C₅×C₃⋊#14523146 #44478013

order	30420
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₃⋊#14523146
unambiguous semidirect product	C₃⋊(C₅×#14523146)

Splittings

Fitting, socle	C₃×C₅×C₁₃² ⋊ C₃×C₄
derived	C₃×C₁₃² ⋊ C₃×C₄×C₅
Hall	C₃×C₅×#92788 ⋊ C₄
Hall	C₃×#92788 ⋊ C₄×C₅
Hall	C₅×C₁₃² ⋊ C₃×Dic₃
Sylow	C₁₃² ⋊ C₃×C₅×Dic₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊#14523146
commutator, hypocenter	C₃×C₁₃²	C₃×C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃²	C₃×C₄
prime radical	C₃×C₅×#11759815	C₂
cosocle	C₃×C₁₃⋊D₁₃	C₂×C₃×C₅
LCS	G ▷ C₃×C₁₃²
UCS	C₁ ◁ C₅

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

normal subgroups only

This group has 256 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) = 34
transitive degree	μ_t(G) = 390

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 602);