C₅×#2810⋊C₃ #14511013

order	1920
automorphism	?
homology	Abelianization (H₁) = C₃×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₅×#2810⋊C₃
unambiguous semidirect product	#2810⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#2810)⋊C₃

Splittings

Fitting, Hall	C₅×#2810 ⋊ C₃
derived, Sylow	#2810 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	#564⋊C₃
commutator, cosocle, hypocenter	#2810	C₃×C₅
Frattini	C₂³	C₅×C₂²⋊A₄
Fitting	C₅×#2810	C₃
socle	C₂³×C₅	C₂²⋊A₄
LCS	G ▷ #2810
UCS	C₁ ◁ C₂×C₅

158 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 22 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) = 29
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(6,7,11)(8,15,21)(9,18,22)(10,12,23)(13,24,17)(14,26,20)(16,25,28)(19,27,29);(6,20,10,9)(7,14,12,18)(8,19,17,16)(11,23)(13,25,15,27)(21,24)(22,26)(28,29);(6,17,10,8)(7,13,12,15)(9,19,20,16)(11,23)(14,27,18,25)(21,24)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236711);