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Group invariants
Abstract group: | $C_4^2.\GL(2,\mathbb{Z}/4)$ |
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Order: | $1536=2^{9} \cdot 3$ |
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Cyclic: | no |
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Abelian: | no |
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Solvable: | yes |
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Nilpotency class: | not nilpotent |
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Group action invariants
Degree $n$: | $24$ |
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Transitive number $t$: | $4563$ |
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Parity: | $-1$ |
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Primitive: | no |
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$\card{\Aut(F/K)}$: | $2$ |
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Generators: | $(1,7)(2,8)(3,5,4,6)(9,21,16,19,10,22,15,20)(11,24,14,17,12,23,13,18)$, $(1,13,23,3,16,21)(2,14,24,4,15,22)(5,10,20,7,12,17)(6,9,19,8,11,18)$ |
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Low degree resolvents
$\card{(G/N)}$ 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 $384$: 12T135 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 4: None
Degree 6: $S_4$, $S_4\times C_2$ x 2
Degree 8: None
Degree 12: 12T101
Low degree siblings
24T4534 x 2, 24T4537 x 2, 24T4554 x 2, 24T4563, 24T4740 x 4, 24T4745 x 4Siblings 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 | $1^{24}$ | $1$ | $1$ | $0$ | $()$ |
2A | $2^{8},1^{8}$ | $3$ | $2$ | $8$ | $( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$ |
2B | $2^{6},1^{12}$ | $4$ | $2$ | $6$ | $( 1, 2)( 3, 4)(13,14)(15,16)(17,18)(19,20)$ |
2C | $2^{12}$ | $8$ | $2$ | $12$ | $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)$ |
2D | $2^{8},1^{8}$ | $12$ | $2$ | $8$ | $( 9,15)(10,16)(11,14)(12,13)(17,21)(18,22)(19,24)(20,23)$ |
2E | $2^{12}$ | $12$ | $2$ | $12$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16)(10,15)(11,13)(12,14)(17,21)(18,22)(19,24)(20,23)$ |
2F | $2^{10},1^{4}$ | $12$ | $2$ | $10$ | $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,15)(10,16)(11,14)(12,13)(19,20)(23,24)$ |
2G | $2^{10},1^{4}$ | $12$ | $2$ | $10$ | $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,10)(15,16)(17,22)(18,21)(19,23)(20,24)$ |
2H | $2^{8},1^{8}$ | $24$ | $2$ | $8$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)(11,12)(15,16)(19,20)(21,22)$ |
2I | $2^{12}$ | $24$ | $2$ | $12$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,13)(10,14)(11,16)(12,15)(17,24)(18,23)(19,21)(20,22)$ |
2J | $2^{11},1^{2}$ | $48$ | $2$ | $11$ | $( 1,16)( 2,15)( 3,13)( 4,14)( 5,11)( 6,12)( 7,10)( 8, 9)(17,18)(21,22)(23,24)$ |
3A | $3^{8}$ | $128$ | $3$ | $16$ | $( 1,21,12)( 2,22,11)( 3,23,10)( 4,24, 9)( 5,17,16)( 6,18,15)( 7,20,13)( 8,19,14)$ |
4A | $4^{4},2^{4}$ | $6$ | $4$ | $16$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23)$ |
4B | $4^{4},1^{8}$ | $6$ | $4$ | $12$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$ |
4C | $4^{4},2^{2},1^{4}$ | $12$ | $4$ | $14$ | $( 1, 4, 2, 3)( 5, 8, 6, 7)(13,14)(15,16)(17,19,18,20)(21,23,22,24)$ |
4D | $4^{4},2^{2},1^{4}$ | $12$ | $4$ | $14$ | $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,15,10,16)(11,14,12,13)(17,18)(23,24)$ |
4E | $4^{4},2^{2},1^{4}$ | $12$ | $4$ | $14$ | $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)(17,18)(21,22)$ |
4F | $4^{4},2^{2},1^{4}$ | $24$ | $4$ | $14$ | $( 1, 2)( 3, 4)( 9,15,10,16)(11,14,12,13)(17,22,18,21)(19,23,20,24)$ |
4G | $4^{4},2^{4}$ | $24$ | $4$ | $16$ | $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,16,10,15)(11,14,12,13)(17,22,18,21)(19,24,20,23)$ |
4H | $4^{4},2^{4}$ | $24$ | $4$ | $16$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,14)(10,13)(11,16)(12,15)(17,22,18,21)(19,23,20,24)$ |
4I | $4^{4},2^{4}$ | $24$ | $4$ | $16$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,15)(10,16)(11,14)(12,13)(17,23,18,24)(19,21,20,22)$ |
4J | $4^{4},2,1^{6}$ | $48$ | $4$ | $13$ | $( 1,14, 2,13)( 3,16, 4,15)( 5, 9, 6,10)( 7,11, 8,12)(19,20)$ |
4K | $4^{3},2^{6}$ | $96$ | $4$ | $15$ | $( 1,17, 2,18)( 3,20, 4,19)( 5,21)( 6,22)( 7,24)( 8,23)( 9,11)(10,12)(13,16,14,15)$ |
4L | $4^{5},2^{2}$ | $96$ | $4$ | $17$ | $( 1,13, 7,12)( 2,14, 8,11)( 3,15, 6, 9)( 4,16, 5,10)(17,22)(18,21)(19,23,20,24)$ |
4M | $4^{5},2^{2}$ | $96$ | $4$ | $17$ | $( 1,21, 6,18)( 2,22, 5,17)( 3,24, 7,20)( 4,23, 8,19)( 9,15,10,16)(11,14)(12,13)$ |
6A | $6^{2},3^{4}$ | $128$ | $6$ | $18$ | $( 1,11,21, 2,12,22)( 3, 9,23, 4,10,24)( 5,16,17)( 6,15,18)( 7,13,20)( 8,14,19)$ |
6B1 | $6^{4}$ | $128$ | $6$ | $20$ | $( 1,18,11, 4,20, 9)( 2,17,12, 3,19,10)( 5,21,16, 8,24,13)( 6,22,15, 7,23,14)$ |
6B-1 | $6^{4}$ | $128$ | $6$ | $20$ | $( 1, 9,20, 4,11,18)( 2,10,19, 3,12,17)( 5,13,24, 8,16,21)( 6,14,23, 7,15,22)$ |
8A | $8^{2},2,1^{6}$ | $48$ | $8$ | $15$ | $( 1,15, 4,14, 2,16, 3,13)( 5,11, 8,10, 6,12, 7, 9)(23,24)$ |
8B | $8^{2},2^{3},1^{2}$ | $48$ | $8$ | $17$ | $( 1,14, 3,15, 2,13, 4,16)( 5, 9, 7,12, 6,10, 8,11)(17,18)(19,20)(23,24)$ |
8C | $8^{2},4,2^{2}$ | $96$ | $8$ | $19$ | $( 1,23, 4,22, 2,24, 3,21)( 5,19, 8,18, 6,20, 7,17)( 9,12)(10,11)(13,16,14,15)$ |
8D | $8^{2},4,2^{2}$ | $96$ | $8$ | $19$ | $( 1, 9, 8,15, 2,10, 7,16)( 3,12, 5,13, 4,11, 6,14)(17,23,18,24)(19,21)(20,22)$ |
8E | $8^{2},4,2^{2}$ | $96$ | $8$ | $19$ | $( 1,11, 6,16, 2,12, 5,15)( 3, 9, 8,14, 4,10, 7,13)(17,21,18,22)(19,24)(20,23)$ |
Malle's constant $a(G)$: $1/6$
Character table
33 x 33 character table
Regular extensions
Data not computed