C₃₇×AΣL(1,𝔽₂₇)⋊C₉ #41004509

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₃₇ · Schur multiplier (H₂) = C₃³

Aliases

direct product	C₃₇×AΣL(1,𝔽₂₇)⋊C₉
unambiguous semidirect product	AΣL(1,𝔽₂₇)⋊(C₉×C₃₇)

Subgroups

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₃₇
Frattini, cosocle	C₃³	C₃³×C₃₇
prime radical	C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

304 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 73
transitive degree	μ_t(G) = 2997

A minimal faithful permutation representation — 3 generators on 73 points:

(1,2,5,4,3,6,9,8,7)(10,11,14)(12,18,23)(13,17,24)(15,25,21)(16,28,22)(19,30,33)(20,27,34)(26,36,32)(29,35,31)(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);(10,12,19)(11,15,26)(13,21,31)(14,23,33)(16,18,29)(17,27,30)(20,32,22)(24,25,35)(28,34,36);(1,3,7,4,8,5,9,2,6)(12,20,21)(15,18,27)(19,31,32)(23,34,25)(26,30,29)(33,35,36)

Conjugacy classes & character table

3885 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 280);