C₅×C₃⋊#14523161 #44477996

order	30420
automorphism	?
homology	Abelianization (H₁) = C₄×C₅ · Schur multiplier (H₂) = C₃×C₁₃

Aliases

direct product	C₅×C₃⋊#14523161
unambiguous semidirect product	C₃⋊(C₅×#14523161)

Splittings

Fitting, socle	C₃×C₅×C₁₃² ⋊ Dic₃
derived, Hall	C₃×#92789 ⋊ C₄×C₅
Hall	C₃×C₅×#92789 ⋊ C₄
Hall	C₅×C₁₃² ⋊ C₃⋊Dic₃
Sylow	C₁₃² ⋊ C₅×C₃⋊Dic₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	C₃⋊#14523161
commutator, hypocenter	C₃×#92789	C₄×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃²	Dic₃
prime radical	C₃×C₅×#11759816	C₂
cosocle	C₃×#11759816	C₂×C₅
LCS	G ▷ C₃×#92789
UCS	C₁ ◁ C₅

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 390

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 585);

C5×C3⋊#14523161 #44477996