C₅×#83938 #14439276

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#83938

Splittings

Fitting	C₂×C₃×C₅×AGL(1,ℤ/8) ⋊ C₂
Hall	C₃×C₅ ⋊ #3153
Sylow	C₃ ⋊ C₅×#3153

Subgroups

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×AGL(1,ℤ/8)	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₅×AGL(1,ℤ/8)	D₃
hypocenter	C₃	C₅×#3153
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×AGL(1,ℤ/8)

1144 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)(15,22)(18,26)(21,27)(23,28);(7,8)(9,10,11,12)(14,19)(18,26)(20,24)(21,27);(13,15,16,23,17,22,25,28)(14,18,20,21,19,26,24,27);(10,12)(13,14)(15,21)(16,24)(17,19)(18,23)(20,25)(22,27)(26,28)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 164974);

C5×#83938 #14439276