ES-(3,1):D9 - GroupNames

G = 3_-¹⁺²⋊D₉ order 486 = 2·3⁵

The semidirect product of 3_-¹⁺² and D₉ acting via D₉/C₃=S₃

Aliases: 3_-¹⁺²⋊D₉, 3_-¹⁺²⋊C₉⋊C₂, C₃²⋊C₉.₁₃S₃, C₃².₂(C₉⋊S₃), (C₃²×C₉).₁₃S₃, C₃³.₂₄(C₃⋊S₃), C₃.₈(C₃³⋊S₃), C₃.₈(C₃²⋊₂D₉), C₃.₈(He₃.₃S₃), (C₃×3_-¹⁺²).₃S₃, C₃².₂₄(He₃⋊C₂), C₃.₅(3_-¹⁺².S₃), SmallGroup(486,57)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — 3_-¹⁺²⋊C₉ — 3_-¹⁺²⋊D₉

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₉ — 3_-¹⁺²⋊C₉ — 3_-¹⁺²⋊D₉

Lower central

3_-¹⁺²⋊C₉ — 3_-¹⁺²⋊D₉

Upper central

C₁

Generators and relations for 3_-¹⁺²⋊D₉
G = < a,b,c,d | a⁹=b³=c⁹=d²=1, bab^-1=a⁴, cac^-1=a⁴b^-1, dad=a^-1, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 682 in 73 conjugacy classes, 17 normal (14 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₃×C₉, 3_-¹⁺², 3_-¹⁺², C₃³, C₃×D₉, C₉⋊C₆, C₉⋊S₃, C₃×C₃⋊S₃, C₃²⋊C₉, C₃²×C₉, C₃×3_-¹⁺², C₃²⋊D₉, C₃×C₉⋊S₃, C₃³.S₃, 3_-¹⁺²⋊C₉, 3_-¹⁺²⋊D₉
Quotients: C₁, C₂, S₃, D₉, C₃⋊S₃, C₉⋊S₃, He₃⋊C₂, C₃²⋊₂D₉, C₃³⋊S₃, He₃.₃S₃, 3_-¹⁺².S₃, 3_-¹⁺²⋊D₉

