C3^3.D9 - GroupNames

G = C₃³.D₉ order 486 = 2·3⁵

3^rd non-split extension by C₃³ of D₉ acting via D₉/C₃=S₃

Aliases: C₃³.₃D₉, C₂₇⋊C₃⋊₁S₃, (C₃×C₉).₃D₉, C₉.₄He₃⋊C₂, (C₃²×C₉).₁₇S₃, C₃².₁₀(C₉⋊S₃), C₉.₂(He₃⋊C₂), C₃.₆(C₃²⋊₂D₉), (C₃×C₉).₉(C₃⋊S₃), SmallGroup(486,55)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃² — C₉.₄He₃ — C₃³.D₉

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃²×C₉ — C₉.₄He₃ — C₃³.D₉

Lower central

C₉.₄He₃ — C₃³.D₉

Upper central

C₁

Generators and relations for C₃³.D₉
G = < a,b,c,d,e | a³=b³=c³=e²=1, d⁹=c, ab=ba, ac=ca, dad^-1=ab^-1c, eae=a^-1bc^-1, ebe=bc=cb, dbd^-1=bc^-1, cd=dc, ece=c^-1, ede=c^-1d⁸ >

Subgroups: 538 in 56 conjugacy classes, 14 normal (11 characteristic)
C₁, C₂, C₃, C₃, S₃, C₆, C₉, C₉, C₃², C₃², D₉, C₃×S₃, C₃⋊S₃, C₂₇, C₃×C₉, C₃×C₉, C₃³, D₂₇, C₃×D₉, C₉⋊S₃, C₃×C₃⋊S₃, C₂₇⋊C₃, C₃²×C₉, C₂₇⋊C₆, C₃×C₉⋊S₃, C₉.₄He₃, C₃³.D₉
Quotients: C₁, C₂, S₃, D₉, C₃⋊S₃, C₉⋊S₃, He₃⋊C₂, C₃²⋊₂D₉, C₃³.D₉

