C₃₇×#10624762 #41004551

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₃₇ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

C₃₇×#10624762

Subgroups

	subgroup	G/H
center, socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₃₇
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₃₇ ◁ C₃³×C₃₇ ◁ G

446 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 64
transitive degree	μ_t(G) = 999

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6)(3,9,5)(4,11,20)(7,17,14)(8,19,15)(10,16,13)(12,18,25)(21,27,23)(22,26,24)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(1,3,9,5,8,19,15,2,6)(4,10,20,14,11,17,13,7,16)(12,21,27,23,22,26,24,18,25);(1,4,12)(2,7,18)(3,10,21)(5,14,23)(6,16,25)(8,11,22)(9,20,27)(13,24,15)(17,26,19)

Conjugacy classes & character table

2109 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 322);