LMFDB - Galois group: 30T146

Show commands: Magma

magma:G := TransitiveGroup(30, 146);

Group invariants

Abstract group:	$S_3\times C_5:F_5$	magma:IdentifyGroup(G);
Order:	$600=2^{3} \cdot 3 \cdot 5^{2}$	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 $n$:	$30$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$146$	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:	$(1,19,7,25,13)(2,21,8,27,14,3,20,9,26,15)(4,22,10,28,16)(5,24,11,30,17,6,23,12,29,18)$, $(1,17,13,29)(2,16,14,28)(3,18,15,30)(4,26,10,20)(5,25,11,19)(6,27,12,21)(7,23)(8,22)(9,24)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_4$ x 2, $C_2^2$
$6$: $S_3$
$8$: $C_4\times C_2$
$12$: $D_{6}$
$20$: $F_5$ x 2
$24$: $S_3 \times C_4$
$40$: $F_{5}\times C_2$ x 2
$100$: $C_5^2 : C_4$
$120$: $F_5 \times S_3$ x 2
$200$: 20T49

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$6$:	$S_3$
$8$:	$C_4\times C_2$
$12$:	$D_{6}$
$20$:	$F_5$ x 2
$24$:	$S_3 \times C_4$
$40$:	$F_{5}\times C_2$ x 2
$100$:	$C_5^2 : C_4$
$120$:	$F_5 \times S_3$ x 2
$200$:	20T49

Subfields

Degree 2: $C_2$
Degree 3: $S_3$
Degree 5: None
Degree 6: $D_{6}$
Degree 10: $C_5^2 : C_4$
Degree 15: None

