C₂×C₅×C₃⋊((C₁₃⋊C₃)⋊D₁₃) #44478168

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

Aliases

direct product	C₂×C₅×C₃⋊((C₁₃⋊C₃)⋊D₁₃)
unambiguous semidirect product	(C₂×C₃)⋊(C₅×(C₁₃⋊C₃)⋊D₁₃)
unambiguous semidirect product	(C₂×C₁₃⋊C₃)⋊(C₅×D₃₉)

Splittings

Fitting, socle	C₂×C₃×C₅×C₁₃² ⋊ C₂×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	C₂×C₅	C₃⋊((C₁₃⋊C₃)⋊D₁₃)
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

256 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 1170

A minimal faithful permutation representation — 2 generators on 36 points:

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

Conjugacy classes & character table

1410 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 757);