C3^2:3GL(2,3) - GroupNames

G = C₃²⋊₃GL₂(𝔽₃) order 432 = 2⁴·3³

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

Aliases: C₃²⋊₃GL₂(𝔽₃), (C₃×C₆).₈S₄, Q₈⋊He₃⋊₂C₂, 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 C₃²⋊₃GL₂(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, Q₈⋊₂S₃, C₃×SD₁₆, GL₂(𝔽₃), He₃⋊C₂, C₂×He₃, C₃×C₃⋊C₈, C₃×SL₂(𝔽₃), C₃×D₁₂, Q₈×C₃², C₂×He₃⋊C₂, C₃×Q₈⋊₂S₃, C₃×GL₂(𝔽₃), Q₈⋊He₃, C₃²⋊₃GL₂(𝔽₃)
Quotients: C₁, C₂, S₃, C₃⋊S₃, S₄, GL₂(𝔽₃), He₃⋊C₂, C₃⋊S₄, C₆.₆S₄, C₃²⋊S₄, C₃²⋊₃GL₂(𝔽₃)

Character table of C₃²⋊₃GL₂(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 C₃²⋊₃GL₂(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 C₃²⋊₃GL₂(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] >;

C₃²⋊₃GL₂(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

Export

Character table of C₃²⋊₃GL₂(𝔽₃) in TeX

G = C32⋊3GL2(𝔽3) order 432 = 24·33

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

G = C₃²⋊₃GL₂(𝔽₃) order 432 = 2⁴·3³

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