D₃×C₅²⋊#3862 #36348004

order	22050
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₅×C₇

Aliases

direct product	D₃×C₅²⋊#3862
unambiguous semidirect product	C₃⋊(C₂×C₅²⋊#3862)
unambiguous semidirect product	(C₅²×D₃)⋊#3862

Splittings

Fitting, socle, derived	C₃×C₅²×C₇² ⋊ C₂×C₃
Hall	C₃×C₅²⋊#3862 ⋊ C₂
Hall	C₅²×C₇² ⋊ C₃×D₃
Sylow	C₇² ⋊ D₃×C₅²⋊C₃
Sylow	C₅² ⋊ D₃×#3862

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₃×C₅²×C₇²
UCS	C₁

168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 315

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

1233 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22050, 441);

D3×C52⋊#3862 #36348004