Low degree siblings

30T146
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	$1^{30}$	$1$	$1$	$0$	$()$
2A	$2^{10},1^{10}$	$3$	$2$	$10$	$( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)$
2B	$2^{12},1^{6}$	$25$	$2$	$12$	$( 1, 7)( 2, 8)( 3, 9)( 4,22)( 5,23)( 6,24)(10,16)(11,17)(12,18)(13,25)(14,26)(15,27)$
2C	$2^{14},1^{2}$	$75$	$2$	$14$	$( 2, 3)( 4,22)( 5,24)( 6,23)( 7,25)( 8,27)( 9,26)(10,16)(11,18)(12,17)(13,19)(14,21)(15,20)(29,30)$
3A	$3^{10}$	$2$	$3$	$20$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
4A1	$4^{6},2^{3}$	$25$	$4$	$21$	$( 1,10, 7,16)( 2,11, 8,17)( 3,12, 9,18)( 4,13,22,25)( 5,14,23,26)( 6,15,24,27)(19,28)(20,29)(21,30)$
4A-1	$4^{6},2^{3}$	$25$	$4$	$21$	$( 1,16, 7,10)( 2,17, 8,11)( 3,18, 9,12)( 4,25,22,13)( 5,26,23,14)( 6,27,24,15)(19,28)(20,29)(21,30)$
4B1	$4^{6},2^{3}$	$75$	$4$	$21$	$( 1,23,13, 5)( 2,22,14, 4)( 3,24,15, 6)( 7,29)( 8,28)( 9,30)(10,26,16,20)(11,25,17,19)(12,27,18,21)$
4B-1	$4^{6},2^{3}$	$75$	$4$	$21$	$( 1, 5,13,23)( 2, 4,14,22)( 3, 6,15,24)( 7,29)( 8,28)( 9,30)(10,20,16,26)(11,19,17,25)(12,21,18,27)$
5A	$5^{6}$	$4$	$5$	$24$	$( 1,25,19,13, 7)( 2,26,20,14, 8)( 3,27,21,15, 9)( 4,16,28,10,22)( 5,17,29,11,23)( 6,18,30,12,24)$
5B	$5^{6}$	$4$	$5$	$24$	$( 1,13,25, 7,19)( 2,14,26, 8,20)( 3,15,27, 9,21)( 4,28,22,16,10)( 5,29,23,17,11)( 6,30,24,18,12)$
5C1	$5^{6}$	$4$	$5$	$24$	$( 1, 7,13,19,25)( 2, 8,14,20,26)( 3, 9,15,21,27)( 4,10,16,22,28)( 5,11,17,23,29)( 6,12,18,24,30)$
5C2	$5^{6}$	$4$	$5$	$24$	$( 1,13,25, 7,19)( 2,14,26, 8,20)( 3,15,27, 9,21)( 4,16,28,10,22)( 5,17,29,11,23)( 6,18,30,12,24)$
5D1	$5^{3},1^{15}$	$4$	$5$	$12$	$( 1,19, 7,25,13)( 2,20, 8,26,14)( 3,21, 9,27,15)$
5D2	$5^{3},1^{15}$	$4$	$5$	$12$	$( 1, 7,13,19,25)( 2, 8,14,20,26)( 3, 9,15,21,27)$
6A	$6^{4},3^{2}$	$50$	$6$	$24$	$( 1, 8, 3, 7, 2, 9)( 4,23, 6,22, 5,24)(10,17,12,16,11,18)(13,26,15,25,14,27)(19,20,21)(28,29,30)$
10A	$10^{2},5^{2}$	$12$	$10$	$26$	$( 1,13,25, 7,19)( 2,15,26, 9,20, 3,14,27, 8,21)( 4,10,16,22,28)( 5,12,17,24,29, 6,11,18,23,30)$
10B	$10^{2},5^{2}$	$12$	$10$	$26$	$( 1, 7,13,19,25)( 2, 9,14,21,26, 3, 8,15,20,27)( 4,16,28,10,22)( 5,18,29,12,23, 6,17,30,11,24)$
10C1	$10^{2},5^{2}$	$12$	$10$	$26$	$( 1,19, 7,25,13)( 2,21, 8,27,14, 3,20, 9,26,15)( 4,22,10,28,16)( 5,24,11,30,17, 6,23,12,29,18)$
10C3	$10^{2},5^{2}$	$12$	$10$	$26$	$( 1, 8,13,20,25, 2, 7,14,19,26)( 3, 9,15,21,27)( 4,29,22,17,10, 5,28,23,16,11)( 6,30,24,18,12)$
10D1	$10,5,2^{5},1^{5}$	$12$	$10$	$18$	$( 1,27,19,15, 7, 3,25,21,13, 9)( 2,26,20,14, 8)( 4, 6)(10,12)(16,18)(22,24)(28,30)$
10D3	$10,5,2^{5},1^{5}$	$12$	$10$	$18$	$( 2, 3)( 4,16,28,10,22)( 5,18,29,12,23, 6,17,30,11,24)( 8, 9)(14,15)(20,21)(26,27)$
12A1	$12^{2},6$	$50$	$12$	$27$	$( 1,18, 8,10, 3,17, 7,12, 2,16, 9,11)( 4,27,23,13, 6,26,22,15, 5,25,24,14)(19,30,20,28,21,29)$
12A-1	$12^{2},6$	$50$	$12$	$27$	$( 1,11, 9,16, 2,12, 7,17, 3,10, 8,18)( 4,14,24,25, 5,15,22,26, 6,13,23,27)(19,29,21,28,20,30)$
15A	$15^{2}$	$8$	$15$	$28$	$( 1,15,26, 7,21, 2,13,27, 8,19, 3,14,25, 9,20)( 4,30,23,16,12, 5,28,24,17,10, 6,29,22,18,11)$
15B	$15^{2}$	$8$	$15$	$28$	$( 1,26,21,13, 8, 3,25,20,15, 7, 2,27,19,14, 9)( 4,17,30,10,23, 6,16,29,12,22, 5,18,28,11,24)$
15C1	$15^{2}$	$8$	$15$	$28$	$( 1, 9,14,19,27, 2, 7,15,20,25, 3, 8,13,21,26)( 4,12,17,22,30, 5,10,18,23,28, 6,11,16,24,29)$
15C2	$15^{2}$	$8$	$15$	$28$	$( 1,14,27, 7,20, 3,13,26, 9,19, 2,15,25, 8,21)( 4,17,30,10,23, 6,16,29,12,22, 5,18,28,11,24)$
15D1	$15,3^{5}$	$8$	$15$	$24$	$( 1,20, 9,25,14, 3,19, 8,27,13, 2,21, 7,26,15)( 4, 5, 6)(10,11,12)(16,17,18)(22,23,24)(28,29,30)$
15D2	$15,3^{5}$	$8$	$15$	$24$	$( 1,27,20,13, 9, 2,25,21,14, 7, 3,26,19,15, 8)( 4, 6, 5)(10,12,11)(16,18,17)(22,24,23)(28,30,29)$

