C₅×#85474 #14511212

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

solvable

Aliases

direct product

C₅×#85474

Splittings

Fitting

C₅×#578 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	D₄×S₄
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#578	D₃
hypercenter	C₅×D₈	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×D₈

500 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 64 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)(6,24,23,7,22,29)(8,26,35,17,30,36)(9,28,11,14,21,12)(10,15)(13,18,37,16,32,34)(19,33)(20,25)(27,31);(7,13)(8,16)(12,17)(14,26)(15,27)(18,30)(19,31)(24,32)(25,33)(28,36)(29,37)(34,35);(7,12)(9,20)(10,22)(13,17)(14,15)(18,31)(19,30)(21,23)(24,25)(26,27)(32,33)(34,35)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236910);