Galois group: 24T1353

• Label: 24T1353
• Degree: $24$
• Order: $480$
• Cyclic: no
• Abelian: no
• Solvable: no
• Transitivity: $1$
• Primitive: no
• $p$-group: no
• Group: $\GL(2,5)$

Group invariants

Abstract group:		$\GL(2,5)$
Order:		$480=2^{5} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$1353$
Parity:		$-1$
Transitivity:		1
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,15,10,21)(2,16,9,22)(3,5,4,6)(7,19,23,17)(8,20,24,18)(11,12)(13,14)$, $(1,11,24,13,2,12,23,14)(3,15,6,18,4,16,5,17)(7,21,10,20,8,22,9,19)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$120$:	$S_5$
$240$:	12T124

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: $\PGL(2,5)$

Degree 8: None

Degree 12: 12T124

Low degree siblings

24T1353 x 3

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{24}$	$1$	$1$	$0$	$()$
2A	$2^{12}$	$1$	$2$	$12$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
2B	$2^{10},1^{4}$	$30$	$2$	$10$	$( 1,18)( 2,17)( 3, 9)( 4,10)( 5, 8)( 6, 7)(11,12)(13,14)(15,23)(16,24)$
3A	$3^{8}$	$20$	$3$	$16$	$( 1,11,20)( 2,12,19)( 3,18, 8)( 4,17, 7)( 5,15,10)( 6,16, 9)(13,22,24)(14,21,23)$
4A1	$4^{6}$	$1$	$4$	$18$	$( 1,23, 2,24)( 3, 5, 4, 6)( 7, 9, 8,10)(11,14,12,13)(15,17,16,18)(19,22,20,21)$
4A-1	$4^{6}$	$1$	$4$	$18$	$( 1,24, 2,23)( 3, 6, 4, 5)( 7,10, 8, 9)(11,13,12,14)(15,18,16,17)(19,21,20,22)$
4B	$4^{6}$	$30$	$4$	$18$	$( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,24,10,23)(11,14,12,13)(15,19,16,20)(17,22,18,21)$
4C1	$4^{5},2^{2}$	$30$	$4$	$17$	$( 1,10,18, 4)( 2, 9,17, 3)( 5,24, 8,16)( 6,23, 7,15)(11,13,12,14)(19,20)(21,22)$
4C-1	$4^{5},2^{2}$	$30$	$4$	$17$	$( 1, 4,18,10)( 2, 3,17, 9)( 5,16, 8,24)( 6,15, 7,23)(11,14,12,13)(19,20)(21,22)$
4D1	$4^{5},1^{4}$	$30$	$4$	$15$	$( 3, 5, 4, 6)( 7,14,20,18)( 8,13,19,17)( 9,12,21,15)(10,11,22,16)$
4D-1	$4^{5},1^{4}$	$30$	$4$	$15$	$( 3, 6, 4, 5)( 7,18,20,14)( 8,17,19,13)( 9,15,21,12)(10,16,22,11)$
5A	$5^{4},1^{4}$	$24$	$5$	$16$	$( 1,11,10,22,16)( 2,12, 9,21,15)( 7,20,18,23,14)( 8,19,17,24,13)$
6A	$6^{4}$	$20$	$6$	$20$	$( 1,19,11, 2,20,12)( 3, 7,18, 4, 8,17)( 5, 9,15, 6,10,16)(13,23,22,14,24,21)$
8A1	$8^{3}$	$20$	$8$	$21$	$( 1,16,23,18, 2,15,24,17)( 3,19, 5,22, 4,20, 6,21)( 7,11, 9,14, 8,12,10,13)$
8A-1	$8^{3}$	$20$	$8$	$21$	$( 1,17,24,15, 2,18,23,16)( 3,21, 6,20, 4,22, 5,19)( 7,13,10,12, 8,14, 9,11)$
10A	$10^{2},2^{2}$	$24$	$10$	$20$	$( 1,21,11,15,10, 2,22,12,16, 9)( 3, 4)( 5, 6)( 7,24,20,13,18, 8,23,19,14,17)$
12A1	$12^{2}$	$20$	$12$	$22$	$( 1,13,19,23,11,22, 2,14,20,24,12,21)( 3,16, 7, 5,18, 9, 4,15, 8, 6,17,10)$
12A-1	$12^{2}$	$20$	$12$	$22$	$( 1,21,12,24,20,14, 2,22,11,23,19,13)( 3,10,17, 6, 8,15, 4, 9,18, 5, 7,16)$
20A1	$20,4$	$24$	$20$	$22$	$( 1,17,21, 7,11,24,15,20,10,13, 2,18,22, 8,12,23,16,19, 9,14)( 3, 6, 4, 5)$
20A-1	$20,4$	$24$	$20$	$22$	$( 1,14, 9,19,16,23,12, 8,22,18, 2,13,10,20,15,24,11, 7,21,17)( 3, 5, 4, 6)$
24A1	$24$	$20$	$24$	$23$	$( 1, 3,13,16,19, 7,23, 5,11,18,22, 9, 2, 4,14,15,20, 8,24, 6,12,17,21,10)$
24A-1	$24$	$20$	$24$	$23$	$( 1,20,16, 4,14, 7,24,22,17, 5,11,10, 2,19,15, 3,13, 8,23,21,18, 6,12, 9)$
24A7	$24$	$20$	$24$	$23$	$( 1,10,17,22, 4,14,24, 8,15,19, 5,11, 2, 9,18,21, 3,13,23, 7,16,20, 6,12)$
24A-7	$24$	$20$	$24$	$23$	$( 1, 9, 4,14,22,18,23, 8, 6,12,20,15, 2,10, 3,13,21,17,24, 7, 5,11,19,16)$