Malle's constant $a(G)$: $1/10$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	3A	4A1	4A-1	4B1	4B-1	5A	5B	5C1	5C2	5D1	5D2	6A	10A	10B	10C1	10C3	10D1	10D3	12A1	12A-1	15A	15B	15C1	15C2	15D1	15D2
	Size	1	3	25	75	2	25	25	75	75	4	4	4	4	4	4	50	12	12	12	12	12	12	50	50	8	8	8	8	8	8
	2 P	1A	1A	1A	1A	3A	2B	2B	2B	2B	5A	5B	5C2	5C1	5D2	5D1	3A	5A	5B	5C1	5C2	5D1	5D2	6A	6A	15A	15B	15C2	15C1	15D2	15D1
	3 P	1A	2A	2B	2C	1A	4A-1	4A1	4B-1	4B1	5A	5B	5C2	5C1	5D2	5D1	2B	10A	10B	10C3	10C1	10D3	10D1	4A1	4A-1	5B	5A	5C2	5C1	5D2	5D1
	5 P	1A	2A	2B	2C	3A	4A1	4A-1	4B1	4B-1	1A	1A	1A	1A	1A	1A	6A	2A	2A	2A	2A	2A	2A	12A1	12A-1	3A	3A	3A	3A	3A	3A
	Type
600.161.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
600.161.1b	R	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$
600.161.1c	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
600.161.1d	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$
600.161.1e1	C	$1$	$- 1$	$- 1$	$1$	$1$	$- i$	$i$	$i$	$- i$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$1$	$1$	$1$	$1$	$1$	$1$
600.161.1e2	C	$1$	$- 1$	$- 1$	$1$	$1$	$i$	$- i$	$- i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$
600.161.1f1	C	$1$	$1$	$- 1$	$- 1$	$1$	$- i$	$i$	$- i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$i$	$- i$	$1$	$1$	$1$	$1$	$1$	$1$
600.161.1f2	C	$1$	$1$	$- 1$	$- 1$	$1$	$i$	$- i$	$i$	$- i$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$
600.161.2a	R	$2$	$0$	$2$	$0$	$- 1$	$2$	$2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
600.161.2b	R	$2$	$0$	$2$	$0$	$- 1$	$- 2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
600.161.2c1	C	$2$	$0$	$- 2$	$0$	$- 1$	$- 2 i$	$2 i$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$- i$	$i$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
600.161.2c2	C	$2$	$0$	$- 2$	$0$	$- 1$	$2 i$	$- 2 i$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$i$	$- i$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
600.161.4a	R	$4$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$4$	$- 1$	$0$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$4$
600.161.4b	R	$4$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$- 1$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$- 1$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
600.161.4c	R	$4$	$- 4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$4$	$- 1$	$0$	$- 4$	$1$	$1$	$1$	$1$	$1$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$4$
600.161.4d	R	$4$	$- 4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$- 1$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$1$	$- 4$	$1$	$1$	$1$	$1$	$0$	$0$	$4$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
600.161.4e1	R	$4$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$- 1$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- 1$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$0$	$- 1$	$- 1$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$0$	$0$	$- 1$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$- 1$
600.161.4e2	R	$4$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- 1$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$- 1$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$0$	$- 1$	$- 1$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$0$	$0$	$- 1$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$- 1$
600.161.4f1	R	$4$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$- 1$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$- 1$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$0$	$- 1$	$- 1$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$0$	$0$	$- 1$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- 1$
600.161.4f2	R	$4$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$- 1$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$- 1$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$0$	$- 1$	$- 1$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$0$	$0$	$- 1$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$- 1$
