Finite Group Databasebeta

← Prev (#67097) Next (#67099) →

C₂×Dic₄₅ #67098

order	360
isoclinism	isoclinic class of D₄₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×Dic₄₅
unambiguous semidirect product	(C₂×C₉)⋊Dic₅

Splittings

derived, Hall	C₅×C₉ ⋊ C₂×C₄
Sylow	C₉ ⋊ C₂×Dic₅
Sylow	C₅ ⋊ C₂×Dic₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	D₄₅
commutator, hypocenter	C₅×C₉	C₂×C₄
Frattini	C₂×C₃	C₂×D₁₅
Fitting	C₂²×C₅×C₉	C₂
socle, prime radical	C₂²×C₃×C₅	D₃
cosocle	C₂×C₅×C₉	C₂²
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₂²

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 2 generators on 20 points:

(1,2)(3,4,5,6)(8,9)(10,11)(13,15)(14,18)(16,17)(19,20);(1,2)(7,8,10,11,9)(12,20,15,18,17,16,14,13,19)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₅×Dic₉, #10623557, Dic₉⋊Dic₅, …
maximal normal subgroup of	Dic₅×Dic₉, #10623557, Dic₉⋊Dic₅, …
maximal quotient of	C₂×C₉⋊#168, #10623602, C₄×Dic₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 28);