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G = C32⋊2GL2(𝔽3)  order 432 = 24·33

1st semidirect product of C32 and GL2(𝔽3) acting via GL2(𝔽3)/Q8=S3

non-abelian, soluble

Aliases: C32⋊2GL2(𝔽3), C6.6S4⋊C3, C6.6(C3×S4), (C3×C6).4S4, Q8⋊He3⋊1C2, Q8⋊(C32⋊C6), (Q8×C32)⋊3S3, (C3×SL2(𝔽3))⋊C6, C2.3(C62⋊S3), C3.3(C3×GL2(𝔽3)), (C3×Q8).6(C3×S3), SmallGroup(432,248)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — Q8 — C3×SL2(𝔽3) — C32⋊2GL2(𝔽3)
Chief series
C1 — C2 — Q8 — C3×Q8 — C3×SL2(𝔽3) — Q8⋊He3 — C32⋊2GL2(𝔽3)
Lower central
C3×SL2(𝔽3) — C32⋊2GL2(𝔽3)
Upper central
C1 — C2

Generators and relations for C32⋊2GL2(𝔽3)
 G = < a,b,c,d,e,f | a3=b3=c4=e3=f2=1, d2=c2, eae-1=ab=ba, ac=ca, ad=da, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, dcd-1=fdf=c-1, ece-1=cd, fcf=c2d, ede-1=c, fef=e-1 >

Subgroups: 512 in 64 conjugacy classes, 13 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, SL2(𝔽3), D12, C3×D4, C3×Q8, C3×Q8, He3, C3×C12, S3×C6, C2×C3⋊S3, Q8⋊2S3, C3×SD16, GL2(𝔽3), C32⋊C6, C2×He3, C3×C3⋊C8, C3×SL2(𝔽3), C3×SL2(𝔽3), C3×D12, Q8×C32, C2×C32⋊C6, C3×Q8⋊2S3, C6.6S4, Q8⋊He3, C32⋊2GL2(𝔽3)
Quotients: C1, C2, C3, S3, C6, C3×S3, S4, GL2(𝔽3), C32⋊C6, C3×S4, C3×GL2(𝔽3), C62⋊S3, C32⋊2GL2(𝔽3)

Character table of C32⋊2GL2(𝔽3)

