Show commands:
Magma
magma: G := TransitiveGroup(38, 28);
Group invariants
| Abstract group: | $C_{19}^2:(C_6\times S_3)$ | magma: IdentifyGroup(G);
| |
| Order: | $12996=2^{2} \cdot 3^{2} \cdot 19^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | magma: NilpotencyClass(G);
|
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| |
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | $(1,24,3,28,6,34)(2,26,14,31,13,29)(4,30,17,37,8,38)(5,32,9,21,15,33)(7,36,12,27,10,23)(11,25,18,20,19,22)(16,35)$, $(1,38,6,23,3,32)(2,35,13,21,14,37)(4,29,8,36,17,28)(5,26,15,34,9,33)(7,20,10,30,12,24)(11,27,19,22,18,25)(16,31)$ | magma: Generators(G);
|
Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $36$: $C_6\times S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: 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
| Label | Cycle Type | Size | Order | Index | Representative |
| 1A | $1^{38}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{19}$ | $57$ | $2$ | $19$ | $( 1,20)( 2,25)( 3,30)( 4,35)( 5,21)( 6,26)( 7,31)( 8,36)( 9,22)(10,27)(11,32)(12,37)(13,23)(14,28)(15,33)(16,38)(17,24)(18,29)(19,34)$ |
| 2B | $2^{19}$ | $57$ | $2$ | $19$ | $( 1,20)( 2,34)( 3,29)( 4,24)( 5,38)( 6,33)( 7,28)( 8,23)( 9,37)(10,32)(11,27)(12,22)(13,36)(14,31)(15,26)(16,21)(17,35)(18,30)(19,25)$ |
| 2C | $2^{18},1^{2}$ | $361$ | $2$ | $18$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(20,24)(21,23)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)$ |
| 3A1 | $3^{6},1^{20}$ | $38$ | $3$ | $12$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$ |
| 3A-1 | $3^{6},1^{20}$ | $38$ | $3$ | $12$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)$ |
| 3B1 | $3^{12},1^{2}$ | $361$ | $3$ | $24$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(20,38,27)(21,30,34)(23,33,29)(24,25,36)(26,28,31)(32,37,35)$ |
| 3B-1 | $3^{12},1^{2}$ | $361$ | $3$ | $24$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(20,27,38)(21,34,30)(23,29,33)(24,36,25)(26,31,28)(32,35,37)$ |
| 3C | $3^{12},1^{2}$ | $722$ | $3$ | $24$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(20,38,27)(21,30,34)(23,33,29)(24,25,36)(26,28,31)(32,37,35)$ |
| 6A1 | $6^{6},1^{2}$ | $361$ | $6$ | $30$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(20,36,38,24,27,25)(21,29,30,23,34,33)(26,32,28,37,31,35)$ |
| 6A-1 | $6^{6},1^{2}$ | $361$ | $6$ | $30$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(20,25,27,24,38,36)(21,33,34,23,30,29)(26,35,31,37,28,32)$ |
| 6B | $6^{3},2^{9},1^{2}$ | $722$ | $6$ | $24$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(20,24)(21,23)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)$ |
| 6C1 | $6^{3},2^{9},1^{2}$ | $722$ | $6$ | $24$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(20,24)(21,23)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)$ |
| 6C-1 | $6^{6},1^{2}$ | $722$ | $6$ | $30$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(20,25,27,24,38,36)(21,33,34,23,30,29)(26,35,31,37,28,32)$ |
| 6D1 | $6^{6},2$ | $1083$ | $6$ | $31$ | $( 1,20, 3,35, 6,29)( 2,37,14,32,13,34)( 4,33,17,26, 8,25)( 5,31, 9,23,15,30)( 7,27,12,36,10,21)(11,38,18,24,19,22)(16,28)$ |
| 6D-1 | $6^{6},2$ | $1083$ | $6$ | $31$ | $( 1,20, 3,33,17,29)( 2,36,10,31, 9,34)( 4,30, 5,27,12,25)( 6,24,19,23,15,35)( 7,21)( 8,37,14,38,18,26)(11,28,16,32,13,22)$ |
