C₅×#3556⋊C₃ #14511399

order	1920
isoclinism	isoclinic class of #3556⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	C₅×#3556⋊C₃
unambiguous semidirect product	#3556⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3556)⋊C₃

Splittings

Fitting, Hall	C₅×#3556 ⋊ C₃
derived	#564 ⋊ C₂×C₃×C₅
Sylow	#3556 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅	#3556⋊C₃
commutator, cosocle, hypocenter	#564	C₂×C₃×C₅
Frattini	C₂²	C₂×C₅×C₂²⋊A₄
Fitting	C₅×#3556	C₃
socle	C₂²×C₅	C₂×C₂²⋊A₄
LCS	G ▷ #564
UCS	C₁ ◁ C₅

350 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 34 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)(6,7,12,11,16,27)(8,14,42)(9,21,45,23,46,53)(10,15,30,26,40,49)(13,52,43)(17,18,36)(19,34,48)(20,39,51)(22,33,47,25,37,28)(24,38,50)(29,41,35)(31,32,44);(6,11)(7,40,16,15)(9,23)(10,26)(12,49,27,30)(13,34,31,35)(14,38,18,39)(21,37,46,33)(22,25)(28,53,47,45)(29,32,48,52)(36,50,42,51);(6,44,11,43)(7,33,16,37)(8,22,17,25)(9,20,23,24)(10,19,26,41)(12,36)(13,14,31,18)(15,21,40,46)(27,42)(28,32)(29,45)(30,51)(34,39,35,38)(47,52)(48,53)(49,50)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237097);