C₃₇×#10624692 #41004481

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

Aliases

direct product

C₃₇×#10624692

Subgroups

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

362 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

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, 252);