C₃₇×#10624466 #41004255

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₉²×C₃₇ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₃₇×#10624466

Subgroups

	subgroup	G/H
center, socle	C₃³×C₃₇	H(3,𝔽₃)
commutator	C₃²	C₉²×C₃₇
Frattini, cosocle	C₃⁴	C₃²×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

434 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 134 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)(19,20,22)(21,23,26)(24,27,25)(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);(10,11,13,12,14,16,15,17,18)(20,23,24)

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, 26);