C₅×#73154 #14510878

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

solvable

Aliases

direct product

C₅×#73154

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#5432
commutator	C₂×SL(2,𝔽₃)	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₂×C₅×2₋¹⁺⁴	D₃
prime radical	C₂×C₅×2₋¹⁺⁴⋊C₃	C₂
cosocle	C₂×#41⋊C₃	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 ◁ C₂×C₄×C₅ ◁ C₂×C₅×D₄

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

normal subgroups only

This group has 380 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)(6,21,23)(7,14,31,12,22,29)(8,17,35,16,30,33)(9,20,11)(13,34,37)(15,19)(18,28)(25,26,36);(6,7,8,13,11,12,16,25)(9,19,17,32,21,15,30,27)(10,22,18,34,24,14,28,26)(20,31,33,37,23,29,35,36)

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, 236576);