| 6E1 | $6^{6},2$ | $1083$ | $6$ | $31$ | $( 1,20,19,36,12,34)( 2,23, 7,38, 4,29)( 3,26,14,21,15,24)( 5,32, 9,25,18,33)( 6,35,16,27,10,28)( 8,22,11,31,13,37)(17,30)$ |
| 6E-1 | $6^{6},2$ | $1083$ | $6$ | $31$ | $( 1,20,19,37, 8,34)( 2,22,11,21,15,29)( 3,24)( 4,26,14,27,10,38)( 5,28, 6,30,17,33)( 7,32, 9,36,12,23)(13,25,18,35,16,31)$ |
| 19A1 | $19,1^{19}$ | $12$ | $19$ | $18$ | $(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| 19A2 | $19,1^{19}$ | $12$ | $19$ | $18$ | $(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$ |
| 19A4 | $19,1^{19}$ | $12$ | $19$ | $18$ | $(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| 19B1 | $19^{2}$ | $18$ | $19$ | $36$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$ |
| 19B2 | $19^{2}$ | $18$ | $19$ | $36$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| 19B4 | $19^{2}$ | $18$ | $19$ | $36$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| 19C1 | $19^{2}$ | $18$ | $19$ | $36$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| 19C2 | $19^{2}$ | $18$ | $19$ | $36$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| 19C4 | $19^{2}$ | $18$ | $19$ | $36$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21)$ |
| 19D1 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,36,33,30,27,24,21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| 19D2 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| 19D4 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| 19E1 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| 19E2 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| 19E4 | $19^{2}$ | $36$ | $19$ | $36$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| 38A1 | $38$ | $342$ | $38$ | $37$ | $( 1,20, 4,24, 7,28,10,32,13,36,16,21,19,25, 3,29, 6,33, 9,37,12,22,15,26,18,30, 2,34, 5,38, 8,23,11,27,14,31,17,35)$ |
| 38A3 | $38$ | $342$ | $38$ | $37$ | $( 1,20, 5,21, 9,22,13,23,17,24, 2,25, 6,26,10,27,14,28,18,29, 3,30, 7,31,11,32,15,33,19,34, 4,35, 8,36,12,37,16,38)$ |
| 38A9 | $38$ | $342$ | $38$ | $37$ | $( 1,20,16,21,12,22, 8,23, 4,24,19,25,15,26,11,27, 7,28, 3,29,18,30,14,31,10,32, 6,33, 2,34,17,35,13,36, 9,37, 5,38)$ |
| 38B1 | $38$ | $342$ | $38$ | $37$ | $( 1,20,19,25,18,30,17,35,16,21,15,26,14,31,13,36,12,22,11,27,10,32, 9,37, 8,23, 7,28, 6,33, 5,38, 4,24, 3,29, 2,34)$ |
| 38B3 | $38$ | $342$ | $38$ | $37$ | $( 1,20, 9,22,17,24, 6,26,14,28, 3,30,11,32,19,34, 8,36,16,38, 5,21,13,23, 2,25,10,27,18,29, 7,31,15,33, 4,35,12,37)$ |
| 38B9 | $38$ | $342$ | $38$ | $37$ | $( 1,20, 3,30, 5,21, 7,31, 9,22,11,32,13,23,15,33,17,24,19,34, 2,25, 4,35, 6,26, 8,36,10,27,12,37,14,28,16,38,18,29)$ |
| 57A1 | $19,3^{6},1$ | $228$ | $57$ | $30$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| 57A-1 | $19,3^{6},1$ | $228$ | $57$ | $30$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$ |
| 57A2 | $19,3^{6},1$ | $228$ | $57$ | $30$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| 57A-2 | $19,3^{6},1$ | $228$ | $57$ | $30$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| 57A4 | $19,3^{6},1$ | $228$ | $57$ | $30$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$ |
| 57A-4 | $19,3^{6},1$ | $228$ | $57$ | $30$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Character table
45 x 45 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed