LMFDB - Galois group: 32T2247

Show commands: Magma

magma:G := TransitiveGroup(32, 2247);

Group invariants

Abstract group:	$(D_4\times C_2^3):C_4$	magma:IdentifyGroup(G);
Order:	$256=2^{8}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	$4$	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$32$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$2247$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$8$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,27,2,28)(3,26,4,25)(5,30,8,31)(6,29,7,32)(9,18,10,17)(11,19,12,20)(13,22,16,23)(14,21,15,24)$, $(1,14,4,16)(2,13,3,15)(5,11,7,9)(6,12,8,10)(17,25)(18,26)(19,28)(20,27)(21,29)(22,30)(23,31)(24,32)$, $(1,26,5,29,4,28,7,31)(2,25,6,30,3,27,8,32)(9,20,14,22,11,17,16,24)(10,19,13,21,12,18,15,23)$, $(1,19,5,23,4,18,7,21)(2,20,6,24,3,17,8,22)(9,27,14,30,11,25,16,32)(10,28,13,29,12,26,15,31)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 15
$4$: $C_4$ x 8, $C_2^2$ x 35
$8$: $D_{4}$ x 8, $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$: $D_4\times C_2$ x 12, $C_2^2:C_4$ x 16, $C_4\times C_2^2$ x 14, $C_2^4$
$32$: $C_2^3 : C_4 $ x 4, $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
$64$: 16T76 x 6, 32T262
$128$: 32T1107

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 15
$4$:	$C_4$ x 8, $C_2^2$ x 35
$8$:	$D_{4}$ x 8, $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$:	$D_4\times C_2$ x 12, $C_2^2:C_4$ x 16, $C_4\times C_2^2$ x 14, $C_2^4$
$32$:	$C_2^3 : C_4 $ x 4, $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
$64$:	16T76 x 6, 32T262
$128$:	32T1107

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4
Degree 8: $C_4\times C_2$, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4
Degree 16: $C_2 \times (C_2^2:C_4)$, 16T452, 16T469

