C₂×C₅×C₃⋊#11759813 #44478165

order	30420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₂×C₅×C₃⋊#11759813
unambiguous semidirect product	(C₂×C₃)⋊(C₅×#11759813)

Splittings

Fitting, socle	C₂×C₃×C₅×C₁₃² ⋊ D₃
derived, Hall	C₃×#92789 ⋊ C₂²×C₅
Hall	C₃×C₅×#92789 ⋊ C₂²
Hall	C₅×C₁₃² ⋊ C₂×C₃⋊D₃
Sylow	C₁₃² ⋊ C₂×C₅×C₃⋊D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	C₃⋊#11759813
commutator, cosocle, hypocenter	C₃×#92789	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₁₃²	D₃
LCS	G ▷ C₃×#92789
UCS	C₁ ◁ C₂×C₅

192 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 36
transitive degree	μ_t(G) = 780

A minimal faithful permutation representation — 3 generators on 36 points:

(4,5)(6,7,8,9,10)(11,17,26,32,34,33,27,18,20,24,35,25,28,14,16,29,30,19,21,12,13,23,22,36,15,31);(1,2)(3,4,5);(13,15,20)(14,17,19)(16,22,30)(18,24,29)(21,27,26)(23,31,25)(28,34,35)(32,36,33)

Conjugacy classes & character table

1080 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 754);