C₅×#85425 #14511163

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

solvable

Aliases

direct product

C₅×#85425

Splittings

Fitting

C₅×#557 ⋊ 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×#557	D₃
prime radical	C₂×C₅×GL(2,𝔽₃)	C₂²
hypercenter	C₅×C₄⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

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

normal subgroups only

This group has 378 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) = 45
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,8,9,12)(7,11,10,13)(14,29,31)(15,22,37,21,30,36)(16,40,42)(17,28,19)(20,34,45,33,41,44)(23,27)(24,39,26)(35,38);(6,7,9,10)(8,13,12,11)(14,16,19,26)(15,20,21,33)(17,24,29,40)(18,25,32,43)(22,34,30,41)(23,35,27,38)(28,39,31,42)(36,44,37,45);(7,10)(8,12)(14,15)(16,20)(17,27)(18,30)(19,21)(22,32)(23,29)(24,38)(25,41)(26,33)(28,37)(31,36)(34,43)(35,40)(39,45)(42,44)

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, 236861);