G = C32:2GL2(F3) order 432 = 24·33
1st semidirect product of C32 and GL2(F3) acting via GL2(F3)/Q8=S3
Aliases: C32:2GL2(F3), C6.6S4:C3, C6.6(C3xS4), (C3xC6).4S4, Q8:He3:1C2, Q8:(C32:C6), (Q8xC32):3S3, (C3xSL2(F3)):C6, C2.3(C62:S3), C3.3(C3xGL2(F3)), (C3xQ8).6(C3xS3), SmallGroup(432,248)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C3xSL2(F3) — C32:2GL2(F3) |
| C3xSL2(F3) — C32:2GL2(F3) |
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, C2xC6, SD16, C3xS3, C3:S3, C3xC6, C3xC6, C3:C8, C24, SL2(F3), D12, C3xD4, C3xQ8, C3xQ8, He3, C3xC12, S3xC6, C2xC3:S3, Q8:2S3, C3xSD16, GL2(F3), C32:C6, C2xHe3, C3xC3:C8, C3xSL2(F3), C3xSL2(F3), C3xD12, Q8xC32, C2xC32:C6, C3xQ8:2S3, C6.6S4, Q8:He3, C32:2GL2(F3)
Quotients: C1, C2, C3, S3, C6, C3xS3, S4, GL2(F3), C32:C6, C3xS4, C3xGL2(F3), C62:S3, C32:2GL2(F3)
Character table of C32:2GL2(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 | 2 | 3 | 3 | 24 | 24 | 24 | 6 | 2 | 3 | 3 | 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 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ65 | ζ6 | -1 | -1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ5 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ6 | ζ65 | -1 | -1 | ζ32 | ζ3 | ζ3 | 1 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 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 |
| ρ8 | 2 | 2 | 0 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 2 | 2 | -1-√-3 | -1+√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | -1-√-3 | -1+√-3 | -1+√-3 | 2 | -1-√-3 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ9 | 2 | 2 | 0 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 2 | 2 | -1+√-3 | -1-√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | -1+√-3 | -1-√-3 | -1-√-3 | 2 | -1+√-3 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ10 | 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) |
| ρ11 | 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) |
| ρ12 | 2 | -2 | 0 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | -2 | 1-√-3 | 1+√-3 | ζ32 | 1 | ζ3 | 0 | 0 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | ζ83ζ3+ζ8ζ3 | ζ87ζ32+ζ85ζ32 | ζ87ζ3+ζ85ζ3 | ζ83ζ32+ζ8ζ32 | complex lifted from C3xGL2(F3) |
| ρ13 | 2 | -2 | 0 | 2 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | -1 | 0 | -2 | 1-√-3 | 1+√-3 | ζ32 | 1 | ζ3 | 0 | 0 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | ζ87ζ3+ζ85ζ3 | ζ83ζ32+ζ8ζ32 | ζ83ζ3+ζ8ζ3 | ζ87ζ32+ζ85ζ32 | complex lifted from C3xGL2(F3) |
| ρ14 | 2 | -2 | 0 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | -2 | 1+√-3 | 1-√-3 | ζ3 | 1 | ζ32 | 0 | 0 | √-2 | -√-2 | 0 | 0 | 0 | 0 | 0 | ζ83ζ32+ζ8ζ32 | ζ87ζ3+ζ85ζ3 | ζ87ζ32+ζ85ζ32 | ζ83ζ3+ζ8ζ3 | complex lifted from C3xGL2(F3) |
| ρ15 | 2 | -2 | 0 | 2 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | -1 | 0 | -2 | 1+√-3 | 1-√-3 | ζ3 | 1 | ζ32 | 0 | 0 | -√-2 | √-2 | 0 | 0 | 0 | 0 | 0 | ζ87ζ32+ζ85ζ32 | ζ83ζ3+ζ8ζ3 | ζ83ζ32+ζ8ζ32 | ζ87ζ3+ζ85ζ3 | complex lifted from C3xGL2(F3) |
| ρ16 | 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 |
| ρ17 | 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 |
| ρ18 | 3 | 3 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | ζ65 | ζ6 | 1 | 1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ3 | ζ32 | ζ3 | ζ32 | complex lifted from C3xS4 |
| ρ19 | 3 | 3 | 1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | ζ32 | ζ3 | -1 | -1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | complex lifted from C3xS4 |
| ρ20 | 3 | 3 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | ζ6 | ζ65 | 1 | 1 | ζ6 | ζ65 | ζ65 | -1 | ζ6 | ζ32 | ζ3 | ζ32 | ζ3 | complex lifted from C3xS4 |
| ρ21 | 3 | 3 | 1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | ζ3 | ζ32 | -1 | -1 | ζ65 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | complex lifted from C3xS4 |
| ρ22 | 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) |
| ρ23 | 4 | -4 | 0 | 4 | -2-2√-3 | -2+2√-3 | ζ32 | ζ3 | 1 | 0 | -4 | 2+2√-3 | 2-2√-3 | ζ65 | -1 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3xGL2(F3) |
| ρ24 | 4 | -4 | 0 | 4 | -2+2√-3 | -2-2√-3 | ζ3 | ζ32 | 1 | 0 | -4 | 2-2√-3 | 2+2√-3 | ζ6 | -1 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C3xGL2(F3) |
| ρ25 | 6 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32:C6 |
| ρ26 | 6 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -2 | 1 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from C62:S3 |
| ρ27 | 6 | 6 | 0 | -3 | 0 | 0 | 0 | 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 | complex lifted from C62:S3 |
| ρ28 | 6 | 6 | 0 | -3 | 0 | 0 | 0 | 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 | complex lifted from C62:S3 |
| ρ29 | 12 | -12 | 0 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►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)
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 69 | 61 | 0 | 0 | 65 | 8 |
| 0 | 0 | 61 | 69 | 0 | 57 | 65 | 65 |
| 0 | 0 | 12 | 12 | 8 | 16 | 8 | 8 |
| 0 | 0 | 0 | 6 | 67 | 69 | 0 | 61 |
| 0 | 0 | 12 | 6 | 6 | 12 | 8 | 12 |
| 0 | 0 | 61 | 67 | 67 | 61 | 0 | 69 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 35 | 0 | 0 | 71 | 0 | 0 |
| 0 | 0 | 0 | 35 | 0 | 0 | 71 | 0 |
| 0 | 0 | 0 | 0 | 35 | 0 | 0 | 71 |
| 45 | 44 | 0 | 0 | 0 | 0 | 0 | 0 |
| 17 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 72 | 72 | 72 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 45 | 56 | 0 | 0 | 0 | 0 | 0 | 0 |
| 29 | 28 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 72 | 72 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 72 | 72 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 0 | 0 | 72 |
| 0 | 0 | 0 | 35 | 0 | 0 | 72 | 0 |
G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,69,61,12,0,12,61,0,0,61,69,12,6,6,67,0,0,0,0,8,67,6,67,0,0,0,57,16,69,12,61,0,0,65,65,8,0,8,0,0,0,8,65,8,61,12,69],[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
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