Finite Group Databasebeta

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D₃×D₅ #307

order	60
isoclinism	isoclinic class of D₃×D₅
automorphism	Aut(G) = D₃×AGL(1,𝔽₅) (order 120) · Inn(G) = G · Out(G) = C₂
holomorph	G ⋊ Aut(G) = D₃²×D₅⋊AGL(1,𝔽₅)
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×D₅
unambiguous semidirect product	(C₅×D₃)⋊C₂
unambiguous semidirect product	(C₃×D₅)⋊C₂
unambiguous semidirect product	D₁₅⋊C₂

Splittings

Fitting, socle, derived, Hall	C₃×C₅ ⋊ C₂²
Sylow	C₅ ⋊ C₂×D₃
Sylow	C₃ ⋊ C₂×D₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁

20 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:

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

Conjugacy classes & character table

	C₁	C₂	C₃	C₄	C₅	C₆	C₇	C₈	C₉	C₁₀	C₁₁	C₁₂
size	1	2	2	5	2	4	4	10	3	6	6	15
order	1	5	5	2	3	15	15	6	2	10	10	2
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	1	1	1	1	-1	-1	-1	-1
χ₆	1	1	1	-1	1	1	1	-1	-1	-1	-1	1
χ₇	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
χ₈	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
χ₉	2	2	2	2	-1	-1	-1	-1	0	0	0	0
χ₁₀	2	2	2	-2	-1	-1	-1	1	0	0	0	0
χ₁₁	4	$- 1 + \sqrt{5}$	$- 1 - \sqrt{5}$	0	-2	$\frac{1 - \sqrt{5}}{2}$	$\frac{1 + \sqrt{5}}{2}$	0	0	0	0	0
χ₁₂	4	$- 1 - \sqrt{5}$	$- 1 + \sqrt{5}$	0	-2	$\frac{1 + \sqrt{5}}{2}$	$\frac{1 - \sqrt{5}}{2}$	0	0	0	0	0

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₅), C₂×D₃×D₅, D₅×D₉, …
maximal normal subgroup of	D₃×AGL(1,𝔽₅), C₂×D₃×D₅, C₃×D₃×D₅, …
maximal quotient of	D₅×Dic₃, D₃×Dic₅, #1200, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(60, 8);