(C3xC9):5D9 - GroupNames

G = (C₃×C₉)⋊₅D₉ order 486 = 2·3⁵

5^th semidirect product of C₃×C₉ and D₉ acting via D₉/C₃=S₃

Aliases: (C₃×C₉)⋊₅D₉, C₃²⋊C₉.₁₁S₃, C₃².₈(C₉⋊S₃), (C₃²×C₉).₁₁S₃, C₃³.₂₁(C₃⋊S₃), C₃.₄(C₃²⋊₂D₉), C₃².₁₉He₃⋊₃C₂, C₃.₆(He₃.₃S₃), C₃².₂₁(He₃⋊C₂), SmallGroup(486,53)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₃².₁₉He₃ — (C₃×C₉)⋊₅D₉

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₉ — C₃².₁₉He₃ — (C₃×C₉)⋊₅D₉

Lower central

C₃².₁₉He₃ — (C₃×C₉)⋊₅D₉

Upper central

C₁

Generators and relations for (C₃×C₉)⋊₅D₉
G = < a,b,c,d | a³=b⁹=c⁹=d²=1, ab=ba, cac^-1=ab³, dad=ab⁶, cbc^-1=ab⁷, dbd=ab², dcd=c^-1 >

Subgroups: 646 in 63 conjugacy classes, 17 normal (8 characteristic)
C₁, C₂, C₃, C₃, C₃, S₃, C₆, C₉, C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₃×C₉, C₃×C₉, C₃³, C₃×D₉, C₉⋊S₃, C₃×C₃⋊S₃, C₃²⋊C₉, C₃²×C₉, C₃²⋊D₉, C₃×C₉⋊S₃, C₃².₁₉He₃, (C₃×C₉)⋊₅D₉
Quotients: C₁, C₂, S₃, D₉, C₃⋊S₃, C₉⋊S₃, He₃⋊C₂, C₃²⋊₂D₉, He₃.₃S₃, (C₃×C₉)⋊₅D₉

Character table of (C₃×C₉)⋊₅D₉

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	2	-1	-1	-1	2	2	-1	-1	0	0	-1	-1	-1	2	2	2	-1	-1	-1	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₈	2	0	2	-1	-1	-1	2	2	-1	-1	0	0	2	2	2	-1	-1	-1	-1	-1	-1	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₉	2	0	2	-1	-1	-1	2	2	-1	-1	0	0	2	2	2	-1	-1	-1	-1	-1	-1	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₀	2	0	2	-1	-1	-1	2	2	-1	-1	0	0	-1	-1	-1	-1	-1	-1	2	2	2	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₁	2	0	2	-1	-1	-1	2	2	-1	-1	0	0	-1	-1	-1	-1	-1	-1	2	2	2	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₂	2	0	2	-1	-1	-1	2	2	-1	-1	0	0	2	2	2	-1	-1	-1	-1	-1	-1	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₃	2	0	2	-1	-1	-1	2	2	-1	-1	0	0	-1	-1	-1	2	2	2	-1	-1	-1	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₄	2	0	2	-1	-1	-1	2	2	-1	-1	0	0	-1	-1	-1	2	2	2	-1	-1	-1	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₅	2	0	2	-1	-1	-1	2	2	-1	-1	0	0	-1	-1	-1	-1	-1	-1	2	2	2	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	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	-3	6	-3	-3	0	0	0	0	0	0	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	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
ρ₂₁	6	0	-3	-3	6	-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
ρ₂₂	6	0	-3	-3	-3	6	0	0	0	0	0	0	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	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
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	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	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
ρ₂₄	6	0	-3	-3	6	-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
ρ₂₅	6	0	-3	-3	6	-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
ρ₂₆	6	0	-3	-3	-3	6	0	0	0	0	0	0	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	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
ρ₂₇	6	0	-3	6	-3	-3	0	0	0	0	0	0	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	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
ρ₂₈	6	0	-3	-3	-3	6	0	0	0	0	0	0	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	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
ρ₂₉	6	0	6	-3	-3	-3	-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	6	-3	-3	-3	-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 (C₃×C₉)⋊₅D₉
►On 81 points

Generators in S₈₁

(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(73 79 76)(74 80 77)(75 81 78)

(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)

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

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

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

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

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

Matrix representation of (C₃×C₉)⋊₅D₉ ►in GL₈(𝔽₁₉)

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	18	1	0	0	0	0
0	0	18	0	0	0	0	0
0	0	7	10	1	0	0	0
0	0	7	10	0	1	0	0
0	0	18	8	0	0	18	18
0	0	12	18	0	0	1	0

0	18	0	0	0	0	0	0
1	18	0	0	0	0	0	0
0	0	5	12	0	0	0	0
0	0	7	17	0	0	0	0
0	0	6	5	7	5	0	0
0	0	2	3	14	2	0	0
0	0	11	7	0	0	17	12
0	0	7	1	0	0	7	5

14	17	0	0	0	0	0	0
2	12	0	0	0	0	0	0
0	0	0	0	18	1	0	0
0	0	3	11	17	18	0	0
0	0	9	12	17	10	1	0
0	0	9	12	17	10	0	1
0	0	16	1	11	18	0	0
0	0	15	2	11	18	0	0

7	14	0	0	0	0	0	0
2	12	0	0	0	0	0	0
0	0	3	11	17	18	0	0
0	0	0	0	18	1	0	0
0	0	9	17	7	9	0	0
0	0	9	18	7	9	0	0
0	0	10	9	12	1	18	18
0	0	4	0	12	1	0	1

