝔽3)
 Lower central Q8⋊He3 — C32⋊3GL2(𝔽3)
 Upper central C1 — C6

Generators and relations for C323GL2(𝔽3)
G = < a,b,c,d,e,f | a3=b3=c4=e3=f2=1, d2=c2, ab=ba, ac=ca, ad=da, eae-1=ab-1, faf=a-1, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fdf=c-1, ece-1=cd, fcf=c2d, ede-1=c, fef=e-1 >

Subgroups: 616 in 86 conjugacy classes, 14 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, C12, D6, C2×C6, SD16, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, SL2(𝔽3), D12, C3×D4, C3×Q8, C3×Q8, He3, C3×C12, S3×C6, Q82S3, C3×SD16, GL2(𝔽3), He3⋊C2, C2×He3, C3×C3⋊C8, C3×SL2(𝔽3), C3×D12, Q8×C32, C2×He3⋊C2, C3×Q82S3, C3×GL2(𝔽3), Q8⋊He3, C323GL2(𝔽3)
Quotients: C1, C2, S3, C3⋊S3, S4, GL2(𝔽3), He3⋊C2, C3⋊S4, C6.6S4, C32⋊S4, C323GL2(𝔽3)

Character table of C323GL2(𝔽3)

 class 1 2A 2B 3A 3B 3C 3D 3E 3F 4 6A 6B 6C 6D 6E 6F 6G 6H 8A 8B 12A 12B 12C 12D 12E 24A 24B 24C 24D size 1 1 36 1 1 6 24 24 24 6 1 1 6 24 24 24 36 36 18 18 6 6 12 12 12 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 2 2 0 2 2 -1 -1 -1 2 2 2 2 -1 -1 -1 2 0 0 0 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ4 2 2 0 2 2 -1 2 -1 -1 2 2 2 -1 -1 2 -1 0 0 0 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ5 2 2 0 2 2 2 -1 -1 -1 2 2 2 2 -1 -1 -1 0 0 0 0 2 2 2 2 2 0 0 0 0 orthogonal lifted from S3 ρ6 2 2 0 2 2 -1 -1 2 -1 2 2 2 -1 2 -1 -1 0 0 0 0 2 2 -1 -1 -1 0 0 0 0 orthogonal lifted from S3 ρ7 2 -2 0 2 2 2 -1 -1 -1 0 -2 -2 -2 1 1 1 0 0 √-2 -√-2 0 0 0 0 0 -√-2 √-2 √-2 -√-2 complex lifted from GL2(𝔽3) ρ8 2 -2 0 2 2 2 -1 -1 -1 0 -2 -2 -2 1 1 1 0 0 -√-2 √-2 0 0 0 0 0 √-2 -√-2 -√-2 √-2 complex lifted from GL2(𝔽3) ρ9 3 3 -1 3 3 3 0 0 0 -1 3 3 3 0 0 0 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from S4 ρ10 3 3 1 3 3 3 0 0 0 -1 3 3 3 0 0 0 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S4 ρ11 3 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ65 ζ6 1 1 ζ65 ζ6 -1-√-3 -1+√-3 2 ζ32 ζ32 ζ3 ζ3 complex lifted from C32⋊S4 ρ12 3 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ32 ζ3 -1 -1 ζ6 ζ65 -1+√-3 -1-√-3 2 ζ65 ζ65 ζ6 ζ6 complex lifted from C32⋊S4 ρ13 3 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ3 ζ32 -1 -1 ζ65 ζ6 -1-√-3 -1+√-3 2 ζ6 ζ6 ζ65 ζ65 complex lifted from C32⋊S4 ρ14 3 3 1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ32 ζ3 1 1 -3-3√-3/2 -3+3√-3/2 0 0 0 ζ3 ζ3 ζ32 ζ32 complex lifted from He3⋊C2 ρ15 3 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ6 ζ65 1 1 ζ6 ζ65 -1+√-3 -1-√-3 2 ζ3 ζ3 ζ32 ζ32 complex lifted from C32⋊S4 ρ16 3 3 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ65 ζ6 -1 -1 -3+3√-3/2 -3-3√-3/2 0 0 0 ζ6 ζ6 ζ65 ζ65 complex lifted from He3⋊C2 ρ17 3 3 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 0 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 ζ6 ζ65 -1 -1 -3-3√-3/2 -3+3√-3/2 0 0 0 ζ65 ζ65 ζ6 ζ6 complex lifted from He3⋊C2 ρ18 3 3 1 -3+3√-3/2 -3-3√-3/2 0 0 0 0 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 ζ3 ζ32 1 1 -3+3√-3/2 -3-3√-3/2 0 0 0 ζ32 ζ32 ζ3 ζ3 complex lifted from He3⋊C2 ρ19 4 -4 0 4 4 -2 1 -2 1 0 -4 -4 2 2 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C6.6S4 ρ20 4 -4 0 4 4 4 1 1 1 0 -4 -4 -4 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from GL2(𝔽3) ρ21 4 -4 0 4 4 -2 1 1 -2 0 -4 -4 2 -1 -1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C6.6S4 ρ22 4 -4 0 4 4 -2 -2 1 1 0 -4 -4 2 -1 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C6.6S4 ρ23 6 6 0 6 6 -3 0 0 0 -2 6 6 -3 0 0 0 0 0 0 0 -2 -2 1 1 1 0 0 0 0 orthogonal lifted from C3⋊S4 ρ24 6 6 0 -3+3√-3 -3-3√-3 0 0 0 0 -2 -3-3√-3 -3+3√-3 0 0 0 0 0 0 0 0 1-√-3 1+√-3 1+√-3 1-√-3 -2 0 0 0 0 complex lifted from C32⋊S4 ρ25 6 6 0 -3-3√-3 -3+3√-3 0 0 0 0 -2 -3+3√-3 -3-3√-3 0 0 0 0 0 0 0 0 1+√-3 1-√-3 1-√-3 1+√-3 -2 0 0 0 0 complex lifted from C32⋊S4 ρ26 6 -6 0 -3-3√-3 -3+3√-3 0 0 0 0 0 3-3√-3 3+3√-3 0 0 0 0 0 0 √-2 -√-2 0 0 0 0 0 ζ87ζ3+ζ85ζ3 ζ83ζ3+ζ8ζ3 ζ83ζ32+ζ8ζ32 ζ87ζ32+ζ85ζ32 complex faithful ρ27 6 -6 0 -3+3√-3 -3-3√-3 0 0 0 0 0 3+3√-3 3-3√-3 0 0 0 0 0 0 √-2 -√-2 0 0 0 0 0 ζ87ζ32+ζ85ζ32 ζ83ζ32+ζ8ζ32 ζ83ζ3+ζ8ζ3 ζ87ζ3+ζ85ζ3 complex faithful ρ28 6 -6 0 -3-3√-3 -3+3√-3 0 0 0 0 0 3-3√-3 3+3√-3 0 0 0 0 0 0 -√-2 √-2 0 0 0 0 0 ζ83ζ3+ζ8ζ3 ζ87ζ3+ζ85ζ3 ζ87ζ32+ζ85ζ32 ζ83ζ32+ζ8ζ32 complex faithful ρ29 6 -6 0 -3+3√-3 -3-3√-3 0 0 0 0 0 3+3√-3 3-3√-3 0 0 0 0 0 0 -√-2 √-2 0 0 0 0 0 ζ83ζ32+ζ8ζ32 ζ87ζ32+ζ85ζ32 ζ87ζ3+ζ85ζ3 ζ83ζ3+ζ8ζ3 complex faithful

