C₃×C₃₇×C₉⋊H(3,𝔽₃) #41004656

order	26973
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₃₇ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₃×C₃₇×C₉⋊H(3,𝔽₃)

Subgroups

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

1328 conjugacy classes of subgroups; only the 548 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 548 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) = 8991

A minimal faithful permutation representation — 4 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);(5,7,10)(8,12,11)(13,14,16)(15,18,20)(17,21,19);(4,5,8,6,7,11,9,10,12);(14,17,18)(16,20,21)

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, 427);