C₅×#73131 #14510855

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#73131

Splittings

Fitting

C₅×#568 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	(C₂×S₄)⋊C₄
commutator	C₂×SL(2,𝔽₃)	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₅×#568	D₃
prime radical	C₂×C₅×#986	C₂
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₅×M₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₅×M₁₆

330 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 48 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 — 2 generators on 37 points:

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236553);