Smallest permutation representation of C323GL2(𝔽3)
On 72 points
Generators in S72
(1 56 32)(2 53 29)(3 54 30)(4 55 31)(5 51 27)(6 52 28)(7 49 25)(8 50 26)(9 57 33)(10 58 34)(11 59 35)(12 60 36)(13 61 37)(14 62 38)(15 63 39)(16 64 40)(17 65 41)(18 66 42)(19 67 43)(20 68 44)(21 69 45)(22 70 46)(23 71 47)(24 72 48)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 67 59)(6 68 60)(7 65 57)(8 66 58)(9 25 17)(10 26 18)(11 27 19)(12 28 20)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(33 49 41)(34 50 42)(35 51 43)(36 52 44)(53 69 61)(54 70 62)(55 71 63)(56 72 64)
(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 11 3 9)(2 10 4 12)(5 70 7 72)(6 69 8 71)(13 18 15 20)(14 17 16 19)(21 26 23 28)(22 25 24 27)(29 34 31 36)(30 33 32 35)(37 42 39 44)(38 41 40 43)(45 50 47 52)(46 49 48 51)(53 58 55 60)(54 57 56 59)(61 66 63 68)(62 65 64 67)
(2 11 10)(4 9 12)(5 66 53)(6 63 57)(7 68 55)(8 61 59)(13 19 18)(15 17 20)(21 27 26)(23 25 28)(29 43 50)(30 38 46)(31 41 52)(32 40 48)(33 44 47)(34 37 51)(35 42 45)(36 39 49)(54 70 62)(56 72 64)(58 69 67)(60 71 65)
(1 3)(2 11)(4 9)(5 45)(6 52)(7 47)(8 50)(13 19)(14 16)(15 17)(21 27)(22 24)(23 25)(29 59)(30 56)(31 57)(32 54)(33 55)(34 58)(35 53)(36 60)(37 67)(38 64)(39 65)(40 62)(41 63)(42 66)(43 61)(44 68)(46 72)(48 70)(49 71)(51 69)

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

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

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

Matrix representation of C323GL2(𝔽3) in GL5(𝔽73)

 1 0 0 0 0 0 1 0 0 0 0 0 69 5 68 0 0 33 41 40 0 0 28 45 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 8
,
 57 44 0 0 0 29 16 0 0 0 0 0 0 72 1 0 0 0 72 0 0 0 1 72 0
,
 45 56 0 0 0 29 28 0 0 0 0 0 72 0 0 0 0 72 0 1 0 0 72 1 0
,
 28 28 0 0 0 57 44 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0
,
 72 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,69,33,28,0,0,5,41,45,0,0,68,40,36],[1,0,0,0,0,0,1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,8],[57,29,0,0,0,44,16,0,0,0,0,0,0,0,1,0,0,72,72,72,0,0,1,0,0],[45,29,0,0,0,56,28,0,0,0,0,0,72,72,72,0,0,0,0,1,0,0,0,1,0],[28,57,0,0,0,28,44,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

C323GL2(𝔽3) in GAP, Magma, Sage, TeX

C_3^2\rtimes_3{\rm GL}_2({\mathbb F}_3)
% in TeX

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

G:=SmallGroup(432,258);
// by ID

G=gap.SmallGroup(432,258);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,57,254,261,3784,5681,172,2273,3414,285,124]);
// Polycyclic

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

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