#72228179

order	2160
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Splittings

derived

C₃²×SL(2,𝔽₅) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊D₃×A₅
commutator, hypocenter	C₃²×SL(2,𝔽₅)	C₂
Fitting, socle, cosocle	C₂×C₃²	C₂×A₅
LCS	G ▷ C₃²×SL(2,𝔽₅)
UCS	C₁ ◁ C₂

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 30 points:

(4,5,6);(1,2)(5,6)(7,17)(8,27)(9,29)(10,19)(11,21)(13,23)(14,25)(16,18)(20,28)(24,30);(1,3)(5,6)(7,22,18,19,12,29,17,26,23,10,15,27)(8,20,30,16,11,25,9,24,28,13,14,21)

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228180
maximal normal subgroup of	#72228180

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,5,6), (1,2)(5,6)(7,17)(8,27)(9,29)(10,19)(11,21)(13,23)(14,25)(16,18)(20,28)(24,30), (1,3)(5,6)(7,22,18,19,12,29,17,26,23,10,15,27)(8,20,30,16,11,25,9,24,28,13,14,21));