C₃₇×#10624563 #41004352

order	26973
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃³×C₃₇ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

C₃₇×#10624563

Subgroups

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

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

normal subgroups only

This group has 758 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) = 64
transitive degree	μ_t(G) = 8991

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,4,3,6,8,7,5,9)(10,11,13)(12,15,17)(14,18,16)(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);(2,5,6)(4,8,9)(19,20,22)(21,24,26)(23,27,25);(11,14,15)(13,17,18)(20,23,24)(22,26,27)

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, 123);