C₃×C₃₇×#6858 #41004369

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₃₇ · Schur multiplier (H₂) = C₃²

Aliases

direct product

C₃×C₃₇×#6858

Subgroups

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

380 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 76
transitive degree	μ_t(G) = 8991

A minimal faithful permutation representation — 3 generators on 76 points:

(1,2,3)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76);(5,7,9)(13,14,17,16,20,26,24,19,30)(15,21,25,23,18,27,34,29,37)(22,32,35,33,31,36,39,28,38);(4,5,8)(6,9,12)(7,11,10)(13,15,22,16,23,33,24,34,39)(14,18,28,20,29,32,19,21,31)(17,25,35,26,27,36,30,37,38)

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, 140);