C₃₇×#10624934 #41004723

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₃₇ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₃₇×#10624934

Subgroups

	subgroup	G/H
center, socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₃₇ ◁ C₃×3₋¹⁺²×C₃₇ ◁ G

980 conjugacy classes of subgroups; only the 470 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 470 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 — 4 generators on 64 points:

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

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, 494);