 class 12A2B3A3B3C3D3E3F46A6B6C6D6E6F6G6H8A8B12A12B12C12D12E24A24B24C24D
 size 11362332424246233242424363618186612121218181818
ρ111111111111111111111111111111    trivial
ρ211-11111111111111-1-1-1-111111-1-1-1-1    linear of order 2
ρ311-11ζ3ζ32ζ3ζ32111ζ3ζ32ζ321ζ3ζ65ζ6-1-1ζ3ζ32ζ321ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ41111ζ32ζ3ζ32ζ3111ζ32ζ3ζ31ζ32ζ32ζ311ζ32ζ3ζ31ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ511-11ζ32ζ3ζ32ζ3111ζ32ζ3ζ31ζ32ζ6ζ65-1-1ζ32ζ3ζ31ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ61111ζ3ζ32ζ3ζ32111ζ3ζ32ζ321ζ3ζ3ζ3211ζ3ζ32ζ321ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ7220222-1-1-12222-1-1-10000222220000    orthogonal lifted from S3
ρ82202-1-√-3-1+√-3ζ6ζ65-122-1-√-3-1+√-3ζ65-1ζ60000-1-√-3-1+√-3-1+√-32-1-√-30000    complex lifted from C3×S3
ρ92202-1+√-3-1-√-3ζ65ζ6-122-1+√-3-1-√-3ζ6-1ζ650000-1+√-3-1-√-3-1-√-32-1+√-30000    complex lifted from C3×S3
ρ102-20222-1-1-10-2-2-211100√-2-√-200000√-2-√-2-√-2√-2    complex lifted from GL2(𝔽3)
ρ112-20222-1-1-10-2-2-211100-√-2√-200000-√-2√-2√-2-√-2    complex lifted from GL2(𝔽3)
ρ122-202-1+√-3-1-√-3ζ65ζ6-10-21-√-31+√-3ζ321ζ300√-2-√-200000ζ83ζ3+ζ8ζ3ζ87ζ32+ζ85ζ32ζ87ζ3+ζ85ζ3ζ83ζ32+ζ8ζ32    complex lifted from C3×GL2(𝔽3)
ρ132-202-1+√-3-1-√-3ζ65ζ6-10-21-√-31+√-3ζ321ζ300-√-2√-200000ζ87ζ3+ζ85ζ3ζ83ζ32+ζ8ζ32ζ83ζ3+ζ8ζ3ζ87ζ32+ζ85ζ32    complex lifted from C3×GL2(𝔽3)
ρ142-202-1-√-3-1+√-3ζ6ζ65-10-21+√-31-√-3ζ31ζ3200√-2-√-200000ζ83ζ32+ζ8ζ32ζ87ζ3+ζ85ζ3ζ87ζ32+ζ85ζ32ζ83ζ3+ζ8ζ3    complex lifted from C3×GL2(𝔽3)
ρ152-202-1-√-3-1+√-3ζ6ζ65-10-21+√-31-√-3ζ31ζ3200-√-2√-200000ζ87ζ32+ζ85ζ32ζ83ζ3+ζ8ζ3ζ83ζ32+ζ8ζ32ζ87ζ3+ζ85ζ3    complex lifted from C3×GL2(𝔽3)
ρ1633-1333000-1333000-1-111-1-1-1-1-11111    orthogonal lifted from S4
ρ17331333000-133300011-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S4
ρ1833-13-3+3√-3/2-3-3√-3/2000-13-3+3√-3/2-3-3√-3/2000ζ65ζ611ζ65ζ6ζ6-1ζ65ζ3ζ32ζ3ζ32    complex lifted from C3×S4
ρ193313-3-3√-3/2-3+3√-3/2000-13-3-3√-3/2-3+3√-3/2000ζ32ζ3-1-1ζ6ζ65ζ65-1ζ6ζ6ζ65ζ6ζ65    complex lifted from C3×S4
ρ2033-13-3-3√-3/2-3+3√-3/2000-13-3-3√-3/2-3+3√-3/2000ζ6ζ6511ζ6ζ65ζ65-1ζ6ζ32ζ3ζ32ζ3    complex lifted from C3×S4
ρ213313-3+3√-3/2-3-3√-3/2000-13-3+3√-3/2-3-3√-3/2000ζ3ζ32-1-1ζ65ζ6ζ6-1ζ65ζ65ζ6ζ65ζ6    complex lifted from C3×S4
ρ224-404441110-4-4-4-1-1-10000000000000    orthogonal lifted from GL2(𝔽3)
ρ234-404-2-2√-3-2+2√-3ζ32ζ310-42+2√-32-2√-3ζ65-1ζ60000000000000    complex lifted from C3×GL2(𝔽3)
ρ244-404-2+2√-3-2-2√-3ζ3ζ3210-42-2√-32+2√-3ζ6-1ζ650000000000000    complex lifted from C3×GL2(𝔽3)
ρ25660-3000006-3000000000000-300000    orthogonal lifted from C32⋊C6
ρ26660-300000-2-300000000044-21-20000    orthogonal lifted from C62⋊S3
ρ27660-300000-2-3000000000-2-2√-3-2+2√-31-√-311+√-30000    complex lifted from C62⋊S3
ρ28660-300000-2-3000000000-2+2√-3-2-2√-31+√-311-√-30000    complex lifted from C62⋊S3
ρ2912-120-60000006000000000000000000    orthogonal faithful

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

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

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

Matrix representation of C32⋊2GL2(𝔽3) ►in GL8(𝔽73)

640000000
064000000
00696100658
0061690576565
00121281688
00066769061
00126612812
0061676761069
,
10000000
01000000
00100200
00010020
00001002
0035007100
0003500710
0000350071
,
4544000000
1728000000
00001000
00727272000
00100000
00000001
00000727272
00000100
,
4556000000
2928000000
00010000
00100000
00727272000
00000010
00000100
00000727272
,
072000000
172000000
00100000
00001000
00727272000
00000100
00000001
00000727272
,
01000000
10000000
00100000
00001000
00010000
0035007200
0000350072
0003500720

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,69,61,12,0,12,61,0,0,61,69,12,6,6,67,0,0,0,0,8,67,6,67,0,0,0,57,16,69,12,61,0,0,65,65,8,0,8,0,0,0,8,65,8,61,12,69],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,35,0,0,0,0,0,1,0,0,35,0,0,0,0,0,1,0,0,35,0,0,2,0,0,71,0,0,0,0,0,2,0,0,71,0,0,0,0,0,2,0,0,71],[45,17,0,0,0,0,0,0,44,28,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0],[45,29,0,0,0,0,0,0,56,28,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,72],[0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,35,0,0,0,0,0,0,1,0,0,35,0,0,0,1,0,0,35,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0] >;
 

C32⋊2GL2(𝔽3) in GAP, Magma, Sage, TeX 

C_3^2\rtimes_2{\rm GL}_2({\mathbb F}_3)
 
% in TeX
 
G:=Group("C3^2:2GL(2,3)");
 
// GroupNames label
 
G:=SmallGroup(432,248);
 
// by ID
 
G=gap.SmallGroup(432,248);
 
# by ID
 
G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,254,261,1011,3784,1908,172,2273,1153,285,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^3=b^3=c^4=e^3=f^2=1,d^2=c^2,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,f*b*f=b^-1,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
 

Export

Character table of C32⋊2GL2(𝔽3) in TeX

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