Character table of 3_-¹⁺²⋊D₉

class	1	2	3A	3B	3C	3D	3E	3F	3G	3H	6A	6B	9A	9B	9C	9D	9E	9F	9G	9H	9I	9J	9K	9L	9M	9N	9O	9P	9Q	9R
size	1	81	2	2	2	2	3	3	6	6	81	81	6	6	6	6	6	6	6	6	6	18	18	18	18	18	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	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	1	linear of order 2
ρ₃	2	0	2	2	2	2	2	2	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	2	2	2	orthogonal lifted from S₃
ρ₄	2	0	2	2	2	2	2	2	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	2	2	2	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₅	2	0	2	2	2	2	2	2	2	2	0	0	2	2	2	2	2	2	2	2	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₆	2	0	2	2	2	2	2	2	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	2	2	2	-1	-1	-1	orthogonal lifted from S₃
ρ₇	2	0	-1	-1	-1	2	2	2	-1	-1	0	0	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	2	-1	-1	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₈	2	0	-1	-1	-1	2	2	2	-1	-1	0	0	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	2	-1	-1	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₉	2	0	-1	-1	-1	2	2	2	-1	-1	0	0	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	-1	2	-1	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₀	2	0	-1	-1	-1	2	2	2	-1	-1	0	0	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	2	-1	-1	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₁	2	0	-1	-1	-1	2	2	2	-1	-1	0	0	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	-1	-1	2	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₂	2	0	-1	-1	-1	2	2	2	-1	-1	0	0	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	-1	-1	2	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₃	2	0	-1	-1	-1	2	2	2	-1	-1	0	0	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	-1	2	-1	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₄	2	0	-1	-1	-1	2	2	2	-1	-1	0	0	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	-1	2	-1	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₅	2	0	-1	-1	-1	2	2	2	-1	-1	0	0	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	-1	-1	2	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₆	3	1	3	3	3	3	^-3+3√-3/₂	^-3-3√-3/₂	^-3+3√-3/₂	^-3-3√-3/₂	ζ₃	ζ₃²	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₁₇	3	-1	3	3	3	3	^-3+3√-3/₂	^-3-3√-3/₂	^-3+3√-3/₂	^-3-3√-3/₂	ζ₆⁵	ζ₆	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₁₈	3	-1	3	3	3	3	^-3-3√-3/₂	^-3+3√-3/₂	^-3-3√-3/₂	^-3+3√-3/₂	ζ₆	ζ₆⁵	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₁₉	3	1	3	3	3	3	^-3-3√-3/₂	^-3+3√-3/₂	^-3-3√-3/₂	^-3+3√-3/₂	ζ₃²	ζ₃	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₂₀	6	0	6	-3	-3	-3	0	0	0	0	0	0	0	3	-3	0	3	-3	0	3	-3	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃³⋊S₃
ρ₂₁	6	0	6	-3	-3	-3	0	0	0	0	0	0	-3	0	3	-3	0	3	-3	0	3	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃³⋊S₃
ρ₂₂	6	0	6	-3	-3	-3	0	0	0	0	0	0	3	-3	0	3	-3	0	3	-3	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₃³⋊S₃
ρ₂₃	6	0	-3	6	-3	-3	0	0	0	0	0	0	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	0	0	0	0	0	0	0	0	0	orthogonal lifted from He₃.₃S₃
ρ₂₄	6	0	-3	-3	6	-3	0	0	0	0	0	0	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	0	0	0	0	0	0	0	0	0	orthogonal lifted from 3_-¹⁺².S₃
ρ₂₅	6	0	-3	-3	6	-3	0	0	0	0	0	0	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	0	0	0	0	0	0	0	0	0	orthogonal lifted from 3_-¹⁺².S₃
ρ₂₆	6	0	-3	6	-3	-3	0	0	0	0	0	0	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	0	0	0	0	0	0	0	0	0	orthogonal lifted from He₃.₃S₃
ρ₂₇	6	0	-3	-3	6	-3	0	0	0	0	0	0	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	0	0	0	0	0	0	0	0	0	orthogonal lifted from 3_-¹⁺².S₃
ρ₂₈	6	0	-3	6	-3	-3	0	0	0	0	0	0	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	0	0	0	0	0	0	0	0	0	orthogonal lifted from He₃.₃S₃
ρ₂₉	6	0	-3	-3	-3	6	-3-3√-3	-3+3√-3	^3+3√-3/₂	^3-3√-3/₂	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₃²⋊₂D₉
ρ₃₀	6	0	-3	-3	-3	6	-3+3√-3	-3-3√-3	^3-3√-3/₂	^3+3√-3/₂	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₃²⋊₂D₉

Smallest permutation representation of 3_-¹⁺²⋊D₉
►On 81 points

Generators in S₈₁

(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)(73 74 75 76 77 78 79 80 81)

(2 8 5)(3 6 9)(10 13 16)(12 18 15)(20 26 23)(21 24 27)(28 34 31)(29 32 35)(37 40 43)(39 45 42)(46 52 49)(47 50 53)(56 62 59)(57 60 63)(65 71 68)(66 69 72)(73 76 79)(75 81 78)

(1 41 64 54 77 61 17 33 25)(2 45 71 46 81 59 18 28 23)(3 37 72 47 73 60 10 29 24)(4 44 67 48 80 55 11 36 19)(5 39 65 49 75 62 12 31 26)(6 40 66 50 76 63 13 32 27)(7 38 70 51 74 58 14 30 22)(8 42 68 52 78 56 15 34 20)(9 43 69 53 79 57 16 35 21)

(1 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 27)(9 26)(10 71)(11 70)(12 69)(13 68)(14 67)(15 66)(16 65)(17 64)(18 72)(28 37)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 63)(53 62)(54 61)(73 81)(74 80)(75 79)(76 78)

G:=sub<Sym(81)| (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)(73,74,75,76,77,78,79,80,81), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(20,26,23)(21,24,27)(28,34,31)(29,32,35)(37,40,43)(39,45,42)(46,52,49)(47,50,53)(56,62,59)(57,60,63)(65,71,68)(66,69,72)(73,76,79)(75,81,78), (1,41,64,54,77,61,17,33,25)(2,45,71,46,81,59,18,28,23)(3,37,72,47,73,60,10,29,24)(4,44,67,48,80,55,11,36,19)(5,39,65,49,75,62,12,31,26)(6,40,66,50,76,63,13,32,27)(7,38,70,51,74,58,14,30,22)(8,42,68,52,78,56,15,34,20)(9,43,69,53,79,57,16,35,21), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,27)(9,26)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,72)(28,37)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,63)(53,62)(54,61)(73,81)(74,80)(75,79)(76,78)>;

