C₃×C₃₇×#6866 #41004474

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₃₇ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₃×C₃₇×#6866

Subgroups

	subgroup	G/H
center, socle	C₃³×C₃₇	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₃₇
Frattini, cosocle	C₃³	C₃³×C₃₇
prime radical	C₃⁵×C₃₇	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

1988 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 1988 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 58
transitive degree	μ_t(G) = 2997

A minimal faithful permutation representation — 3 generators on 58 points:

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

Conjugacy classes & character table

5661 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 245);