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Properties

Label 12T135
Order \(384\)
n \(12\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^4:C_3.D_4$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $135$
Group :  $C_2^4:C_3.D_4$
CHM label :  $[2^{6}]D_{6}=2wrD_{6}(6)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,11)(2,8)(3,9)(4,6)(5,7)(10,12)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
8:  $D_{4}$
12:  $D_{6}$
24:  $S_4$ x 3, $(C_6\times C_2):C_2$
48:  $S_4\times C_2$ x 3
96:  $V_4^2:S_3$, 12T49 x 3
192:  12T100

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_3$

Low degree siblings

12T135, 12T148 x 6, 16T765 x 2, 24T1003 x 6, 24T1004 x 3, 24T1005 x 3, 24T1008 x 3, 24T1010 x 3, 24T1123 x 6, 24T1173 x 6, 24T1174 x 6, 24T1175 x 2, 24T1176 x 12, 24T1177 x 12, 24T1178 x 2, 24T1179 x 6, 24T1180 x 3, 24T1181 x 2, 24T1182 x 2, 24T1183, 24T1250 x 12, 24T1251 x 12, 24T1252 x 3, 24T1253 x 3, 32T9368

Siblings are shown with degree $\leq 47$

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

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 6, 7)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 6, 7)( 8, 9)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 2, 3)( 6, 7)(10,11)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 2, 3)( 6, 7)( 8, 9)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $6$ $2$ $( 2, 3)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 4, 5)( 6, 7)( 8, 9)$
$ 2, 2, 2, 2, 2, 1, 1 $ $6$ $2$ $( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$
$ 2, 2, 2, 2, 2, 2 $ $24$ $2$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12)( 4,10, 5,11)( 6, 8)( 7, 9)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12)( 4,10)( 5,11)( 6, 8, 7, 9)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2)( 3,12)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 4, 2, 2, 2, 2 $ $24$ $4$ $( 1, 2,12, 3)( 4,10)( 5,11)( 6, 8)( 7, 9)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2,12, 3)( 4,10, 5,11)( 6, 8)( 7, 9)$
$ 4, 4, 2, 2 $ $24$ $4$ $( 1, 2,12, 3)( 4,10)( 5,11)( 6, 8, 7, 9)$
$ 4, 4, 4 $ $24$ $4$ $( 1, 2,12, 3)( 4,10, 5,11)( 6, 8, 7, 9)$
$ 3, 3, 3, 3 $ $32$ $3$ $( 1, 4, 8)( 2, 6,10)( 3, 7,11)( 5, 9,12)$
$ 6, 3, 3 $ $32$ $6$ $( 1, 4, 8)( 2, 6,10, 3, 7,11)( 5, 9,12)$
$ 6, 3, 3 $ $32$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 6,10)( 3, 7,11)$
$ 6, 6 $ $32$ $6$ $( 1, 4, 8,12, 5, 9)( 2, 6,10, 3, 7,11)$
$ 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$

Group invariants

Order:  $384=2^{7} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [384, 20100]
Character table: Data not available.