C₅×#85454 #14511192

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

solvable

Aliases

direct product

C₅×#85454

Splittings

Fitting

C₅×#568 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂×C₄⋊S₄
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#568	D₃
hypercenter	C₅×M₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×M₁₆

514 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 72 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)(6,26,35,12,9,29,16,24,22,7,32,37,11,14,34,23,21,31,8,30,20,13,17,36)(10,15,18,25,19,27,33,28);(7,13)(12,23)(14,26)(15,27)(24,30)(25,28)(29,31)(36,37);(7,12)(8,16)(9,19)(10,21)(13,23)(14,28)(15,30)(17,18)(20,22)(24,27)(25,26)(29,37)(31,36)(32,33)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236890);