C₅×#11759813⋊D₃ #44477997

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

Aliases

direct product	C₅×#11759813⋊D₃
unambiguous semidirect product	#11759813⋊(C₅×D₃)

Splittings

Fitting, socle	C₃×C₅×C₁₃² ⋊ C₂×D₃
derived, Hall	C₃×#92789 ⋊ C₂²×C₅
Hall	C₃×C₅×#92789 ⋊ C₂²
Hall	C₅×C₁₃² ⋊ D₃²
Sylow	C₁₃² ⋊ C₅×D₃²

Subgroups

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

154 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 34
transitive degree	μ_t(G) = 390

A minimal faithful permutation representation — 2 generators on 34 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 586);