# Properties

 Label 26T42 Order $$7800$$ n $$26$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSL(2,25)$

## Group action invariants

 Degree $n$ : $26$ Transitive number $t$ : $42$ Group : $\PSL(2,25)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,19,26,4,17,2,6,22,18,5,21,12,16)(3,8,7,14,23,13,24,10,25,9,11,15,20), (1,21,5,9,4,2,10,16,12,18,14,25,3)(6,23,7,24,17,11,8,20,26,15,22,19,13) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 13: None

## 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

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 3, 3, 1, 1$ $650$ $3$ $( 1, 6,11)( 2,13,17)( 3,24, 5)( 4,15,25)( 7,19, 8)( 9,23,10)(12,18,14) (16,26,21)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $325$ $2$ $( 1,16)( 2,25)( 3,23)( 4,13)( 5, 9)( 6,26)( 7,18)( 8,12)(10,24)(11,21)(14,19) (15,17)$ $6, 6, 6, 6, 1, 1$ $650$ $6$ $( 1,21, 6,16,11,26)( 2,15,13,25,17, 4)( 3, 9,24,23, 5,10)( 7,12,19,18, 8,14)$ $4, 4, 4, 4, 4, 4, 1, 1$ $650$ $4$ $( 1,25,16, 2)( 3,19,23,14)( 4,26,13, 6)( 5, 7, 9,18)( 8,10,12,24)(11,15,21,17)$ $12, 12, 1, 1$ $650$ $12$ $( 1,13,21,25, 6,17,16, 4,11, 2,26,15)( 3,12, 9,19,24,18,23, 8, 5,14,10, 7)$ $12, 12, 1, 1$ $650$ $12$ $( 1, 4,21, 2, 6,15,16,13,11,25,26,17)( 3, 8, 9,14,24, 7,23,12, 5,19,10,18)$ $13, 13$ $600$ $13$ $( 1, 2,19, 4,25,26, 3,24, 9,21,22,15,13)( 5, 8,14,20,23,16, 6,11,10,18,12,17, 7)$ $13, 13$ $600$ $13$ $( 1, 9, 4,15, 3, 2,21,25,13,24,19,22,26)( 5,10,20,17, 6, 8,18,23, 7,11,14,12, 16)$ $13, 13$ $600$ $13$ $( 1,21,26, 2,22, 3,19,15,24, 4,13, 9,25)( 5,18,16, 8,12, 6,14,17,11,20, 7,10, 23)$ $13, 13$ $600$ $13$ $( 1,24, 2, 9,19,21, 4,22,25,15,26,13, 3)( 5,11, 8,10,14,18,20,12,23,17,16, 7, 6)$ $13, 13$ $600$ $13$ $( 1, 4, 3,21,13,19,26, 9,15, 2,25,24,22)( 5,20, 6,18, 7,14,16,10,17, 8,23,11, 12)$ $13, 13$ $600$ $13$ $( 1,15,21,24,26, 4, 2,13,22, 9, 3,25,19)( 5,17,18,11,16,20, 8, 7,12,10, 6,23, 14)$ $5, 5, 5, 5, 5, 1$ $312$ $5$ $( 1,26,20, 9,12)( 2,17, 3,16,21)( 4,19,25, 5,18)( 6, 8,24,11,14) ( 7,10,22,13,15)$ $5, 5, 5, 5, 5, 1$ $312$ $5$ $( 1,17,22, 6, 5)( 2,10,14,25,12)( 3,13, 8,18,26)( 4,20,16,15,24) ( 7,11,19, 9,21)$

## Group invariants

 Order: $7800=2^{3} \cdot 3 \cdot 5^{2} \cdot 13$ Cyclic: No Abelian: No Solvable: No GAP id: Data not available
 Character table:  2 3 3 2 2 2 2 2 . . . . . . . . 3 1 1 1 1 1 1 1 . . . . . . . . 5 2 . . . . . . . . . . . . 2 2 13 1 . . . . . . 1 1 1 1 1 1 . . 1a 2a 3a 4a 6a 12a 12b 13a 13b 13c 13d 13e 13f 5a 5b 2P 1a 1a 3a 2a 3a 6a 6a 13f 13e 13b 13a 13d 13c 5a 5b 3P 1a 2a 1a 4a 2a 4a 4a 13e 13f 13a 13b 13c 13d 5a 5b 5P 1a 2a 3a 4a 6a 12b 12a 13b 13a 13d 13c 13f 13e 1a 1a 7P 1a 2a 3a 4a 6a 12b 12a 13d 13c 13f 13e 13b 13a 5a 5b 11P 1a 2a 3a 4a 6a 12a 12b 13f 13e 13b 13a 13d 13c 5a 5b 13P 1a 2a 3a 4a 6a 12a 12b 1a 1a 1a 1a 1a 1a 5a 5b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 13 1 1 -1 1 -1 -1 . . . . . . 3 -2 X.3 13 1 1 -1 1 -1 -1 . . . . . . -2 3 X.4 24 . . . . . . B G F C D E -1 -1 X.5 24 . . . . . . C F E D G B -1 -1 X.6 24 . . . . . . D E B G F C -1 -1 X.7 24 . . . . . . E D G B C F -1 -1 X.8 24 . . . . . . F C D E B G -1 -1 X.9 24 . . . . . . G B C F E D -1 -1 X.10 25 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 . . X.11 26 2 -1 -2 -1 1 1 . . . . . . 1 1 X.12 26 2 -1 2 -1 -1 -1 . . . . . . 1 1 X.13 26 -2 2 . -2 . . . . . . . . 1 1 X.14 26 -2 -1 . 1 A -A . . . . . . 1 1 X.15 26 -2 -1 . 1 -A A . . . . . . 1 1 A = -E(12)^7+E(12)^11 = Sqrt(3) = r3 B = -E(13)-E(13)^12 C = -E(13)^6-E(13)^7 D = -E(13)^3-E(13)^10 E = -E(13)^2-E(13)^11 F = -E(13)^4-E(13)^9 G = -E(13)^5-E(13)^8