C₅×#85365 #14511103

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

solvable

Aliases

direct product

C₅×#85365

Splittings

Fitting

C₅×#520 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₄×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#520	D₃
prime radical	C₂×C₅×#985	C₂
cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
hypercenter	C₅×C₄⋊C₄	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

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

normal subgroups only

This group has 530 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)(12,21,39,20,34,37)(13,24,15)(16,38,41)(19,23)(22,35)(29,40,31);(10,12,15,20)(11,16,17,31)(13,21,25,34)(14,22,28,35)(18,29,26,38)(19,30,23,36)(24,37,27,39)(32,40,33,41);(10,11)(12,16)(13,23)(14,26)(15,17)(18,28)(19,25)(20,31)(21,36)(22,38)(24,33)(27,32)(29,35)(30,34)(37,41)(39,40)

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, 236801);