# Properties

 Label 12T38 Order $$72$$ n $$12$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_3^2:D_4$

# Related objects

## Group action invariants

 Degree $n$ : $12$ Transitive number $t$ : $38$ Group : $C_3^2:D_4$ CHM label : $1/2[3^{2}:2]cD(4)$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,11)(2,10)(3,9)(4,8)(5,7), (2,6,10)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
8:  $D_{4}$
12:  $D_{6}$ x 2
24:  $D_{12}$, $(C_6\times C_2):C_2$
36:  $S_3^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: $S_3^2$

## Low degree siblings

12T38, 24T74, 36T33, 36T38

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 1, 1$ $18$ $2$ $( 2, 4)( 3,11)( 5, 9)( 6,12)( 8,10)$ $3, 3, 1, 1, 1, 1, 1, 1$ $4$ $3$ $( 2, 6,10)( 4, 8,12)$ $2, 2, 2, 2, 2, 2$ $6$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$ $12$ $6$ $12$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$ $6, 6$ $6$ $6$ $( 1, 2, 5,10, 9, 6)( 3,12, 7, 8,11, 4)$ $4, 4, 4$ $6$ $4$ $( 1, 2, 7, 8)( 3, 4, 9,10)( 5, 6,11,12)$ $6, 6$ $6$ $6$ $( 1, 2, 9, 6, 5,10)( 3,12,11, 4, 7, 8)$ $12$ $6$ $12$ $( 1, 2,11,12, 9,10, 7, 8, 5, 6, 3, 4)$ $6, 6$ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ $6, 2, 2, 2$ $4$ $6$ $( 1, 3, 5, 7, 9,11)( 2, 8)( 4,10)( 6,12)$ $6, 6$ $2$ $6$ $( 1, 3, 5, 7, 9,11)( 2,12,10, 8, 6, 4)$ $3, 3, 3, 3$ $2$ $3$ $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ $3, 3, 3, 3$ $2$ $3$ $( 1, 5, 9)( 2,10, 6)( 3, 7,11)( 4,12, 8)$ $2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 23]
 Character table:  2 3 2 1 2 2 2 2 2 2 2 1 2 2 2 3 3 2 . 2 1 1 1 1 1 1 2 2 2 2 2 2 1a 2a 3a 2b 12a 6a 4a 6b 12b 6c 6d 6e 3b 3c 2c 2P 1a 1a 3a 1a 6c 3c 2c 3c 6c 3b 3a 3c 3b 3c 1a 3P 1a 2a 1a 2b 4a 2b 4a 2b 4a 2c 2c 2c 1a 1a 2c 5P 1a 2a 3a 2b 12b 6b 4a 6a 12a 6c 6d 6e 3b 3c 2c 7P 1a 2a 3a 2b 12b 6a 4a 6b 12a 6c 6d 6e 3b 3c 2c 11P 1a 2a 3a 2b 12a 6b 4a 6a 12b 6c 6d 6e 3b 3c 2c X.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 X.3 1 -1 1 1 -1 1 -1 1 -1 1 1 1 1 1 1 X.4 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 X.5 2 . -1 -2 . 1 . 1 . 2 -1 -1 2 -1 2 X.6 2 . -1 2 . -1 . -1 . 2 -1 -1 2 -1 2 X.7 2 . 2 . . . . . . -2 -2 -2 2 2 -2 X.8 2 . -1 . -1 . 2 . -1 -1 -1 2 -1 2 2 X.9 2 . -1 . 1 . -2 . 1 -1 -1 2 -1 2 2 X.10 2 . -1 . . B . -B . -2 1 1 2 -1 -2 X.11 2 . -1 . . -B . B . -2 1 1 2 -1 -2 X.12 2 . -1 . A . . . -A 1 1 -2 -1 2 -2 X.13 2 . -1 . -A . . . A 1 1 -2 -1 2 -2 X.14 4 . 1 . . . . . . -2 1 -2 -2 -2 4 X.15 4 . 1 . . . . . . 2 -1 2 -2 -2 -4 A = -E(12)^7+E(12)^11 = Sqrt(3) = r3 B = -E(3)+E(3)^2 = -Sqrt(-3) = -i3