C₅×#87496 #14511430

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

solvable

Aliases

direct product

C₅×#87496

Splittings

Fitting

C₂×C₅×2₋¹⁺⁴ ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂³×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×S₄
Fitting	C₂×C₅×2₋¹⁺⁴	D₃
hypercenter	C₂×C₅×D₄	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₅×D₄

1476 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1476 conjugacy classes of subgroups — too many to draw. Only its 200 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 — 4 generators on 37 points:

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

Conjugacy classes & character table

215 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237128);