C₃₇×#10624450 #41004239

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃×C₉×C₃₇ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₃₇×#10624450

Subgroups

	subgroup	G/H
center, socle	C₃³×C₃₇	H(3,𝔽₃)
commutator	C₃³	C₃×C₉×C₃₇
Frattini, cosocle	C₃⁴	C₃²×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

254 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 8991

A minimal faithful permutation representation — 2 generators on 64 points:

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

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, 10);