C₅×#84102 #14424651

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₅×#84102

Splittings

Fitting	C₂×C₃×C₅×H(5,𝔽₂) ⋊ C₂
Hall	C₃×C₅ ⋊ #3012
Sylow	C₃ ⋊ C₅×#3012

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×H(5,𝔽₂)	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₅×H(5,𝔽₂)	D₃
hypocenter	C₃	C₅×#3012
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×H(5,𝔽₂)

1528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 4 generators on 28 points:

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

Conjugacy classes & character table

330 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 150349);

C5×#84102 #14424651