C₅×#73009 #14510733

order	1920
isoclinism	isoclinic class of #5386
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₁

solvable

Aliases

direct product

C₅×#73009

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×S₄
commutator	C₂×SL(2,𝔽₃)	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₅×C₈×Q₈	D₃
socle	C₂²×C₅	C₄×S₄
prime radical	C₂×C₅×SL(2,𝔽₃)	C₂×C₄
cosocle	C₄×SL(2,𝔽₃)	C₂²×C₅
hypercenter	C₂×C₅×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₅×M₁₆
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

186 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 29
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 2 generators on 29 points:

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

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, 236431);