G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,7,7,18,12,0,0,1,0,10,10,8,18,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0],[0,1,0,0,0,0,0,0,18,18,0,0,0,0,0,0,0,0,5,7,6,2,11,7,0,0,12,17,5,3,7,1,0,0,0,0,7,14,0,0,0,0,0,0,5,2,0,0,0,0,0,0,0,0,17,7,0,0,0,0,0,0,12,5],[14,2,0,0,0,0,0,0,17,12,0,0,0,0,0,0,0,0,0,3,9,9,16,15,0,0,0,11,12,12,1,2,0,0,18,17,17,17,11,11,0,0,1,18,10,10,18,18,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[7,2,0,0,0,0,0,0,14,12,0,0,0,0,0,0,0,0,3,0,9,9,10,4,0,0,11,0,17,18,9,0,0,0,17,18,7,7,12,12,0,0,18,1,9,9,1,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,1] >;

(C₃×C₉)⋊₅D₉ in GAP, Magma, Sage, TeX

(C_3\times C_9)\rtimes_5D_9

% in TeX

G:=Group("(C3xC9):5D9");

// GroupNames label

G:=SmallGroup(486,53);

// by ID

G=gap.SmallGroup(486,53);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,223,6050,548,500,867,11344,3250,11669]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=c^9=d^2=1,a*b=b*a,c*a*c^-1=a*b^3,d*a*d=a*b^6,c*b*c^-1=a*b^7,d*b*d=a*b^2,d*c*d=c^-1>;

// generators/relations

Export

Character table of (C₃×C₉)⋊₅D₉ in TeX

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	18	1	0	0	0	0
0	0	18	0	0	0	0	0
0	0	7	10	1	0	0	0
0	0	7	10	0	1	0	0
0	0	18	8	0	0	18	18
0	0	12	18	0	0	1	0

0	18	0	0	0	0	0	0
1	18	0	0	0	0	0	0
0	0	5	12	0	0	0	0
0	0	7	17	0	0	0	0
0	0	6	5	7	5	0	0
0	0	2	3	14	2	0	0
0	0	11	7	0	0	17	12
0	0	7	1	0	0	7	5

14	17	0	0	0	0	0	0
2	12	0	0	0	0	0	0
0	0	0	0	18	1	0	0
0	0	3	11	17	18	0	0
0	0	9	12	17	10	1	0
0	0	9	12	17	10	0	1
0	0	16	1	11	18	0	0
0	0	15	2	11	18	0	0

7	14	0	0	0	0	0	0
2	12	0	0	0	0	0	0
0	0	3	11	17	18	0	0
0	0	0	0	18	1	0	0
0	0	9	17	7	9	0	0
0	0	9	18	7	9	0	0
0	0	10	9	12	1	18	18
0	0	4	0	12	1	0	1

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	18	1	0	0	0	0
0	0	18	0	0	0	0	0
0	0	7	10	1	0	0	0
0	0	7	10	0	1	0	0
0	0	18	8	0	0	18	18
0	0	12	18	0	0	1	0

0	18	0	0	0	0	0	0
1	18	0	0	0	0	0	0
0	0	5	12	0	0	0	0
0	0	7	17	0	0	0	0
0	0	6	5	7	5	0	0
0	0	2	3	14	2	0	0
0	0	11	7	0	0	17	12
0	0	7	1	0	0	7	5

14	17	0	0	0	0	0	0
2	12	0	0	0	0	0	0
0	0	0	0	18	1	0	0
0	0	3	11	17	18	0	0
0	0	9	12	17	10	1	0
0	0	9	12	17	10	0	1
0	0	16	1	11	18	0	0
0	0	15	2	11	18	0	0

7	14	0	0	0	0	0	0
2	12	0	0	0	0	0	0
0	0	3	11	17	18	0	0
0	0	0	0	18	1	0	0
0	0	9	17	7	9	0	0
0	0	9	18	7	9	0	0
0	0	10	9	12	1	18	18
0	0	4	0	12	1	0	1

G = (C3×C9)⋊5D9 order 486 = 2·35

5th semidirect product of C3×C9 and D9 acting via D9/C3=S3

G = (C₃×C₉)⋊₅D₉ order 486 = 2·3⁵

5^th semidirect product of C₃×C₉ and D₉ acting via D₉/C₃=S₃

1	0	0	0	0	0	0	0
0	1	0	0	0	0	0	0
0	0	18	1	0	0	0	0
0	0	18	0	0	0	0	0
0	0	7	10	1	0	0	0
0	0	7	10	0	1	0	0
0	0	18	8	0	0	18	18
0	0	12	18	0	0	1	0

0	18	0	0	0	0	0	0
1	18	0	0	0	0	0	0
0	0	5	12	0	0	0	0
0	0	7	17	0	0	0	0
0	0	6	5	7	5	0	0
0	0	2	3	14	2	0	0
0	0	11	7	0	0	17	12
0	0	7	1	0	0	7	5

14	17	0	0	0	0	0	0
2	12	0	0	0	0	0	0
0	0	0	0	18	1	0	0
0	0	3	11	17	18	0	0
0	0	9	12	17	10	1	0
0	0	9	12	17	10	0	1
0	0	16	1	11	18	0	0
0	0	15	2	11	18	0	0

7	14	0	0	0	0	0	0
2	12	0	0	0	0	0	0
0	0	3	11	17	18	0	0
0	0	0	0	18	1	0	0
0	0	9	17	7	9	0	0
0	0	9	18	7	9	0	0
0	0	10	9	12	1	18	18
0	0	4	0	12	1	0	1