C₃₇×#10624557 #41004346

order	26973
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₃₇ · Schur multiplier (H₂) = C₃²×C₉

Aliases

direct product

C₃₇×#10624557

Subgroups

	subgroup	G/H
center, socle	C₃³×C₃₇	C₃³
commutator	C₃²	C₃²×C₉×C₃₇
Frattini, cosocle	C₃³	C₃³×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³×C₃₇ ◁ G

410 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 170 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)(5,9,7)(10,11,14,13,12,15,18,17,16)(19,20,21,23,26,24,22,25,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(10,12,16,13,17,14,18,11,15)(19,20,24,23,26,27,22,25,21);(2,5,6)(4,8,9)(19,21,25,22,27,26,23,24,20)

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, 117);