Low degree siblings

16T452 x 4, 16T468 x 4, 16T469 x 4, 32T2247 x 3, 32T2248 x 2, 32T2249 x 2, 32T2250 x 2, 32T2251 x 4, 32T2252 x 2, 32T2253 x 2, 32T2321 x 4, 32T2322 x 2, 32T2323 x 2, 32T2324 x 2, 32T2325 x 2, 32T2326 x 2, 32T2327 x 2, 32T2328 x 2, 32T2329 x 2, 32T2330 x 8, 32T2331 x 8, 32T2332 x 2, 32T2333 x 2, 32T7415 x 4
Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{32}$	$1$	$1$	$0$	$()$
2A	$2^{16}$	$1$	$2$	$16$	$( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2B	$2^{8},1^{16}$	$2$	$2$	$8$	$( 9,11)(10,12)(13,15)(14,16)(25,27)(26,28)(29,31)(30,32)$
2C	$2^{8},1^{16}$	$2$	$2$	$8$	$( 1, 4)( 2, 3)( 5, 7)( 6, 8)(25,27)(26,28)(29,31)(30,32)$
2D	$2^{16}$	$2$	$2$	$16$	$( 1,11)( 2,12)( 3,10)( 4, 9)( 5,16)( 6,15)( 7,14)( 8,13)(17,28)(18,27)(19,25)(20,26)(21,30)(22,29)(23,32)(24,31)$
2E	$2^{16}$	$2$	$2$	$16$	$( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)(17,28)(18,27)(19,25)(20,26)(21,30)(22,29)(23,32)(24,31)$
2F	$2^{8},1^{16}$	$2$	$2$	$8$	$( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2G	$2^{16}$	$4$	$2$	$16$	$( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,22)(20,21)(25,29)(26,30)(27,31)(28,32)$
2H	$2^{16}$	$4$	$2$	$16$	$( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,13)(10,14)(11,15)(12,16)(17,23)(18,24)(19,22)(20,21)(25,31)(26,32)(27,29)(28,30)$
2I	$2^{16}$	$4$	$2$	$16$	$( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)(17,20)(18,19)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2J	$2^{16}$	$4$	$2$	$16$	$( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)$
2K	$2^{16}$	$4$	$2$	$16$	$( 1,13)( 2,14)( 3,16)( 4,15)( 5,10)( 6, 9)( 7,12)( 8,11)(17,32)(18,31)(19,29)(20,30)(21,26)(22,25)(23,28)(24,27)$
2L	$2^{8},1^{16}$	$4$	$2$	$8$	$( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,15)(10,16)(11,13)(12,14)$
2M	$2^{16}$	$4$	$2$	$16$	$( 1,16)( 2,15)( 3,13)( 4,14)( 5, 9)( 6,10)( 7,11)( 8,12)(17,31)(18,32)(19,30)(20,29)(21,27)(22,28)(23,25)(24,26)$
2N	$2^{16}$	$4$	$2$	$16$	$( 1,10)( 2, 9)( 3,11)( 4,12)( 5,15)( 6,16)( 7,13)( 8,14)(17,27)(18,28)(19,26)(20,25)(21,31)(22,32)(23,29)(24,30)$
2O	$2^{16}$	$4$	$2$	$16$	$( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,32)(30,31)$
2P	$2^{12},1^{8}$	$8$	$2$	$12$	$( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,15)(10,16)(11,13)(12,14)(17,20)(18,19)(21,23)(22,24)$
2Q	$2^{16}$	$8$	$2$	$16$	$( 1,15)( 2,16)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(17,26)(18,25)(19,27)(20,28)(21,32)(22,31)(23,30)(24,29)$
4A	$4^{8}$	$2$	$4$	$24$	$( 1,11, 4, 9)( 2,12, 3,10)( 5,16, 7,14)( 6,15, 8,13)(17,28,20,26)(18,27,19,25)(21,30,23,32)(22,29,24,31)$
4B	$4^{8}$	$2$	$4$	$24$	$( 1, 9, 4,11)( 2,10, 3,12)( 5,14, 7,16)( 6,13, 8,15)(17,28,20,26)(18,27,19,25)(21,30,23,32)(22,29,24,31)$
4C	$4^{8}$	$4$	$4$	$24$	$( 1,13, 4,15)( 2,14, 3,16)( 5,10, 7,12)( 6, 9, 8,11)(17,32,20,30)(18,31,19,29)(21,26,23,28)(22,25,24,27)$
4D	$4^{8}$	$4$	$4$	$24$	$( 1,10, 4,12)( 2, 9, 3,11)( 5,15, 7,13)( 6,16, 8,14)(17,27,20,25)(18,28,19,26)(21,31,23,29)(22,32,24,30)$
4E	$4^{8}$	$4$	$4$	$24$	$( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,16,11,14)(10,15,12,13)(17,24,20,22)(18,23,19,21)(25,32,27,30)(26,31,28,29)$
4F	$4^{8}$	$4$	$4$	$24$	$( 1,16, 4,14)( 2,15, 3,13)( 5, 9, 7,11)( 6,10, 8,12)(17,31,20,29)(18,32,19,30)(21,27,23,25)(22,28,24,26)$
