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D₁₅ #91

order	30
isoclinism	isoclinic class of D₁₅
automorphism	Aut(G) = D₃×AGL(1,𝔽₅) (order 120) · Inn(G) = G · Out(G) = C₄
holomorph	G ⋊ Aut(G) = #72210392
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅
unambiguous semidirect product	C₃⋊D₅
unambiguous semidirect product	C₅⋊D₃

Splittings

Fitting, socle, derived, Hall	C₃×C₅ ⋊ C₂
Sylow	C₅ ⋊ D₃
Sylow	C₃ ⋊ D₅

Subgroups

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

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

Polycyclic presentation:

2,3,5|b^a=b^2,c^a=c^4

Conjugacy classes & character table

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

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

Relationships

maximal subgroup of	D₃×D₅, C₂×D₁₅, D₄₅, …
maximal normal subgroup of	D₃×D₅, C₂×D₁₅, C₃×D₁₅, …
maximal quotient of	Dic₁₅, C₂×D₁₅, D₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30, 3);