Character table of C₃³.D₉

class	1	2	3A	3B	3C	3D	3E	3F	6A	6B	9A	9B	9C	9D	9E	9F	9G	9H	9I	9J	9K	27A	27B	27C	27D	27E	27F	27G	27H	27I
size	1	81	2	3	3	6	6	6	81	81	2	2	2	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	-1	-1	-1	0	0	2	2	2	-1	-1	-1	-1	-1	2	2	-1	-1	-1	-1	2	2	2	-1	-1	-1	orthogonal lifted from S₃
ρ₄	2	0	2	2	2	2	2	2	0	0	2	2	2	2	2	2	2	2	2	2	2	-1	-1	-1	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₅	2	0	2	2	2	-1	-1	-1	0	0	2	2	2	-1	-1	-1	-1	-1	2	2	-1	-1	-1	-1	-1	-1	-1	2	2	2	orthogonal lifted from S₃
ρ₆	2	0	2	2	2	-1	-1	-1	0	0	2	2	2	-1	-1	-1	-1	-1	2	2	-1	2	2	2	-1	-1	-1	-1	-1	-1	orthogonal lifted from S₃
ρ₇	2	0	2	2	2	-1	-1	-1	0	0	-1	-1	-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
ρ₈	2	0	2	2	2	-1	-1	-1	0	0	-1	-1	-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
ρ₉	2	0	2	2	2	2	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₀	2	0	2	2	2	2	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	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	2	2	-1	-1	-1	0	0	-1	-1	-1	2	2	-1	-1	-1	-1	-1	2	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₂	2	0	2	2	2	-1	-1	-1	0	0	-1	-1	-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
ρ₁₃	2	0	2	2	2	2	2	2	0	0	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	orthogonal lifted from D₉
ρ₁₄	2	0	2	2	2	-1	-1	-1	0	0	-1	-1	-1	2	2	-1	-1	-1	-1	-1	2	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	orthogonal lifted from D₉
ρ₁₅	2	0	2	2	2	-1	-1	-1	0	0	-1	-1	-1	2	2	-1	-1	-1	-1	-1	2	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	ζ₉⁸+ζ₉	ζ₉⁸+ζ₉	ζ₉⁷+ζ₉²	ζ₉⁵+ζ₉⁴	orthogonal lifted from D₉
ρ₁₆	3	1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	ζ₃	ζ₃²	3	3	3	0	0	0	0	0	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₁₇	3	-1	3	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	ζ₆⁵	ζ₆	3	3	3	0	0	0	0	0	^-3-3√-3/₂	^-3+3√-3/₂	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₁₈	3	1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	ζ₃²	ζ₃	3	3	3	0	0	0	0	0	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₁₉	3	-1	3	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	ζ₆	ζ₆⁵	3	3	3	0	0	0	0	0	^-3+3√-3/₂	^-3-3√-3/₂	0	0	0	0	0	0	0	0	0	0	complex lifted from He₃⋊C₂
ρ₂₀	6	0	-3	0	0	-3	0	3	0	0	3ζ₉⁸+3ζ₉	3ζ₉⁷+3ζ₉²	3ζ₉⁵+3ζ₉⁴	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	0	0	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₁	6	0	-3	0	0	3	-3	0	0	0	3ζ₉⁸+3ζ₉	3ζ₉⁷+3ζ₉²	3ζ₉⁵+3ζ₉⁴	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	0	0	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₂	6	0	-3	0	0	0	3	-3	0	0	3ζ₉⁵+3ζ₉⁴	3ζ₉⁸+3ζ₉	3ζ₉⁷+3ζ₉²	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	0	0	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₃	6	0	-3	0	0	0	3	-3	0	0	3ζ₉⁸+3ζ₉	3ζ₉⁷+3ζ₉²	3ζ₉⁵+3ζ₉⁴	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	0	0	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₄	6	0	-3	0	0	3	-3	0	0	0	3ζ₉⁵+3ζ₉⁴	3ζ₉⁸+3ζ₉	3ζ₉⁷+3ζ₉²	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	0	0	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₅	6	0	-3	0	0	3	-3	0	0	0	3ζ₉⁷+3ζ₉²	3ζ₉⁵+3ζ₉⁴	3ζ₉⁸+3ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	0	0	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₆	6	0	-3	0	0	-3	0	3	0	0	3ζ₉⁵+3ζ₉⁴	3ζ₉⁸+3ζ₉	3ζ₉⁷+3ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	0	0	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₇	6	0	-3	0	0	-3	0	3	0	0	3ζ₉⁷+3ζ₉²	3ζ₉⁵+3ζ₉⁴	3ζ₉⁸+3ζ₉	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	0	0	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₈	6	0	-3	0	0	0	3	-3	0	0	3ζ₉⁷+3ζ₉²	3ζ₉⁵+3ζ₉⁴	3ζ₉⁸+3ζ₉	ζ₉⁵+2ζ₉⁴-ζ₉²+ζ₉	2ζ₉⁸-ζ₉⁴+ζ₉²+ζ₉	ζ₉⁸+ζ₉⁴-ζ₉²+2ζ₉	2ζ₉⁵+ζ₉⁴+ζ₉²-ζ₉	ζ₉⁸+ζ₉⁷-ζ₉⁴+2ζ₉²	0	0	-ζ₉⁸+2ζ₉⁷+ζ₉⁴+ζ₉²	0	0	0	0	0	0	0	0	0	orthogonal faithful
ρ₂₉	6	0	6	-3+3√-3	-3-3√-3	0	0	0	0	0	-3	-3	-3	0	0	0	0	0	^3-3√-3/₂	^3+3√-3/₂	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	0	0	0	0	0	-3	-3	-3	0	0	0	0	0	^3+3√-3/₂	^3-3√-3/₂	0	0	0	0	0	0	0	0	0	0	complex lifted from C₃²⋊₂D₉

Permutation representations of C₃³.D₉
►On 27 points - transitive group 27T156

Generators in S₂₇

(1 19 10)(2 11 20)(3 12 21)(4 22 13)(5 14 23)(6 15 24)(7 25 16)(8 17 26)(9 18 27)

(1 19 10)(2 11 20)(4 22 13)(5 14 23)(7 25 16)(8 17 26)

(1 10 19)(2 11 20)(3 12 21)(4 13 22)(5 14 23)(6 15 24)(7 16 25)(8 17 26)(9 18 27)

(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)

(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 27)(20 26)(21 25)(22 24)

G:=sub<Sym(27)| (1,19,10)(2,11,20)(3,12,21)(4,22,13)(5,14,23)(6,15,24)(7,25,16)(8,17,26)(9,18,27), (1,19,10)(2,11,20)(4,22,13)(5,14,23)(7,25,16)(8,17,26), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,27)(20,26)(21,25)(22,24)>;

G:=Group( (1,19,10)(2,11,20)(3,12,21)(4,22,13)(5,14,23)(6,15,24)(7,25,16)(8,17,26)(9,18,27), (1,19,10)(2,11,20)(4,22,13)(5,14,23)(7,25,16)(8,17,26), (1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,27)(20,26)(21,25)(22,24) );

