C₅×#85452 #14511190

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₅×#85452

Splittings

Fitting

C₅×#568 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂²×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#568	D₃
socle	C₂×C₅	C₂×C₄×S₄
prime radical	C₂×C₅×#985	C₂
cosocle	#41⋊C₃	C₂³×C₅
hypercenter	C₅×M₁₆	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₅×M₁₆

500 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,24,33,13,9,28,18,26,22,7,32,37,11,14,31,23,21,29,8,34,20,12,16,36)(10,15,17,27,19,25,30,35);(7,12)(13,23)(14,24)(15,25)(26,34)(27,35)(28,29)(36,37);(9,19)(10,21)(14,25)(15,24)(16,30)(17,32)(20,22)(26,35)(27,34)(28,29)(31,33)(36,37)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236888);