Finite Group Databasebeta

← Prev (#10597092) Next (#10597094) →

C₃×#6033 #10597093

order	600
isoclinism	isoclinic class of #1043
automorphism	?
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

Splittings

derived, Sylow	C₅² ⋊ C₃×C₈
Hall	C₃×C₅² ⋊ C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#1043
commutator, hypocenter	C₅²	C₃×C₈
Frattini	C₂	C₃×#1043
Fitting, socle, prime radical	C₂×C₃×C₅²	C₄
cosocle	#1039	C₂×C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₃

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 600

A minimal faithful permutation representation — 3 generators on 21 points:

(1,2,3)(4,5,6,8,7,9,10,11)(13,14,15,16)(18,19,20,21);(17,18,19,21,20);(12,13,14,16,15)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#87822, C₃⋊#87822, C₃×D₅⋊#170, …
maximal normal subgroup of	C₃×#87822, C₃⋊#87822, C₃×D₅⋊#170, …
maximal quotient of	C₃×#87764, C₂×C₃×#6033, C₉×#6033, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(600, 74);