C₅×#85356 #14511094

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#85356

Splittings

Fitting

C₅×#519 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂³×C₅
Frattini	C₂²	C₂²×C₅×S₄
Fitting	C₅×#519	D₃
prime radical	C₅×#5421	C₂
hypercenter	C₅×C₂²⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊C₄

416 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236792);