C₅×#69317 #14315304

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#69317

Splittings

Fitting	C₃×C₅×#366 ⋊ C₂
Hall	C₃×C₅ ⋊ C₄×D₁₆
Sylow	C₃ ⋊ C₄×C₅×D₁₆

Subgroups

	subgroup	G/H
center	C₂²×C₅	D₃×D₈
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#366	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₅×D₄₈	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#366	D₃
hypocenter	C₃	C₄×C₅×D₁₆
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#366

348 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 28
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 28 points:

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

Conjugacy classes & character table

330 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 41002);