# Properties

 Label 27T8 Degree $27$ Order $54$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{27}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(27, 8);

## Group action invariants

 Degree $n$: $27$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $8$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_{27}$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,22)(2,24)(3,23)(4,21)(5,20)(6,19)(7,17)(8,16)(9,18)(10,15)(11,14)(12,13)(25,26), (1,18,6,19,9,22,12,26,14,2,16,4,20,7,23,10,27,15,3,17,5,21,8,24,11,25,13) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$18$:  $D_{9}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 9: $D_{9}$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Label Cycle Type Size Order Index Representative 1A $1^{27}$ $1$ $1$ $0$ $()$ 2A $2^{13},1$ $27$ $2$ $13$ $( 2, 3)( 4,27)( 5,26)( 6,25)( 7,23)( 8,22)( 9,24)(10,20)(11,19)(12,21)(13,18)(14,17)(15,16)$ 3A $3^{9}$ $2$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)$ 9A1 $9^{3}$ $2$ $9$ $24$ $( 1,20,11, 2,21,12, 3,19,10)( 4,24,14, 5,22,15, 6,23,13)( 7,25,16, 8,26,17, 9,27,18)$ 9A2 $9^{3}$ $2$ $9$ $24$ $( 1,19,12, 2,20,10, 3,21,11)( 4,23,15, 5,24,13, 6,22,14)( 7,27,17, 8,25,18, 9,26,16)$ 9A4 $9^{3}$ $2$ $9$ $24$ $( 1,21,10, 2,19,11, 3,20,12)( 4,22,13, 5,23,14, 6,24,15)( 7,26,18, 8,27,16, 9,25,17)$ 27A1 $27$ $2$ $27$ $26$ $( 1,16, 5,19, 7,24,12,27,13, 2,17, 6,20, 8,22,10,25,14, 3,18, 4,21, 9,23,11,26,15)$ 27A2 $27$ $2$ $27$ $26$ $( 1,18, 6,19, 9,22,12,26,14, 2,16, 4,20, 7,23,10,27,15, 3,17, 5,21, 8,24,11,25,13)$ 27A4 $27$ $2$ $27$ $26$ $( 1,27,22,21,16,13,10, 9, 5, 2,25,23,19,17,14,11, 7, 6, 3,26,24,20,18,15,12, 8, 4)$ 27A5 $27$ $2$ $27$ $26$ $( 1, 9,14,20,27, 5,11,18,22, 2, 7,15,21,25, 6,12,16,23, 3, 8,13,19,26, 4,10,17,24)$ 27A7 $27$ $2$ $27$ $26$ $( 1,25,24,21,17,15,10, 7, 4, 2,26,22,19,18,13,11, 8, 5, 3,27,23,20,16,14,12, 9, 6)$ 27A8 $27$ $2$ $27$ $26$ $( 1,26,23,21,18,14,10, 8, 6, 2,27,24,19,16,15,11, 9, 4, 3,25,22,20,17,13,12, 7, 5)$ 27A10 $27$ $2$ $27$ $26$ $( 1,17, 4,19, 8,23,12,25,15, 2,18, 5,20, 9,24,10,26,13, 3,16, 6,21, 7,22,11,27,14)$ 27A11 $27$ $2$ $27$ $26$ $( 1, 8,15,20,26, 6,11,17,23, 2, 9,13,21,27, 4,12,18,24, 3, 7,14,19,25, 5,10,16,22)$ 27A13 $27$ $2$ $27$ $26$ $( 1, 7,13,20,25, 4,11,16,24, 2, 8,14,21,26, 5,12,17,22, 3, 9,15,19,27, 6,10,18,23)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/13$

## Group invariants

 Order: $54=2 \cdot 3^{3}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 54.1 magma: IdentifyGroup(G); Character table:

