C₅×#85348 #14511086

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

solvable

Aliases

direct product

C₅×#85348

Splittings

Fitting

C₄×C₅×#41 ⋊ D₃

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
hypercenter	C₄²×C₅	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅

458 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

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

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, 236784);