C₂×C₅×C₉⋊(C₁₃⋊D₁₃) #44477789

order	30420
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂×C₁₃

Aliases

direct product	C₂×C₅×C₉⋊(C₁₃⋊D₁₃)
unambiguous semidirect product	(C₂×C₉)⋊(C₅×C₁₃⋊D₁₃)
unambiguous semidirect product	(C₂×C₁₃)⋊(C₅×D₁₁₇)

Splittings

Fitting	C₂×C₅×C₉×C₁₃² ⋊ C₂
derived, Hall	C₉×C₁₃² ⋊ C₂²×C₅
Hall	C₅×C₉×C₁₃² ⋊ C₂²
Hall	C₅×C₁₃² ⋊ C₂×D₉
Sylow	C₁₃² ⋊ C₂×C₅×D₉
Hall	C₅×C₉ ⋊ C₂×C₁₃⋊D₁₃
Sylow	C₉ ⋊ C₂×C₅×C₁₃⋊D₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	C₉⋊(C₁₃⋊D₁₃)
commutator, cosocle, hypocenter	C₉×C₁₃²	C₂²×C₅
Frattini	C₃	C₂×C₅×C₃⋊(C₁₃⋊D₁₃)
Fitting	C₂×C₅×C₉×C₁₃²	C₂
socle	C₂×C₃×C₅×C₁₃²	D₃
LCS	G ▷ C₉×C₁₃²
UCS	C₁ ◁ C₂×C₅

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 15210

A minimal faithful permutation representation — 3 generators on 42 points:

(3,4,5,6,7)(9,11)(10,14)(12,13)(15,16)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42);(1,2)(8,16,11,14,13,12,10,9,15)(30,31,33,35,37,39,41,42,40,38,36,34,32);(17,18,20,22,24,26,28,29,27,25,23,21,19)

Conjugacy classes & character table

7620 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30420, 378);

C2×C5×C9⋊(C13⋊D13) #44477789