C₅×#85352 #14511090

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

solvable

Aliases

direct product

C₅×#85352

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×#519	D₃
prime radical	C₂²×C₅×SL(2,𝔽₃)	C₂²
hypercenter	C₅×C₂²⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₅×#41
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₂²⋊C₄

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

normal subgroups only

This group has 386 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)(7,11)(10,13)(14,29,31)(15,22,37,21,30,36)(16,25,43,24,38,41)(17,28,19)(20,42,45)(23,27)(26,39)(33,44,35);(6,8,9,12)(7,10,11,13)(14,16)(15,20)(17,25)(18,26)(19,24)(21,35)(22,33)(23,34)(27,40)(28,41)(29,38)(30,42)(31,43)(32,39)(36,44)(37,45);(6,7)(8,10)(9,11)(12,13)(14,15,19,21)(16,20,24,35)(17,27,29,23)(18,30,32,22)(25,40,38,34)(26,42,39,33)(28,37,31,36)(41,45,43,44)

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, 236788);