4G	$4^{8}$	$4$	$4$	$24$	$( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,14,11,16)(10,13,12,15)(17,24,20,22)(18,23,19,21)(25,30,27,32)(26,29,28,31)$
4H	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,16,11,14)(10,15,12,13)(17,19)(18,20)(21,22)(23,24)(25,28)(26,27)(29,30)(31,32)$
4I	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1,14)( 2,13)( 3,15)( 4,16)( 5,11)( 6,12)( 7, 9)( 8,10)(17,25,20,27)(18,26,19,28)(21,29,23,31)(22,30,24,32)$
4J	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,14,11,16)(10,13,12,15)(17,19)(18,20)(21,22)(23,24)(25,26)(27,28)(29,32)(30,31)$
4K	$4^{4},2^{8}$	$8$	$4$	$20$	$( 1,14, 4,16)( 2,13, 3,15)( 5,11, 7, 9)( 6,12, 8,10)(17,25)(18,26)(19,28)(20,27)(21,29)(22,30)(23,31)(24,32)$
4L	$4^{8}$	$8$	$4$	$24$	$( 1,15, 4,13)( 2,16, 3,14)( 5,12, 7,10)( 6,11, 8, 9)(17,26,20,28)(18,25,19,27)(21,32,23,30)(22,31,24,29)$
4M1	$4^{8}$	$8$	$4$	$24$	$( 1,23, 3,22)( 2,24, 4,21)( 5,18, 6,17)( 7,19, 8,20)( 9,30,12,31)(10,29,11,32)(13,26,14,25)(15,28,16,27)$
4M-1	$4^{8}$	$8$	$4$	$24$	$( 1,22, 3,23)( 2,21, 4,24)( 5,17, 6,18)( 7,20, 8,19)( 9,31,12,30)(10,32,11,29)(13,25,14,26)(15,27,16,28)$
4N1	$4^{8}$	$8$	$4$	$24$	$( 1,32, 2,31)( 3,29, 4,30)( 5,27, 8,26)( 6,28, 7,25)( 9,23,10,24)(11,21,12,22)(13,17,16,19)(14,18,15,20)$
4N-1	$4^{8}$	$8$	$4$	$24$	$( 1,29, 2,30)( 3,32, 4,31)( 5,28, 8,25)( 6,27, 7,26)( 9,22,10,21)(11,24,12,23)(13,18,16,20)(14,17,15,19)$
4O1	$4^{8}$	$8$	$4$	$24$	$( 1,23, 3,22)( 2,24, 4,21)( 5,18, 6,17)( 7,19, 8,20)( 9,32,12,29)(10,31,11,30)(13,28,14,27)(15,26,16,25)$
4O-1	$4^{8}$	$8$	$4$	$24$	$( 1,22, 3,23)( 2,21, 4,24)( 5,17, 6,18)( 7,20, 8,19)( 9,29,12,32)(10,30,11,31)(13,27,14,28)(15,25,16,26)$
4P1	$4^{8}$	$8$	$4$	$24$	$( 1,32, 3,29)( 2,31, 4,30)( 5,27, 6,28)( 7,25, 8,26)( 9,21,12,24)(10,22,11,23)(13,20,14,19)(15,17,16,18)$
4P-1	$4^{8}$	$8$	$4$	$24$	$( 1,29, 3,32)( 2,30, 4,31)( 5,28, 6,27)( 7,26, 8,25)( 9,24,12,21)(10,23,11,22)(13,19,14,20)(15,18,16,17)$
8A1	$8^{4}$	$8$	$8$	$28$	$( 1,24, 5,20, 4,22, 7,17)( 2,23, 6,19, 3,21, 8,18)( 9,29,14,28,11,31,16,26)(10,30,13,27,12,32,15,25)$
8A-1	$8^{4}$	$8$	$8$	$28$	$( 1,21, 5,19, 4,23, 7,18)( 2,22, 6,20, 3,24, 8,17)( 9,32,14,27,11,30,16,25)(10,31,13,28,12,29,15,26)$
8B1	$8^{4}$	$8$	$8$	$28$	$( 1,31, 7,28, 4,29, 5,26)( 2,32, 8,27, 3,30, 6,25)( 9,24,16,17,11,22,14,20)(10,23,15,18,12,21,13,19)$
8B-1	$8^{4}$	$8$	$8$	$28$	$( 1,30, 7,27, 4,32, 5,25)( 2,29, 8,28, 3,31, 6,26)( 9,21,16,18,11,23,14,19)(10,22,15,17,12,24,13,20)$
8C1	$8^{4}$	$8$	$8$	$28$	$( 1,24, 5,20, 4,22, 7,17)( 2,23, 6,19, 3,21, 8,18)( 9,31,14,26,11,29,16,28)(10,32,13,25,12,30,15,27)$
8C-1	$8^{4}$	$8$	$8$	$28$	$( 1,21, 5,19, 4,23, 7,18)( 2,22, 6,20, 3,24, 8,17)( 9,30,14,25,11,32,16,27)(10,29,13,26,12,31,15,28)$
8D1	$8^{4}$	$8$	$8$	$28$	$( 1,31, 5,26, 4,29, 7,28)( 2,32, 6,25, 3,30, 8,27)( 9,22,14,17,11,24,16,20)(10,21,13,18,12,23,15,19)$
8D-1	$8^{4}$	$8$	$8$	$28$	$( 1,30, 5,25, 4,32, 7,27)( 2,29, 6,26, 3,31, 8,28)( 9,23,14,18,11,21,16,19)(10,24,13,17,12,22,15,20)$

Malle's constant $a(G)$: $1/8$

magma:ConjugacyClasses(G);

Character table

46 x 46 character table

magma:CharacterTable(G);

Regular extensions

Data not computed

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	32T2247

Degree	$32$
Order	$256$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$(D_4\times C_2^3):C_4$