Malle's constant $a(G)$:     $1/10$

Character table

		1A	2A	2B	3A	4A1	4A-1	4B	4C1	4C-1	4D1	4D-1	5A	6A	8A1	8A-1	10A	12A1	12A-1	20A1	20A-1	24A1	24A-1	24A7	24A-7
	Size	1	1	30	20	1	1	30	30	30	30	30	24	20	20	20	24	20	20	24	24	20	20	20	20
	2 P	1A	1A	1A	3A	2A	2A	2A	2B	2B	2B	2B	5A	3A	4A1	4A-1	5A	6A	6A	10A	10A	12A1	12A-1	12A-1	12A1
	3 P	1A	2A	2B	1A	4A-1	4A1	4B	4C-1	4C1	4D-1	4D1	5A	2A	8A-1	8A1	10A	4A1	4A-1	20A-1	20A1	8A1	8A-1	8A-1	8A1
	5 P	1A	2A	2B	3A	4A1	4A-1	4B	4C1	4C-1	4D1	4D-1	1A	6A	8A1	8A-1	2A	12A1	12A-1	4A-1	4A1	24A1	24A-1	24A7	24A-7
	Type
480.218.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
480.218.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
480.218.1c1	C	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$	$1$	$1$	$1$	$i$	$-i$	$1$	$-1$	$-1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$
480.218.1c2	C	$1$	$1$	$-1$	$1$	$-1$	$-1$	$i$	$-i$	$-i$	$i$	$1$	$1$	$1$	$-i$	$i$	$1$	$-1$	$-1$	$-1$	$-1$	$-i$	$i$	$i$	$-i$
480.218.4a	R	$4$	$4$	$0$	$1$	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$	$2$	$2$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
480.218.4b	R	$4$	$4$	$0$	$1$	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$	$-2$	$-2$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
480.218.4c1	C	$4$	$4$	$0$	$1$	$-4$	$-4$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$	$-2i$	$2i$	$-1$	$-1$	$-1$	$1$	$1$	$i$	$-i$	$-i$	$i$
480.218.4c2	C	$4$	$4$	$0$	$1$	$-4$	$-4$	$0$	$0$	$0$	$0$	$0$	$-1$	$1$	$2i$	$-2i$	$-1$	$-1$	$-1$	$1$	$1$	$-i$	$i$	$i$	$-i$
480.218.4d1	C	$4$	$-4$	$0$	$-2$	$-4i$	$4i$	$0$	$0$	$0$	$0$	$0$	$-1$	$2$	$0$	$0$	$1$	$-2i$	$2i$	$-i$	$i$	$0$	$0$	$0$	$0$
480.218.4d2	C	$4$	$-4$	$0$	$-2$	$4i$	$-4i$	$0$	$0$	$0$	$0$	$0$	$-1$	$2$	$0$	$0$	$1$	$2i$	$-2i$	$i$	$-i$	$0$	$0$	$0$	$0$
480.218.4e1	C	$4$	$-4$	$0$	$1$	$-4{\zeta }_{24}^6$	$4{\zeta }_{24}^6$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$0$	$0$	$1$	${\zeta }_{24}^6$	$-{\zeta }_{24}^6$	$-{\zeta }_{24}^6$	${\zeta }_{24}^6$	$-{\zeta }_{24}-{\zeta }_{24}^5$	$-{\zeta }_{24}^3+2{\zeta }_{24}^7$	${\zeta }_{24}^3-2{\zeta }_{24}^7$	${\zeta }_{24}+{\zeta }_{24}^5$
