C₅×#85467 #14511205

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

solvable

Aliases

direct product

C₅×#85467

Splittings

Fitting

C₅×#576 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	D₄×S₄
commutator, cosocle	#41⋊C₃	C₂³×C₅
Frattini	C₄	C₂²×C₅×S₄
Fitting	C₅×#576	D₃
hypercenter	C₅×Q₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₂×C₅×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×Q₁₆

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

normal subgroups only

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

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

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236903);