C₅×#85339 #14511077

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

solvable

Aliases

direct product

C₅×#85339

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₄×C₅×#41	D₃
prime radical	C₂×C₅×#41⋊C₃	C₂²
hypercenter	C₄²×C₅	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×Q₈
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅

390 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 76 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)(6,8,9,13)(7,10,11,12)(14,28,33)(15,22,37,21,32,36)(16,40,43)(17,18,31)(20,34,45,24,38,44)(23,30)(25,26,42)(35,39);(14,16,17,25)(15,20,21,24)(18,26,28,40)(19,27,29,41)(22,34,32,38)(23,35,30,39)(31,42,33,43)(36,44,37,45);(6,7,9,11)(8,12,13,10)(14,15,17,21)(16,24,25,20)(18,30,28,23)(19,32,29,22)(26,35,40,39)(27,34,41,38)(31,37,33,36)(42,44,43,45)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236775);