C₅×#85362 #14511100

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

solvable

Aliases

direct product

C₅×#85362

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#520	D₃
prime radical	C₂×C₅×#41⋊C₃	C₂²
hypercenter	C₅×C₄⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

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

normal subgroups only

This group has 384 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) = 45
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(7,11)(10,13)(14,38,28,24,29,43,19,25,31,16,17,41)(15,42,36,33,30,45,23,34,37,20,21,44)(18,26,32,39)(22,35,27,40);(6,8,9,12)(7,10,11,13)(16,24)(20,33)(25,38)(26,39)(34,42)(35,40)(41,43)(44,45);(6,7)(8,10)(9,11)(12,13)(14,15,19,23)(16,20,24,33)(17,27,29,22)(18,30,32,21)(25,40,38,35)(26,42,39,34)(28,37,31,36)(41,45,43,44)

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, 236798);