G:=Group( (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)(73,74,75,76,77,78,79,80,81), (2,8,5)(3,6,9)(10,13,16)(12,18,15)(20,26,23)(21,24,27)(28,34,31)(29,32,35)(37,40,43)(39,45,42)(46,52,49)(47,50,53)(56,62,59)(57,60,63)(65,71,68)(66,69,72)(73,76,79)(75,81,78), (1,41,64,54,77,61,17,33,25)(2,45,71,46,81,59,18,28,23)(3,37,72,47,73,60,10,29,24)(4,44,67,48,80,55,11,36,19)(5,39,65,49,75,62,12,31,26)(6,40,66,50,76,63,13,32,27)(7,38,70,51,74,58,14,30,22)(8,42,68,52,78,56,15,34,20)(9,43,69,53,79,57,16,35,21), (1,25)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,27)(9,26)(10,71)(11,70)(12,69)(13,68)(14,67)(15,66)(16,65)(17,64)(18,72)(28,37)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(46,60)(47,59)(48,58)(49,57)(50,56)(51,55)(52,63)(53,62)(54,61)(73,81)(74,80)(75,79)(76,78) );

G=PermutationGroup([[(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),(73,74,75,76,77,78,79,80,81)], [(2,8,5),(3,6,9),(10,13,16),(12,18,15),(20,26,23),(21,24,27),(28,34,31),(29,32,35),(37,40,43),(39,45,42),(46,52,49),(47,50,53),(56,62,59),(57,60,63),(65,71,68),(66,69,72),(73,76,79),(75,81,78)], [(1,41,64,54,77,61,17,33,25),(2,45,71,46,81,59,18,28,23),(3,37,72,47,73,60,10,29,24),(4,44,67,48,80,55,11,36,19),(5,39,65,49,75,62,12,31,26),(6,40,66,50,76,63,13,32,27),(7,38,70,51,74,58,14,30,22),(8,42,68,52,78,56,15,34,20),(9,43,69,53,79,57,16,35,21)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,27),(9,26),(10,71),(11,70),(12,69),(13,68),(14,67),(15,66),(16,65),(17,64),(18,72),(28,37),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(46,60),(47,59),(48,58),(49,57),(50,56),(51,55),(52,63),(53,62),(54,61),(73,81),(74,80),(75,79),(76,78)]])

Matrix representation of 3_-¹⁺²⋊D₉ ►in GL₈(𝔽₁₉)

0	1	0	0	0	0	0	0
18	18	0	0	0	0	0	0
0	0	17	8	7	10	0	0
0	0	4	3	16	7	0	0
0	0	10	2	0	18	12	17
0	0	9	16	0	18	2	14
0	0	18	8	5	17	0	0
0	0	6	3	5	17	0	0

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	16	12	18	18	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	0	0	1
0	0	10	12	0	0	18	18

2	14	0	0	0	0	0	0
5	7	0	0	0	0	0	0
0	0	2	5	0	0	0	0
0	0	14	7	0	0	0	0
0	0	7	2	17	12	0	0
0	0	9	13	7	5	0	0
0	0	4	11	0	0	17	12
0	0	10	3	0	0	7	5

1	0	0	0	0	0	0	0
18	18	0	0	0	0	0	0
0	0	14	7	0	0	0	0
0	0	2	5	0	0	0	0
0	0	4	6	0	0	14	2
0	0	11	3	0	0	7	5
0	0	1	11	14	2	0	0
0	0	16	8	7	5	0	0

G:=sub<GL(8,GF(19))| [0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,17,4,10,9,18,6,0,0,8,3,2,16,8,3,0,0,7,16,0,0,5,5,0,0,10,7,18,18,17,17,0,0,0,0,12,2,0,0,0,0,0,0,17,14,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16,0,0,10,0,0,0,1,12,0,0,12,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[2,5,0,0,0,0,0,0,14,7,0,0,0,0,0,0,0,0,2,14,7,9,4,10,0,0,5,7,2,13,11,3,0,0,0,0,17,7,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,12,5],[1,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,14,2,4,11,1,16,0,0,7,5,6,3,11,8,0,0,0,0,0,0,14,7,0,0,0,0,0,0,2,5,0,0,0,0,14,7,0,0,0,0,0,0,2,5,0,0] >;

