C₅×#85541 #14511279

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

solvable

Aliases

direct product

C₅×#85541

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂²×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#585	D₃
socle	C₂×C₅	C₂²×SL(2,ℤ/4)
prime radical	C₅×#5945	C₂
cosocle	#41⋊C₃	C₂³×C₅
hypercenter	C₅×#41	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₅×#41

608 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 53
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,22,45)(7,26,44)(8,39,53,18,36,51)(9,27,23)(10,43,16)(11,34,42)(12,25,41)(13,47,52,28,40,48)(14,15,24)(17,32,33)(19,20,37,31,35,49)(21,30,50,29,46,38);(6,8)(7,13)(9,19)(10,20)(11,21)(12,18)(14,29)(15,30)(16,31)(17,28)(22,36)(23,37)(24,38)(25,39)(26,40)(27,35)(32,47)(33,48)(34,46)(41,51)(42,50)(43,49)(44,52)(45,53);(6,7,9,14,12,17,16,11)(8,13,19,29,18,28,31,21)(10,23,22,41,27,43,25,45)(15,33,32,42,34,44,26,24)(20,37,36,51,35,49,39,53)(30,48,47,50,46,52,40,38)

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, 236977);