C₂×C₅×D₃×#92788 #44478170

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

Aliases

direct product	C₂×C₅×D₃×#92788
unambiguous semidirect product	C₃⋊(C₂²×C₅×#92788)
unambiguous semidirect product	(C₂×C₃)⋊(C₂×C₅×#92788)

Splittings

Fitting, socle	C₂×C₃×C₅×C₁₃² ⋊ C₂×C₃
derived	C₃×C₁₃² ⋊ C₂²×C₃×C₅
Hall	C₃×C₅×#92788 ⋊ C₂²
Hall	C₃×#92788 ⋊ C₂²×C₅
Hall	C₅×C₁₃² ⋊ C₂×C₃×D₃
Sylow	C₁₃² ⋊ C₂×C₃×C₅×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	D₃×#92788
commutator, cosocle, hypocenter	C₃×C₁₃²	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅×C₁₃²	C₂×C₃
LCS	G ▷ C₃×C₁₃²
UCS	C₁ ◁ C₂×C₅

704 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 238 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) = 5070

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

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

Conjugacy classes & character table

1770 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 759);