C₅×#85448 #14511186

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

solvable

Aliases

direct product

C₅×#85448

Splittings

Fitting

C₂×C₅×#131 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₅×C₈	C₂×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₂×C₅×#131	D₃
socle	C₂×C₅	C₂×C₄×S₄
prime radical	C₅×#5923	C₂
cosocle	#41⋊C₃	C₂³×C₅
hypercenter	C₂×C₅×C₈	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₈

496 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 69 points:

(1,2,3,4,5)(6,45,53,18,10,44,24,40,29,8,51,66,12,19,50,42,27,47,9,64,26,21,22,63)(7,46,61,33,16,58,32,56,39,14,52,69,13,34,60,55,28,59,15,65,38,31,36,68)(11,20,23,41,30,48,54,67)(17,35,37,57,25,43,49,62);(7,13)(14,31)(15,32)(16,28)(17,25)(33,55)(34,46)(35,43)(36,52)(37,49)(38,39)(56,65)(57,62)(58,59)(60,61)(68,69);(6,7)(8,14)(9,15)(10,25)(11,28)(12,13)(16,30)(17,27)(18,33)(19,43)(20,46)(21,31)(22,49)(23,52)(24,32)(26,39)(29,38)(34,48)(35,45)(36,54)(37,51)(40,62)(41,65)(42,55)(44,59)(47,58)(50,61)(53,60)(56,67)(57,64)(63,69)(66,68)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236884);