C₂×C₅×#11759813⋊C₃ #44477899

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

Aliases

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

Splittings

Fitting, socle	C₂×C₅×C₁₃² ⋊ C₃×D₃
derived	#92789 ⋊ C₂²×C₃×C₅
Hall	C₅×C₁₃⋊C₃² ⋊ C₂²
Hall	C₁₃⋊C₃² ⋊ C₂²×C₅
Hall	C₅×C₁₃² ⋊ C₂×C₃×D₃
Sylow	C₁₃² ⋊ C₂×C₃×C₅×D₃

Subgroups

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

160 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 33
transitive degree	μ_t(G) = 260

A minimal faithful permutation representation — 2 generators on 33 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 488);