C₅×#83012 #14438664

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#83012

Splittings

Fitting	C₂×C₃×C₅×#125 ⋊ C₂
Hall	C₃×C₅ ⋊ #3142
Sylow	C₃ ⋊ C₅×#3142

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×#125	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5477	C₂
hypercenter	C₂×C₅×#125	D₃
hypocenter	C₃	C₅×#3142
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#125

784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 960

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,3,4,5)(6,7,8)(17,21,23,22)(18,26,28,27)(19,30,31,20)(24,29,32,25);(7,8)(10,14)(13,16)(17,20,21,31,23,30,22,19)(18,25,27,24,28,29,26,32);(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,30,23,31,21,20)(18,24,26,25,28,32,27,29);(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,24)(21,26)(22,27)(23,28)(25,31)(30,32)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 164362);

C5×#83012 #14438664