C₅×#85341 #14511079

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

solvable

Aliases

direct product

C₅×#85341

Splittings

Fitting

C₄×C₅×#41 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₄×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₄×C₅×#41	D₃
socle	C₂²×C₅	C₂²×S₄
prime radical	C₂×C₅×#985	C₂
cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
hypercenter	C₄²×C₅	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅

534 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 92 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 — 3 generators on 41 points:

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

Conjugacy classes & character table

260 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236777);