C₅×#69841 #14298919

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#69841

Splittings

Hall	C₃×C₅ ⋊ #1838
Sylow	C₃ ⋊ C₅×#1838

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#97	D₃
hypocenter	C₃	C₅×#1838
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#97

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

normal subgroups only

This group has 360 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) = 960

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

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

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, 24617);