480.218.4e2	C	$4$	$-4$	$0$	$1$	$4{\zeta }_{24}^6$	$-4{\zeta }_{24}^6$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$0$	$0$	$1$	$-{\zeta }_{24}^6$	${\zeta }_{24}^6$	${\zeta }_{24}^6$	$-{\zeta }_{24}^6$	$-{\zeta }_{24}^3+2{\zeta }_{24}^7$	$-{\zeta }_{24}-{\zeta }_{24}^5$	${\zeta }_{24}+{\zeta }_{24}^5$	${\zeta }_{24}^3-2{\zeta }_{24}^7$
480.218.4e3	C	$4$	$-4$	$0$	$1$	$-4{\zeta }_{24}^6$	$4{\zeta }_{24}^6$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$0$	$0$	$1$	${\zeta }_{24}^6$	$-{\zeta }_{24}^6$	$-{\zeta }_{24}^6$	${\zeta }_{24}^6$	${\zeta }_{24}+{\zeta }_{24}^5$	${\zeta }_{24}^3-2{\zeta }_{24}^7$	$-{\zeta }_{24}^3+2{\zeta }_{24}^7$	$-{\zeta }_{24}-{\zeta }_{24}^5$
480.218.4e4	C	$4$	$-4$	$0$	$1$	$4{\zeta }_{24}^6$	$-4{\zeta }_{24}^6$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$0$	$0$	$1$	$-{\zeta }_{24}^6$	${\zeta }_{24}^6$	${\zeta }_{24}^6$	$-{\zeta }_{24}^6$	${\zeta }_{24}^3-2{\zeta }_{24}^7$	${\zeta }_{24}+{\zeta }_{24}^5$	$-{\zeta }_{24}-{\zeta }_{24}^5$	$-{\zeta }_{24}^3+2{\zeta }_{24}^7$
480.218.5a	R	$5$	$5$	$1$	$-1$	$5$	$5$	$1$	$1$	$1$	$1$	$1$	$0$	$-1$	$-1$	$-1$	$0$	$-1$	$-1$	$0$	$0$	$-1$	$-1$	$-1$	$-1$
480.218.5b	R	$5$	$5$	$1$	$-1$	$5$	$5$	$-1$	$-1$	$-1$	$-1$	$1$	$0$	$-1$	$1$	$1$	$0$	$-1$	$-1$	$0$	$0$	$1$	$1$	$1$	$1$
480.218.5c1	C	$5$	$5$	$-1$	$-1$	$-5$	$-5$	$-i$	$i$	$i$	$-i$	$1$	$0$	$-1$	$-i$	$i$	$0$	$1$	$1$	$0$	$0$	$-i$	$i$	$i$	$-i$
480.218.5c2	C	$5$	$5$	$-1$	$-1$	$-5$	$-5$	$i$	$-i$	$-i$	$i$	$1$	$0$	$-1$	$i$	$-i$	$0$	$1$	$1$	$0$	$0$	$i$	$-i$	$-i$	$i$
480.218.6a	R	$6$	$6$	$2$	$0$	$-6$	$-6$	$0$	$0$	$0$	$0$	$-2$	$1$	$0$	$0$	$0$	$1$	$0$	$0$	$-1$	$-1$	$0$	$0$	$0$	$0$
480.218.6b	R	$6$	$6$	$-2$	$0$	$6$	$6$	$0$	$0$	$0$	$0$	$-2$	$1$	$0$	$0$	$0$	$1$	$0$	$0$	$1$	$1$	$0$	$0$	$0$	$0$
480.218.6c1	C	$6$	$-6$	$0$	$0$	$-6i$	$6i$	$-1+i$	$1+i$	$-1-i$	$1-i$	$0$	$1$	$0$	$0$	$0$	$-1$	$0$	$0$	$i$	$-i$	$0$	$0$	$0$	$0$
480.218.6c2	C	$6$	$-6$	$0$	$0$	$6i$	$-6i$	$-1-i$	$1-i$	$-1+i$	$1+i$	$0$	$1$	$0$	$0$	$0$	$-1$	$0$	$0$	$-i$	$i$	$0$	$0$	$0$	$0$
480.218.6d1	C	$6$	$-6$	$0$	$0$	$-6i$	$6i$	$1-i$	$-1-i$	$1+i$	$-1+i$	$0$	$1$	$0$	$0$	$0$	$-1$	$0$	$0$	$i$	$-i$	$0$	$0$	$0$	$0$
480.218.6d2	C	$6$	$-6$	$0$	$0$	$6i$	$-6i$	$1+i$	$-1+i$	$1-i$	$-1-i$	$0$	$1$	$0$	$0$	$0$	$-1$	$0$	$0$	$-i$	$i$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed