# Properties

 Label 28T70 Order $$504$$ n $$28$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSL(2,8)$

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## Group action invariants

 Degree $n$ : $28$ Transitive number $t$ : $70$ Group : $\PSL(2,8)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,21,27,15,25,10,24,7,20)(2,19,8,26,17,16,6,5,3)(9,28,18,13,11,22,14,23,12), (1,28,21,4,25,18,10)(2,26,19,17,12,7,11)(3,24,20,16,13,14,9)(5,6,23,15,8,27,22) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: None

## Low degree siblings

9T27, 36T712

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $63$ $2$ $( 3, 7)( 4,28)( 5,14)( 6,27)( 8,11)( 9,10)(12,13)(15,21)(16,22)(17,24)(18,20) (19,23)$ $9, 9, 9, 1$ $56$ $9$ $( 2, 3,27, 8,13,12,11, 6, 7)( 4,14,19, 9,25,10,23, 5,28)(15,20,22,26,16,18,21, 24,17)$ $9, 9, 9, 1$ $56$ $9$ $( 2, 6,12, 8, 3, 7,11,13,27)( 4, 5,10, 9,14,28,23,25,19)(15,24,18,26,20,17,21, 16,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 1$ $56$ $3$ $( 2, 8,11)( 3,13, 6)( 4, 9,23)( 5,14,25)( 7,27,12)(10,28,19)(15,26,21) (16,24,20)(17,22,18)$ $9, 9, 9, 1$ $56$ $9$ $( 2,12, 3,11,27, 6, 8, 7,13)( 4,10,14,23,19, 5, 9,28,25)(15,18,20,21,22,24,26, 17,16)$ $7, 7, 7, 7$ $72$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,21,24,17,12,23,14)( 9,26,22,25,18,13,15) (10,11,28,27,19,20,16)$ $7, 7, 7, 7$ $72$ $7$ $( 1, 2,15,10, 5,20,13)( 3,23, 8, 7,21,16,12)( 4,18, 9, 6,19,14,11) (17,22,24,25,27,28,26)$ $7, 7, 7, 7$ $72$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,17,14,24,23,21,12)( 9,25,15,22,13,26,18) (10,27,16,28,20,11,19)$

## Group invariants

 Order: $504=2^{3} \cdot 3^{2} \cdot 7$ Cyclic: No Abelian: No Solvable: No GAP id: [504, 156]
 Character table:  2 3 3 . . . . . . . 3 2 . 2 2 2 2 . . . 7 1 . . . . . 1 1 1 1a 2a 9a 9b 3a 9c 7a 7b 7c 2P 1a 1a 9b 9c 3a 9a 7b 7c 7a 3P 1a 2a 3a 3a 1a 3a 7c 7a 7b 5P 1a 2a 9c 9a 3a 9b 7b 7c 7a 7P 1a 2a 9b 9c 3a 9a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 X.2 7 -1 1 1 -2 1 . . . X.3 7 -1 A C 1 B . . . X.4 7 -1 B A 1 C . . . X.5 7 -1 C B 1 A . . . X.6 8 . -1 -1 -1 -1 1 1 1 X.7 9 1 . . . . D F E X.8 9 1 . . . . E D F X.9 9 1 . . . . F E D A = -E(9)^4-E(9)^5 B = -E(9)^2-E(9)^7 C = E(9)^2+E(9)^4+E(9)^5+E(9)^7 D = E(7)^3+E(7)^4 E = E(7)^2+E(7)^5 F = E(7)+E(7)^6