C₅×#73155 #14510879

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

solvable

Aliases

direct product

C₅×#73155

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	GL(2,ℤ/4)
commutator	#41⋊C₃	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₂×C₅×2₋¹⁺⁴	D₃
prime radical	C₂×C₅×2₋¹⁺⁴⋊C₃	C₂
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₂×C₅×D₄	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×D₄

400 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 41 points:

(1,2,3,4,5)(10,26,28)(11,32,35,16,27,41)(12,22,39,21,36,38)(13,25,15)(17,18,40,20,30,34)(19,31)(23,37)(24,33);(6,7,8,9)(10,11)(12,16)(13,24)(14,27)(15,20)(17,21)(18,29)(19,26)(22,33)(23,32)(25,35)(28,34)(30,37)(31,36)(38,41)(39,40)

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, 236577);