C₅×#73016 #14510740

order	1920
isoclinism	isoclinic class of #73011
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₁

solvable

Aliases

direct product

C₅×#73016

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₄⋊S₄
commutator, cosocle	C₄×SL(2,𝔽₃)	C₂²×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₅×C₈×Q₈	D₃
prime radical	C₂×C₅×SL(2,𝔽₃)	D₄
hypercenter	C₂×C₅×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₅×Q₁₆
LCS	G ▷ C₄×SL(2,𝔽₃) ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

180 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 640

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236438);