C₅×#71777 #14285339

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

Aliases

direct product

C₅×#71777

Splittings

Hall	C₃×C₅ ⋊ #1558
Sylow	C₃ ⋊ C₅×#1558

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#217
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₂×C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₅×#1558
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×C₄⋊Q₈

456 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 32
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 11037);