C₅×C₉⋊#10621725 #44477607

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

Aliases

direct product	C₅×C₉⋊#10621725
unambiguous semidirect product	C₉⋊(C₅×#10621725)

Splittings

derived, Hall	C₉×C₁₃² ⋊ C₄×C₅
Hall	C₅×C₉×C₁₃² ⋊ C₄
Hall	C₅×C₁₃² ⋊ Dic₉
Sylow	C₁₃² ⋊ C₅×Dic₉
Hall	C₅×C₉ ⋊ #10621725
Sylow	C₉ ⋊ C₅×#10621725

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅	C₉⋊(C₁₃⋊D₁₃)
commutator, hypocenter	C₉×C₁₃²	C₄×C₅
Frattini	C₂×C₃	C₅×C₃⋊(C₁₃⋊D₁₃)
Fitting	C₂×C₅×C₉×C₁₃²	C₂
socle, prime radical	C₂×C₃×C₅×C₁₃²	D₃
cosocle	C₂×C₉×C₁₃²	C₂×C₅
LCS	G ▷ C₉×C₁₃²
UCS	C₁ ◁ C₂×C₅

288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 30420

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,3,4,5)(6,7,8,9)(11,13)(12,16)(14,17)(15,18)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44);(10,15,13,12,17,14,16,11,18)(32,33,35,37,39,41,43,44,42,40,38,36,34);(19,20,22,24,26,28,30,31,29,27,25,23,21)

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, 196);

C5×C9⋊#10621725 #44477607