C₅×#85563 #14511301

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

solvable

Aliases

direct product

C₅×#85563

Splittings

derived

C₂²⋊SL(2,𝔽₃) ⋊ C₄×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₅	#1020
commutator, hypocenter	C₂²⋊SL(2,𝔽₃)	C₄×C₅
Frattini	C₄	C₅×#1020
Fitting	C₂²×C₅×#41	D₃
socle	C₂³×C₅	SL(2,ℤ/4)
prime radical	C₅×#5948	C₂
cosocle	#5948	C₂×C₅
LCS	G ▷ C₂²⋊SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅

370 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 29
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236999);