C₅×#73156 #14510880

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

solvable

Aliases

direct product

C₅×#73156

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#5432
commutator	C₂×SL(2,𝔽₃)	C₂×C₄×C₅
Frattini	C₂³	C₂×C₅×S₄
Fitting	C₂×C₅×2₋¹⁺⁴	D₃
prime radical	C₂×C₅×2₋¹⁺⁴⋊C₃	C₂
cosocle	C₂²×SL(2,𝔽₃)	C₂²×C₅
hypercenter	C₂×C₅×D₄	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂×C₅×D₄

386 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 56 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)(7,30,29,12,23,37)(9,34,11,18,21,17)(10,19)(13,26,28,16,14,36)(15,27)(20,31)(25,33);(6,7)(8,16,17,12)(9,20)(10,23)(11,13)(14,25)(15,22)(18,27,32,31)(19,26,33,30)(21,29)(24,28)(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, 236578);