C₃²×C₅⋊(C₅⋊(C₇⋊D₇)) #36348067

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

Aliases

direct product	C₃²×C₅⋊(C₅⋊(C₇⋊D₇))
unambiguous semidirect product	C₅⋊(C₃²×C₅⋊(C₇⋊D₇))

Splittings

Fitting, socle, Hall	C₃²×C₅²×C₇² ⋊ C₂
derived, Hall	C₅²×C₇² ⋊ C₂×C₃²
Hall	C₃²×C₇² ⋊ C₅⋊D₅
Hall	C₃²×C₅² ⋊ C₇⋊D₇
Sylow	C₇² ⋊ C₃²×C₅⋊D₅
Sylow	C₅² ⋊ C₃²×C₇⋊D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	C₅⋊(C₅⋊(C₇⋊D₇))
commutator, cosocle, hypocenter	C₅²×C₇²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₇²	C₂
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₃²

960 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 11025

A minimal faithful permutation representation — 3 generators on 30 points:

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

Conjugacy classes & character table

5526 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22050, 504);

C32×C5⋊(C5⋊(C7⋊D7)) #36348067