C₅×#85453 #14511191

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

solvable

Aliases

direct product

C₅×#85453

Splittings

Fitting

C₅×#568 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄×C₅	C₂²×S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂²×C₄×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#568	D₃
socle	C₂×C₅	C₂×C₄×S₄
prime radical	C₅×#5923	C₂
cosocle	#41⋊C₃	C₂³×C₅
hypercenter	C₅×M₁₆	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄×C₅ ◁ C₅×M₁₆

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

normal subgroups only

This group has 500 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) = 53
transitive degree	μ_t(G) = 240

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

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

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, 236889);