C₃×C₅⋊(C₅⋊(ΓL(1,𝔽₈)⋊D₇)) #36347982

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

Aliases

direct product	C₃×C₅⋊(C₅⋊(ΓL(1,𝔽₈)⋊D₇))
unambiguous semidirect product	C₅⋊(C₃×C₅⋊(ΓL(1,𝔽₈)⋊D₇))
unambiguous semidirect product	ΓL(1,𝔽₈)⋊(C₃×C₅⋊D₃₅)

	subgroup	G/H
center, hypercenter	C₃	C₅⋊(C₅⋊(ΓL(1,𝔽₈)⋊D₇))
commutator, cosocle, hypocenter	C₅²×C₇²	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇²	C₂×C₃
LCS	G ▷ C₅²×C₇²
UCS	C₁ ◁ C₃

448 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 1575

A minimal faithful permutation representation — 3 generators on 27 points:

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

1326 conjugacy classes. The character table is too large to display.

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22050, 419);