C₅×#71589 #14298403

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

Aliases

direct product

C₅×#71589

Splittings

Hall	C₃×C₅ ⋊ #1822
Sylow	C₃ ⋊ C₅×#1822

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×#217
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#117	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#117	D₃
hypocenter	C₃	C₅×#1822
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂×C₅×#117

544 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 3 generators on 32 points:

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

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, 24101);