 1A 2A 3A 9A1 9A2 9A4 27A1 27A2 27A4 27A5 27A7 27A8 27A10 27A11 27A13 Size 1 27 2 2 2 2 2 2 2 2 2 2 2 2 2 2 P 1A 1A 3A 9A2 9A4 9A1 27A5 27A13 27A8 27A2 27A1 27A10 27A4 27A11 27A7 3 P 1A 2A 1A 3A 3A 3A 9A2 9A2 9A4 9A1 9A4 9A4 9A2 9A1 9A1 Type 54.1.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 54.1.1b R $1$ $−1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 54.1.2a R $2$ $0$ $2$ $2$ $2$ $2$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ 54.1.2b1 R $2$ $0$ $2$ $−1$ $−1$ $−1$ $ζ9−4+ζ94$ $ζ9−1+ζ9$ $ζ9−2+ζ92$ $ζ9−2+ζ92$ $ζ9−1+ζ9$ $ζ9−4+ζ94$ $ζ9−4+ζ94$ $ζ9−1+ζ9$ $ζ9−2+ζ92$ 54.1.2b2 R $2$ $0$ $2$ $−1$ $−1$ $−1$ $ζ9−2+ζ92$ $ζ9−4+ζ94$ $ζ9−1+ζ9$ $ζ9−1+ζ9$ $ζ9−4+ζ94$ $ζ9−2+ζ92$ $ζ9−2+ζ92$ $ζ9−4+ζ94$ $ζ9−1+ζ9$ 54.1.2b3 R $2$ $0$ $2$ $−1$ $−1$ $−1$ $ζ9−1+ζ9$ $ζ9−2+ζ92$ $ζ9−4+ζ94$ $ζ9−4+ζ94$ $ζ9−2+ζ92$ $ζ9−1+ζ9$ $ζ9−1+ζ9$ $ζ9−2+ζ92$ $ζ9−4+ζ94$ 54.1.2c1 R $2$ $0$ $−1$ $ζ27−12+ζ2712$ $ζ27−3+ζ273$ $ζ27−6+ζ276$ $ζ27−13+ζ2713$ $ζ27−1+ζ27$ $ζ27−2+ζ272$ $ζ27−11+ζ2711$ $ζ27−10+ζ2710$ $ζ27−4+ζ274$ $ζ27−5+ζ275$ $ζ27−8+ζ278$ $ζ27−7+ζ277$ 54.1.2c2 R $2$ $0$ $−1$ $ζ27−12+ζ2712$ $ζ27−3+ζ273$ $ζ27−6+ζ276$ $ζ27−5+ζ275$ $ζ27−10+ζ2710$ $ζ27−7+ζ277$ $ζ27−2+ζ272$ $ζ27−8+ζ278$ $ζ27−13+ζ2713$ $ζ27−4+ζ274$ $ζ27−1+ζ27$ $ζ27−11+ζ2711$ 54.1.2c3 R $2$ $0$ $−1$ $ζ27−12+ζ2712$ $ζ27−3+ζ273$ $ζ27−6+ζ276$ $ζ27−4+ζ274$ $ζ27−8+ζ278$ $ζ27−11+ζ2711$ $ζ27−7+ζ277$ $ζ27−1+ζ27$ $ζ27−5+ζ275$ $ζ27−13+ζ2713$ $ζ27−10+ζ2710$ $ζ27−2+ζ272$ 54.1.2c4 R $2$ $0$ $−1$ $ζ27−6+ζ276$ $ζ27−12+ζ2712$ $ζ27−3+ζ273$ $ζ27−11+ζ2711$ $ζ27−5+ζ275$ $ζ27−10+ζ2710$ $ζ27−1+ζ27$ $ζ27−4+ζ274$ $ζ27−7+ζ277$ $ζ27−2+ζ272$ $ζ27−13+ζ2713$ $ζ27−8+ζ278$ 54.1.2c5 R $2$ $0$ $−1$ $ζ27−6+ζ276$ $ζ27−12+ζ2712$ $ζ27−3+ζ273$ $ζ27−7+ζ277$ $ζ27−13+ζ2713$ $ζ27−1+ζ27$ $ζ27−8+ζ278$ $ζ27−5+ζ275$ $ζ27−2+ζ272$ $ζ27−11+ζ2711$ $ζ27−4+ζ274$ $ζ27−10+ζ2710$ 54.1.2c6 R $2$ $0$ $−1$ $ζ27−6+ζ276$ $ζ27−12+ζ2712$ $ζ27−3+ζ273$ $ζ27−2+ζ272$ $ζ27−4+ζ274$ $ζ27−8+ζ278$ $ζ27−10+ζ2710$ $ζ27−13+ζ2713$ $ζ27−11+ζ2711$ $ζ27−7+ζ277$ $ζ27−5+ζ275$ $ζ27−1+ζ27$ 54.1.2c7 R $2$ $0$ $−1$ $ζ27−3+ζ273$ $ζ27−6+ζ276$ $ζ27−12+ζ2712$ $ζ27−10+ζ2710$ $ζ27−7+ζ277$ $ζ27−13+ζ2713$ $ζ27−4+ζ274$ $ζ27−11+ζ2711$ $ζ27−1+ζ27$ $ζ27−8+ζ278$ $ζ27−2+ζ272$ $ζ27−5+ζ275$ 54.1.2c8 R $2$ $0$ $−1$ $ζ27−3+ζ273$ $ζ27−6+ζ276$ $ζ27−12+ζ2712$ $ζ27−8+ζ278$ $ζ27−11+ζ2711$ $ζ27−5+ζ275$ $ζ27−13+ζ2713$ $ζ27−2+ζ272$ $ζ27−10+ζ2710$ $ζ27−1+ζ27$ $ζ27−7+ζ277$ $ζ27−4+ζ274$ 54.1.2c9 R $2$ $0$ $−1$ $ζ27−3+ζ273$ $ζ27−6+ζ276$ $ζ27−12+ζ2712$ $ζ27−1+ζ27$ $ζ27−2+ζ272$ $ζ27−4+ζ274$ $ζ27−5+ζ275$ $ζ27−7+ζ277$ $ζ27−8+ζ278$ $ζ27−10+ζ2710$ $ζ27−11+ζ2711$ $ζ27−13+ζ2713$

magma: CharacterTable(G);