600.161.4g1	R	$4$	$- 4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$- 1$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- 1$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$0$	$1$	$1$	$- 2 ζ_{5}^{- 1} - 2 ζ_{5}$	$ζ_{5}^{- 2} - 1 + ζ_{5}^{2}$	$- 2 ζ_{5}^{- 2} - 2 ζ_{5}^{2}$	$- ζ_{5}^{- 2} - 2 - ζ_{5}^{2}$	$0$	$0$	$- 1$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$- 1$
600.161.4g2	R	$4$	$- 4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- 1$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$- 1$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$0$	$1$	$1$	$- 2 ζ_{5}^{- 2} - 2 ζ_{5}^{2}$	$- ζ_{5}^{- 2} - 2 - ζ_{5}^{2}$	$- 2 ζ_{5}^{- 1} - 2 ζ_{5}$	$ζ_{5}^{- 2} - 1 + ζ_{5}^{2}$	$0$	$0$	$- 1$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$- 1$
600.161.4h1	R	$4$	$- 4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$- 1$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$- 1$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$0$	$1$	$1$	$ζ_{5}^{- 2} - 1 + ζ_{5}^{2}$	$- 2 ζ_{5}^{- 2} - 2 ζ_{5}^{2}$	$- ζ_{5}^{- 2} - 2 - ζ_{5}^{2}$	$- 2 ζ_{5}^{- 1} - 2 ζ_{5}$	$0$	$0$	$- 1$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- 1$
600.161.4h2	R	$4$	$- 4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$- 1$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$- 1$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$0$	$1$	$1$	$- ζ_{5}^{- 2} - 2 - ζ_{5}^{2}$	$- 2 ζ_{5}^{- 1} - 2 ζ_{5}$	$ζ_{5}^{- 2} - 1 + ζ_{5}^{2}$	$- 2 ζ_{5}^{- 2} - 2 ζ_{5}^{2}$	$0$	$0$	$- 1$	$ζ_{5}^{- 2} + 2 + ζ_{5}^{2}$	$2 ζ_{5}^{- 1} + 2 ζ_{5}$	$- ζ_{5}^{- 2} + 1 - ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 2 ζ_{5}^{2}$	$- 1$
600.161.8a	R	$8$	$0$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$- 2$	$- 2$	$8$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$1$	$1$	$1$	$1$	$- 4$
600.161.8b	R	$8$	$0$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$- 2$	$8$	$- 2$	$- 2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 4$	$1$	$1$	$1$	$1$	$1$
600.161.8c1	R	$8$	$0$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$4 ζ_{5}^{- 2} + 4 ζ_{5}^{2}$	$- 2$	$- 2 ζ_{5}^{- 2} + 2 - 2 ζ_{5}^{2}$	$4 ζ_{5}^{- 1} + 4 ζ_{5}$	$- 2$	$2 ζ_{5}^{- 2} + 4 + 2 ζ_{5}^{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$- 2 ζ_{5}^{- 1} - 2 ζ_{5}$	$ζ_{5}^{- 2} - 1 + ζ_{5}^{2}$	$- 2 ζ_{5}^{- 2} - 2 ζ_{5}^{2}$	$- ζ_{5}^{- 2} - 2 - ζ_{5}^{2}$	$1$
600.161.8c2	R	$8$	$0$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$4 ζ_{5}^{- 1} + 4 ζ_{5}$	$- 2$	$2 ζ_{5}^{- 2} + 4 + 2 ζ_{5}^{2}$	$4 ζ_{5}^{- 2} + 4 ζ_{5}^{2}$	$- 2$	$- 2 ζ_{5}^{- 2} + 2 - 2 ζ_{5}^{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$- 2 ζ_{5}^{- 2} - 2 ζ_{5}^{2}$	$- ζ_{5}^{- 2} - 2 - ζ_{5}^{2}$	$- 2 ζ_{5}^{- 1} - 2 ζ_{5}$	$ζ_{5}^{- 2} - 1 + ζ_{5}^{2}$	$1$
600.161.8d1	R	$8$	$0$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$2 ζ_{5}^{- 2} + 4 + 2 ζ_{5}^{2}$	$- 2$	$4 ζ_{5}^{- 2} + 4 ζ_{5}^{2}$	$- 2 ζ_{5}^{- 2} + 2 - 2 ζ_{5}^{2}$	$- 2$	$4 ζ_{5}^{- 1} + 4 ζ_{5}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$ζ_{5}^{- 2} - 1 + ζ_{5}^{2}$	$- 2 ζ_{5}^{- 2} - 2 ζ_{5}^{2}$	$- ζ_{5}^{- 2} - 2 - ζ_{5}^{2}$	$- 2 ζ_{5}^{- 1} - 2 ζ_{5}$	$1$
600.161.8d2	R	$8$	$0$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$- 2 ζ_{5}^{- 2} + 2 - 2 ζ_{5}^{2}$	$- 2$	$4 ζ_{5}^{- 1} + 4 ζ_{5}$	$2 ζ_{5}^{- 2} + 4 + 2 ζ_{5}^{2}$	$- 2$	$4 ζ_{5}^{- 2} + 4 ζ_{5}^{2}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$- ζ_{5}^{- 2} - 2 - ζ_{5}^{2}$	$- 2 ζ_{5}^{- 1} - 2 ζ_{5}$	$ζ_{5}^{- 2} - 1 + ζ_{5}^{2}$	$- 2 ζ_{5}^{- 2} - 2 ζ_{5}^{2}$	$1$

magma:CharacterTable(G);

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Label	30T146

Degree	$30$
Order	$600$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$S_3\times C_5:F_5$