C₅×#85468 #14511206

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

solvable

Aliases

direct product

C₅×#85468

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

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

normal subgroups only

This group has 514 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)(6,34,53,7,25,46,14,39,52,9,29,37)(8,35,49,23,43,44,13,33,45,19,30,50)(10,17,38,21,31,51,26,22,48,16,11,47)(12,36,28,18,20,27,24,42,40,15,32,41);(6,8,14,13)(7,15,9,18)(10,24,26,12)(11,20,31,32)(16,23,21,19)(17,33,22,35)(25,43,29,30)(27,46,41,37)(28,38,40,48)(34,42,39,36)(44,51,50,47)(45,53,49,52);(6,7)(8,18)(9,14)(10,16)(11,27)(12,23)(13,15)(17,28)(19,24)(20,37)(21,26)(22,40)(25,44)(29,50)(30,51)(31,41)(32,46)(33,48)(34,45)(35,38)(36,53)(39,49)(42,52)(43,47)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236904);