C₅×#87497 #14511431

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

solvable

Aliases

direct product

C₅×#87497

Splittings

Fitting

C₅×#585 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂²×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×S₄
Fitting	C₅×#585	D₃
socle	C₂×C₅	C₂³×S₄
hypercenter	C₅×#41	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#41

1476 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1476 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 4 generators on 37 points:

(1,2,3,4,5)(6,21,22)(7,14,29,13,26,28)(8,32,33)(9,20,11)(12,24,37,23,34,36)(15,27)(16,31,18)(25,35);(6,8,11,18)(7,12,13,23)(9,16,21,32)(10,17,19,30)(14,24,26,34)(15,25,27,35)(20,31,22,33)(28,36,29,37);(6,7)(8,12)(9,14)(10,15)(11,13)(16,24)(17,25)(18,23)(19,27)(20,28)(21,26)(22,29)(30,35)(31,36)(32,34)(33,37);(9,19)(10,21)(14,27)(15,26)(16,30)(17,32)(20,22)(24,35)(25,34)(28,29)(31,33)(36,37)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237129);