C₅×#73157 #14510881

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#73157

Subgroups

	subgroup	G/H
center	C₄×C₅	GL(2,ℤ/4)
commutator	#41⋊C₃	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₅×#585	D₃
socle	C₂×C₅	#5432
prime radical	C₅×#5945	C₂
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₅×#41	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₅×#41

380 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 2 generators on 37 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236579);