C₅×#3567⋊C₃ #14511400

order	1920
isoclinism	isoclinic class of #3567⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	C₅×#3567⋊C₃
unambiguous semidirect product	#3567⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3567)⋊C₃

Splittings

Fitting, Hall	C₅×#3567 ⋊ C₃
derived	#564 ⋊ C₂×C₃×C₅
Sylow	#3567 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅	#3567⋊C₃
commutator, cosocle, hypocenter	#564	C₂×C₃×C₅
Frattini	C₂²	C₂×C₅×C₂²⋊A₄
Fitting	C₅×#3567	C₃
socle	C₂²×C₅	C₂×C₂²⋊A₄
LCS	G ▷ #564
UCS	C₁ ◁ C₅

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

normal subgroups only

This group has 370 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)(8,17,40)(9,33,27,21,22,47)(10,24,29)(11,16,30)(13,28,41,31,32,18)(14,35,20)(15,38,26)(19,37,51)(23,46,53,25,45,48)(34,50,42,39,49,43)(36,52,44);(6,10,11,26)(7,16)(8,44,20,19)(9,25,21,23)(12,29,30,38)(13,31)(15,24)(18,42,41,43)(27,53,47,48)(28,49,32,50)(34,39)(35,52,40,51);(6,18,11,41)(7,33)(8,21,20,9)(10,43,26,42)(12,40,30,35)(13,17)(14,31)(15,46)(16,22)(19,23,44,25)(24,45)(27,28,47,32)(29,51,38,52)(34,37)(36,39)(48,50,53,49)

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, 237098);