C₅×#67679 #14277113

order	1920
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#67679

Splittings

Hall	C₃×C₅ ⋊ #1293
Sylow	C₃ ⋊ C₅×#1293

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₃⋊#95
commutator	C₂×C₃×C₄	C₄²×C₅
Frattini	C₂³×C₄	C₂×C₅×D₃
Fitting	C₂×C₃×C₅×C₂²⋊C₈	C₂
socle	C₂²×C₃×C₅	#95
prime radical	C₂⁴×C₃×C₅	Q₈
cosocle	C₂³×C₃×C₄	C₂²×C₅
hypercenter	C₂×C₅×C₂²⋊C₈	D₃
hypocenter	C₃	C₅×#1293
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂×C₅×C₂²⋊C₈

368 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 480

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 2811);