C₅×(C₃⋊(D₁₃⋊C₃))⋊D₁₃ #44478007

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

Aliases

direct product	C₅×(C₃⋊(D₁₃⋊C₃))⋊D₁₃
unambiguous semidirect product	(C₃⋊(D₁₃⋊C₃))⋊(C₅×D₁₃)
unambiguous semidirect product	((C₁₃⋊C₃)⋊D₁₃)⋊(C₅×D₃)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₅	(C₃⋊(D₁₃⋊C₃))⋊D₁₃
commutator, cosocle, hypocenter	C₃×C₁₃²	C₂²×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃²	C₂²×C₃
LCS	G ▷ C₃×C₁₃²
UCS	C₁ ◁ C₅

192 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 1170

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 596);