C₃×#20301292 #36347958

order	22050
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃×C₅×C₇

Aliases

direct product

C₃×#20301292

Splittings

Fitting, socle	C₃×C₅²×C₇² ⋊ C₂×C₃
derived, Hall	C₅²×C₇² ⋊ C₂×C₃²
Hall	C₃×C₅²⋊#3862 ⋊ C₂
Hall	C₅⋊(C₅⋊(C₇⋊D₇)) ⋊ C₃²
Sylow	C₇² ⋊ C₃×(C₅⋊D₅)⋊C₃
Sylow	C₅² ⋊ C₃×#64425

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#20301292
commutator, cosocle, hypocenter	C₅²×C₇²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₃

192 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)(3,13,9,6,7,15)(5,10,14,11,12,8)(17,20,21,18,19,22)(24,28,27,25,26,29);(1,3,8,14,6)(2,5,7,13,11)(4,9,10,12,15)(16,17,19,22,21,20,18)(23,24,27,28,26,29,25)(30,31,32)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22050, 395);

C3×#20301292 #36347958