C₅×#85363 #14511101

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

solvable

Aliases

direct product

C₅×#85363

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×#520	D₃
prime radical	C₂×C₅×#41⋊C₃	C₂²
hypercenter	C₅×C₄⋊C₄	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×Q₈
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×C₄⋊C₄

386 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 640

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

(1,2,3,4,5)(6,8,9,13)(7,10,11,12)(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);(14,16,19,24)(15,20,21,35)(17,25,29,38)(18,26,32,39)(22,33,30,42)(23,34,27,40)(28,41,31,43)(36,44,37,45);(6,7,9,11)(8,12,13,10)(14,15)(16,20)(17,27)(18,30)(19,21)(22,32)(23,29)(24,35)(25,40)(26,42)(28,37)(31,36)(33,39)(34,38)(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, 236799);