# Properties

 Label 37T2 Order $$74$$ n $$37$$ Cyclic No Abelian No Solvable Yes Primitive Yes $p$-group No Group: $D_{37}$

## Group action invariants

 Degree $n$ : $37$ Transitive number $t$ : $2$ Group : $D_{37}$ 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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) $|\Aut(F/K)|$: $1$

## Low degree resolvents

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

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

## Group invariants

 Order: $74=2 \cdot 37$ Cyclic: No Abelian: No Solvable: Yes GAP id: [74, 1]
 Character table:  2 1 1 . . . . . . . . . . . . . . . . 37 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 37a 37b 37c 37d 37e 37f 37g 37h 37i 37j 37k 37l 37m 37n 37o 37p 2P 1a 1a 37b 37d 37f 37h 37j 37l 37n 37p 37r 37q 37o 37m 37k 37i 37g 37e 3P 1a 2a 37c 37f 37i 37l 37o 37r 37p 37m 37j 37g 37d 37a 37b 37e 37h 37k 5P 1a 2a 37e 37j 37o 37q 37l 37g 37b 37c 37h 37m 37r 37n 37i 37d 37a 37f 7P 1a 2a 37g 37n 37p 37i 37b 37e 37l 37r 37k 37d 37c 37j 37q 37m 37f 37a 11P 1a 2a 37k 37o 37d 37g 37r 37h 37c 37n 37l 37a 37j 37p 37e 37f 37q 37i 13P 1a 2a 37m 37k 37b 37o 37i 37d 37q 37g 37f 37r 37e 37h 37p 37c 37j 37n 17P 1a 2a 37q 37c 37n 37f 37k 37i 37h 37l 37e 37o 37b 37r 37a 37p 37d 37m 19P 1a 2a 37r 37a 37q 37b 37p 37c 37o 37d 37n 37e 37m 37f 37l 37g 37k 37h 23P 1a 2a 37n 37i 37e 37r 37d 37j 37m 37a 37o 37h 37f 37q 37c 37k 37l 37b 29P 1a 2a 37h 37p 37m 37e 37c 37k 37r 37j 37b 37f 37n 37o 37g 37a 37i 37q 31P 1a 2a 37f 37l 37r 37m 37g 37a 37e 37k 37q 37n 37h 37b 37d 37j 37p 37o 37P 1a 2a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a X.1 1 1 1 1 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 1 1 1 1 1 X.3 2 . A P B J I C R G M H O E K L D F X.4 2 . B C M E D Q F K H R J A P I G O X.5 2 . C E Q K R A I O N L G P J H F D X.6 2 . D R G L A F C M K P N I H E B Q X.7 2 . E K A O L P H D B M F J G N I R X.8 2 . F I O H C D A N J E M R L P Q B X.9 2 . G F K I B O Q H P C L D R A M N X.10 2 . H N R B K L J C F O A M Q G P E X.11 2 . I H D N E R P B G K Q L M J A C X.12 2 . J G E F N K M I A B R O D Q L H X.13 2 . K O P D M J N R C Q I G F B H L X.14 2 . L M I Q J H K A D G C N B O E P X.15 2 . M Q H A G N O P R F E B C D K J X.16 2 . N B L C O M G E I D P Q A F J K X.17 2 . O D J R Q G B L E A H F I C N M X.18 2 . P J C G H E L F Q N D K O M R I X.19 2 . Q A N P F B D J L I K C E R O G X.20 2 . R L F M P I E Q O J B H N K C A 2 . . 37 1 1 37q 37r 2P 37c 37a 3P 37n 37q 5P 37k 37p 7P 37h 37o 11P 37b 37m 13P 37a 37l 17P 37g 37j 19P 37j 37i 23P 37p 37g 29P 37l 37d 31P 37i 37c 37P 1a 1a X.1 1 1 X.2 1 1 X.3 N Q X.4 L N X.5 M B X.6 J O X.7 Q C X.8 K G X.9 E J X.10 D I X.11 O F X.12 C P X.13 A E X.14 F R X.15 I L X.16 R H X.17 P K X.18 B A X.19 H M X.20 G D A = E(37)^2+E(37)^35 B = E(37)^6+E(37)^31 C = E(37)^12+E(37)^25 D = E(37)^7+E(37)^30 E = E(37)^13+E(37)^24 F = E(37)^5+E(37)^32 G = E(37)^16+E(37)^21 H = E(37)^17+E(37)^20 I = E(37)^10+E(37)^27 J = E(37)^8+E(37)^29 K = E(37)^11+E(37)^26 L = E(37)^9+E(37)^28 M = E(37)^18+E(37)^19 N = E(37)^3+E(37)^34 O = E(37)^15+E(37)^22 P = E(37)^4+E(37)^33 Q = E(37)+E(37)^36 R = E(37)^14+E(37)^23