C₅×#73074 #14510798

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

solvable

Aliases

direct product

C₅×#73074

Splittings

Fitting

C₅×#519 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	(C₂×S₄)⋊C₄
commutator	C₂×SL(2,𝔽₃)	C₂×C₄×C₅
Frattini	C₂³	C₂×C₅×S₄
Fitting	C₅×#519	D₃
prime radical	C₂×C₅×#983	C₂
cosocle	C₂²×SL(2,𝔽₃)	C₂²×C₅
hypercenter	C₅×C₂²⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₅×C₂²⋊C₄

324 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,3,4,5)(6,32,24,8,22,35)(7,30,28,16,23,37,13,26,29,12,14,36)(9,34,11,18,21,17)(10,19)(15,27,20,31)(25,33);(6,7)(8,16)(9,20)(10,23)(11,13)(12,17)(14,25)(15,22)(18,27)(19,26)(21,29)(24,28)(30,33)(31,32)(34,36)(35,37)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236496);