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C₃×D₅ #90

order	30
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₂×AGL(1,𝔽₅) (order 40) · Inn(G) = D₅ · Out(G) = C₂²
holomorph	G ⋊ Aut(G) = D₃×D₅⋊AGL(1,𝔽₅)
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅
unambiguous semidirect product	C₅⋊(C₂×C₃)

Splittings

Fitting, socle, Hall	C₃×C₅ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 15

A minimal faithful permutation representation — 2 generators on 8 points:

(1,2,3)(5,6)(7,8);(4,5,7,8,6)

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	2	2	5	1	2	2	5	1	2	2	5
order	1	5	5	2	3	15	15	6	3	15	15	6
χ₁	1	1	1	1	1	1	1	1	1	1	1	1
χ₂	1	1	1	-1	1	1	1	-1	1	1	1	-1
χ₃	2	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$	0	2	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$	0	2	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$	0
χ₄	2	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$	0	2	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$	0	2	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$	0
χ₅	1	1	1	1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$
χ₆	1	1	1	-1	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$
χ₇	2	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$	0	$- 1 + i \sqrt{3}$	ζ²+ζ⁸	ζ¹¹+ζ¹⁴	0	$- 1 - i \sqrt{3}$	ζ⁷+ζ¹³	ζ+ζ⁴	0
χ₈	2	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$	0	$- 1 + i \sqrt{3}$	ζ¹¹+ζ¹⁴	ζ²+ζ⁸	0	$- 1 - i \sqrt{3}$	ζ+ζ⁴	ζ⁷+ζ¹³	0
χ₉	1	1	1	1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$
χ₁₀	1	1	1	-1	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{- 1 - i \sqrt{3}}{2}$	$\frac{1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{- 1 + i \sqrt{3}}{2}$	$\frac{1 - i \sqrt{3}}{2}$
χ₁₁	2	$\frac{- 1 + \sqrt{5}}{2}$	$\frac{- 1 - \sqrt{5}}{2}$	0	$- 1 - i \sqrt{3}$	ζ⁷+ζ¹³	ζ+ζ⁴	0	$- 1 + i \sqrt{3}$	ζ²+ζ⁸	ζ¹¹+ζ¹⁴	0
χ₁₂	2	$\frac{- 1 - \sqrt{5}}{2}$	$\frac{- 1 + \sqrt{5}}{2}$	0	$- 1 - i \sqrt{3}$	ζ+ζ⁴	ζ⁷+ζ¹³	0	$- 1 + i \sqrt{3}$	ζ¹¹+ζ¹⁴	ζ²+ζ⁸	0

Here ζ is a primitive root of unity of order 15 (ζ¹⁵ = 1); i = ζ³.

Relationships

maximal subgroup of	C₃×AGL(1,𝔽₅), ΓL(1,𝔽₁₆), D₃×D₅, …
maximal normal subgroup of	C₃×AGL(1,𝔽₅), ΓL(1,𝔽₁₆), D₃×D₅, …
maximal quotient of	C₃×Dic₅, C₂×C₃×D₅, C₉×D₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30, 2);