C₅₃×H(3,𝔽₃)⋊C₂₇ #50311332

order	38637
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃×C₂₇×C₅₃ · Schur multiplier (H₂) = C₃²

Aliases

direct product	C₅₃×H(3,𝔽₃)⋊C₂₇
unambiguous semidirect product	H(3,𝔽₃)⋊(C₂₇×C₅₃)
unambiguous semidirect product	(H(3,𝔽₃)×C₅₃)⋊C₂₇

Subgroups

	subgroup	G/H
center	C₃×C₉×C₅₃	H(3,𝔽₃)
commutator	C₃²	C₃×C₂₇×C₅₃
Frattini, cosocle	C₃²×C₉	C₃²×C₅₃
socle	C₃²×C₅₃	C₃²⋊C₉
prime radical	C₃×H(3,𝔽₃)×C₅₃	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₅₃ ◁ C₃²×C₉×C₅₃ ◁ G

144 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 12879

A minimal faithful permutation representation — 2 generators on 89 points:

(1,2,5,3,7,6,4,8,9)(10,11,14,12,15,20,17,22,27,13,16,21,18,23,28,25,30,33,19,24,29,26,31,34,32,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,6)(3,7,9)(4,8,5)

Conjugacy classes & character table

8109 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38637, 7);