3_-¹⁺²⋊D₉ in GAP, Magma, Sage, TeX

3_-^{1+2}\rtimes D_9

% in TeX

G:=Group("ES-(3,1):D9");

// GroupNames label

G:=SmallGroup(486,57);

// by ID

G=gap.SmallGroup(486,57);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,697,655,1190,224,338,8211,2169,3244,11669]);

// Polycyclic

G:=Group<a,b,c,d|a^9=b^3=c^9=d^2=1,b*a*b^-1=a^4,c*a*c^-1=a^4*b^-1,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

Export

Character table of 3_-¹⁺²⋊D₉ in TeX

0	1	0	0	0	0	0	0
18	18	0	0	0	0	0	0
0	0	17	8	7	10	0	0
0	0	4	3	16	7	0	0
0	0	10	2	0	18	12	17
0	0	9	16	0	18	2	14
0	0	18	8	5	17	0	0
0	0	6	3	5	17	0	0

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	16	12	18	18	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	0	0	1
0	0	10	12	0	0	18	18

2	14	0	0	0	0	0	0
5	7	0	0	0	0	0	0
0	0	2	5	0	0	0	0
0	0	14	7	0	0	0	0
0	0	7	2	17	12	0	0
0	0	9	13	7	5	0	0
0	0	4	11	0	0	17	12
0	0	10	3	0	0	7	5

1	0	0	0	0	0	0	0
18	18	0	0	0	0	0	0
0	0	14	7	0	0	0	0
0	0	2	5	0	0	0	0
0	0	4	6	0	0	14	2
0	0	11	3	0	0	7	5
0	0	1	11	14	2	0	0
0	0	16	8	7	5	0	0

0	1	0	0	0	0	0	0
18	18	0	0	0	0	0	0
0	0	17	8	7	10	0	0
0	0	4	3	16	7	0	0
0	0	10	2	0	18	12	17
0	0	9	16	0	18	2	14
0	0	18	8	5	17	0	0
0	0	6	3	5	17	0	0

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	16	12	18	18	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	0	0	1
0	0	10	12	0	0	18	18

2	14	0	0	0	0	0	0
5	7	0	0	0	0	0	0
0	0	2	5	0	0	0	0
0	0	14	7	0	0	0	0
0	0	7	2	17	12	0	0
0	0	9	13	7	5	0	0
0	0	4	11	0	0	17	12
0	0	10	3	0	0	7	5

1	0	0	0	0	0	0	0
18	18	0	0	0	0	0	0
0	0	14	7	0	0	0	0
0	0	2	5	0	0	0	0
0	0	4	6	0	0	14	2
0	0	11	3	0	0	7	5
0	0	1	11	14	2	0	0
0	0	16	8	7	5	0	0

G = 3- 1+2⋊D9 order 486 = 2·35

The semidirect product of 3- 1+2 and D9 acting via D9/C3=S3

G = 3_-¹⁺²⋊D₉ order 486 = 2·3⁵

The semidirect product of 3_-¹⁺² and D₉ acting via D₉/C₃=S₃

0	1	0	0	0	0	0	0
18	18	0	0	0	0	0	0
0	0	17	8	7	10	0	0
0	0	4	3	16	7	0	0
0	0	10	2	0	18	12	17
0	0	9	16	0	18	2	14
0	0	18	8	5	17	0	0
0	0	6	3	5	17	0	0

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	16	12	18	18	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	0	0	1
0	0	10	12	0	0	18	18

2	14	0	0	0	0	0	0
5	7	0	0	0	0	0	0
0	0	2	5	0	0	0	0
0	0	14	7	0	0	0	0
0	0	7	2	17	12	0	0
0	0	9	13	7	5	0	0
0	0	4	11	0	0	17	12
0	0	10	3	0	0	7	5

1	0	0	0	0	0	0	0
18	18	0	0	0	0	0	0
0	0	14	7	0	0	0	0
0	0	2	5	0	0	0	0
0	0	4	6	0	0	14	2
0	0	11	3	0	0	7	5
0	0	1	11	14	2	0	0
0	0	16	8	7	5	0	0