C₃₇×#10624851 #41004640

order	26973
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₃³×C₃₇ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₃₇×#10624851

Subgroups

	subgroup	G/H
center, socle	C₃×C₃₇	#6911
commutator, Frattini, cosocle	C₃×C₉	C₃³×C₃₇
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₃₇ ◁ C₃²×C₃₇ ◁ C₃×3₋¹⁺²×C₃₇ ◁ G

200 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 999

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

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

Conjugacy classes & character table

1517 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 411);