G=PermutationGroup([[(1,19,10),(2,11,20),(3,12,21),(4,22,13),(5,14,23),(6,15,24),(7,25,16),(8,17,26),(9,18,27)], [(1,19,10),(2,11,20),(4,22,13),(5,14,23),(7,25,16),(8,17,26)], [(1,10,19),(2,11,20),(3,12,21),(4,13,22),(5,14,23),(6,15,24),(7,16,25),(8,17,26),(9,18,27)], [(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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,27),(20,26),(21,25),(22,24)]])

G:=TransitiveGroup(27,156);

Matrix representation of C₃³.D₉ ►in GL₆(𝔽₁₀₉)

0	108	0	0	0	0
1	108	0	0	0	0
35	39	0	1	0	0
35	35	108	108	0	0
43	24	0	0	0	1
42	43	0	0	108	108

0	108	0	0	0	0
1	108	0	0	0	0
35	39	1	0	0	0
35	39	0	1	0	0
43	24	0	0	0	1
42	43	0	0	108	108

108	1	0	0	0	0
108	0	0	0	0	0
74	74	0	1	0	0
74	70	108	108	0	0
67	66	0	0	0	1
66	85	0	0	108	108

0	0	108	1	0	0
0	105	107	108	0	0
73	108	39	74	1	0
73	108	39	74	0	1
44	5	24	66	0	0
71	55	24	66	0	0

0	108	104	86	0	0
0	18	86	91	0	0
6	41	67	24	0	0
65	14	67	24	0	0
71	37	74	39	27	77
44	5	74	39	50	82

G:=sub<GL(6,GF(109))| [0,1,35,35,43,42,108,108,39,35,24,43,0,0,0,108,0,0,0,0,1,108,0,0,0,0,0,0,0,108,0,0,0,0,1,108],[0,1,35,35,43,42,108,108,39,39,24,43,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,108,0,0,0,0,1,108],[108,108,74,74,67,66,1,0,74,70,66,85,0,0,0,108,0,0,0,0,1,108,0,0,0,0,0,0,0,108,0,0,0,0,1,108],[0,0,73,73,44,71,0,105,108,108,5,55,108,107,39,39,24,24,1,108,74,74,66,66,0,0,1,0,0,0,0,0,0,1,0,0],[0,0,6,65,71,44,108,18,41,14,37,5,104,86,67,67,74,74,86,91,24,24,39,39,0,0,0,0,27,50,0,0,0,0,77,82] >;

C₃³.D₉ in GAP, Magma, Sage, TeX

C_3^3.D_9

% in TeX

G:=Group("C3^3.D9");

// GroupNames label

G:=SmallGroup(486,55);

// by ID

G=gap.SmallGroup(486,55);

# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,1195,218,548,4755,453,11344,3250,11669]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^9=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1*c,e*a*e=a^-1*b*c^-1,e*b*e=b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c,e*c*e=c^-1,e*d*e=c^-1*d^8>;

// generators/relations

Export

Character table of C₃³.D₉ in TeX

0	0	108	1	0	0
0	105	107	108	0	0
73	108	39	74	1	0
73	108	39	74	0	1
44	5	24	66	0	0
71	55	24	66	0	0

0	108	104	86	0	0
0	18	86	91	0	0
6	41	67	24	0	0
65	14	67	24	0	0
71	37	74	39	27	77
44	5	74	39	50	82

0	0	108	1	0	0
0	105	107	108	0	0
73	108	39	74	1	0
73	108	39	74	0	1
44	5	24	66	0	0
71	55	24	66	0	0

0	108	104	86	0	0
0	18	86	91	0	0
6	41	67	24	0	0
65	14	67	24	0	0
71	37	74	39	27	77
44	5	74	39	50	82

G = C33.D9 order 486 = 2·35

3rd non-split extension by C33 of D9 acting via D9/C3=S3

G = C₃³.D₉ order 486 = 2·3⁵

3^rd non-split extension by C₃³ of D₉ acting via D₉/C₃=S₃

0	0	108	1	0	0
0	105	107	108	0	0
73	108	39	74	1	0
73	108	39	74	0	1
44	5	24	66	0	0
71	55	24	66	0	0

0	108	104	86	0	0
0	18	86	91	0	0
6	41	67	24	0	0
65	14	67	24	0	0
71	37	74	39	27	77
44	5	74	39	50	82