# Properties

 Label 37T4 Order $$148$$ n $$37$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $C_{37}:C_{4}$

## Group action invariants

 Degree $n$ : $37$ Transitive number $t$ : $4$ Group : $C_{37}:C_{4}$ Parity: $-1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (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), (1,31,36,6)(2,25,35,12)(3,19,34,18)(4,13,33,24)(5,7,32,30)(8,26,29,11)(9,20,28,17)(10,14,27,23)(15,21,22,16) $|\Aut(F/K)|$: $1$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 1$ $37$ $4$ $( 2, 7,37,32)( 3,13,36,26)( 4,19,35,20)( 5,25,34,14)( 6,31,33, 8)( 9,12,30,27) (10,18,29,21)(11,24,28,15)(16,17,23,22)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 1$ $37$ $4$ $( 2,32,37, 7)( 3,26,36,13)( 4,20,35,19)( 5,14,34,25)( 6, 8,33,31)( 9,27,30,12) (10,21,29,18)(11,15,28,24)(16,22,23,17)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1$ $37$ $2$ $( 2,37)( 3,36)( 4,35)( 5,34)( 6,33)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,27) (13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ $37$ $4$ $37$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$ $37$ $4$ $37$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$ $37$ $4$ $37$ $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$ $37$ $4$ $37$ $( 1, 5, 9,13,17,21,25,29,33,37, 4, 8,12,16,20,24,28,32,36, 3, 7,11,15,19,23, 27,31,35, 2, 6,10,14,18,22,26,30,34)$ $37$ $4$ $37$ $( 1, 6,11,16,21,26,31,36, 4, 9,14,19,24,29,34, 2, 7,12,17,22,27,32,37, 5,10, 15,20,25,30,35, 3, 8,13,18,23,28,33)$ $37$ $4$ $37$ $( 1, 9,17,25,33, 4,12,20,28,36, 7,15,23,31, 2,10,18,26,34, 5,13,21,29,37, 8, 16,24,32, 3,11,19,27,35, 6,14,22,30)$ $37$ $4$ $37$ $( 1,10,19,28,37, 9,18,27,36, 8,17,26,35, 7,16,25,34, 6,15,24,33, 5,14,23,32, 4,13,22,31, 3,12,21,30, 2,11,20,29)$ $37$ $4$ $37$ $( 1,11,21,31, 4,14,24,34, 7,17,27,37,10,20,30, 3,13,23,33, 6,16,26,36, 9,19, 29, 2,12,22,32, 5,15,25,35, 8,18,28)$ $37$ $4$ $37$ $( 1,16,31, 9,24, 2,17,32,10,25, 3,18,33,11,26, 4,19,34,12,27, 5,20,35,13,28, 6,21,36,14,29, 7,22,37,15,30, 8,23)$

## Group invariants

 Order: $148=2^{2} \cdot 37$ Cyclic: No Abelian: No Solvable: Yes GAP id: [148, 3]
 Character table:  2 2 2 2 2 . . . . . . . . . 37 1 . . . 1 1 1 1 1 1 1 1 1 1a 4a 4b 2a 37a 37b 37c 37d 37e 37f 37g 37h 37i 2P 1a 2a 2a 1a 37b 37d 37a 37f 37h 37i 37c 37g 37e 3P 1a 4b 4a 2a 37c 37a 37g 37b 37i 37d 37h 37e 37f 5P 1a 4a 4b 2a 37e 37h 37i 37g 37b 37c 37f 37d 37a 7P 1a 4b 4a 2a 37e 37h 37i 37g 37b 37c 37f 37d 37a 11P 1a 4b 4a 2a 37f 37i 37d 37e 37c 37h 37b 37a 37g 13P 1a 4a 4b 2a 37d 37f 37b 37i 37g 37e 37a 37c 37h 17P 1a 4a 4b 2a 37g 37c 37h 37a 37f 37b 37e 37i 37d 19P 1a 4b 4a 2a 37c 37a 37g 37b 37i 37d 37h 37e 37f 23P 1a 4b 4a 2a 37h 37g 37e 37c 37d 37a 37i 37f 37b 29P 1a 4a 4b 2a 37f 37i 37d 37e 37c 37h 37b 37a 37g 31P 1a 4b 4a 2a 37a 37b 37c 37d 37e 37f 37g 37h 37i 37P 1a 4a 4b 2a 1a 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 1 1 1 1 1 X.3 1 A -A -1 1 1 1 1 1 1 1 1 1 X.4 1 -A A -1 1 1 1 1 1 1 1 1 1 X.5 4 . . . B F D E I C H J G X.6 4 . . . C G E I D J F B H X.7 4 . . . D B H F G E J I C X.8 4 . . . E C F G H I B D J X.9 4 . . . F E B C J G D H I X.10 4 . . . G I C J B H E F D X.11 4 . . . H D J B C F I G E X.12 4 . . . I J G H F D C E B X.13 4 . . . J H I D E B G C F A = -E(4) = -Sqrt(-1) = -i B = E(37)^9+E(37)^17+E(37)^20+E(37)^28 C = E(37)^2+E(37)^12+E(37)^25+E(37)^35 D = E(37)^10+E(37)^14+E(37)^23+E(37)^27 E = E(37)+E(37)^6+E(37)^31+E(37)^36 F = E(37)^3+E(37)^18+E(37)^19+E(37)^34 G = E(37)^4+E(37)^13+E(37)^24+E(37)^33 H = E(37)^5+E(37)^7+E(37)^30+E(37)^32 I = E(37)^8+E(37)^11+E(37)^26+E(37)^29 J = E(37)^15+E(37)^16+E(37)^21+E(37)^22