C₅×#73284 #14511008

order	1920
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₂²×C₄

solvable

Aliases

direct product

C₅×#73284

Splittings

Fitting, Hall	C₄²×C₅×Q₈ ⋊ C₃
derived, Sylow	C₄²×Q₈ ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	C₂²⋊(C₄²⋊C₃)
commutator, cosocle, hypocenter	C₄²×Q₈	C₃×C₅
Frattini	C₂³	C₅×C₂²⋊A₄
Fitting	C₄²×C₅×Q₈	C₃
socle	C₂³×C₅	C₂²⋊A₄
LCS	G ▷ C₄²×Q₈
UCS	C₁ ◁ C₂×C₅

278 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 25
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236706);