C₃×C₃₇×#6913 #41004717

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₃₇ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₃×C₃₇×#6913

Subgroups

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

992 conjugacy classes of subgroups; only the 476 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 476 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) = 67
transitive degree	μ_t(G) = 2997

A minimal faithful permutation representation — 4 generators on 67 points:

(1,2,3)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67);(4,5,8)(6,10,15)(7,9,16)(11,20,14)(12,17,24)(13,19,25)(18,27,21)(22,26,29)(23,28,30);(5,9,11)(6,12,21)(10,18,17)(13,23,22)(15,24,27)(25,30,29);(4,6,13,7,12,22,14,21,23)(5,10,19,9,17,26,11,18,28)(8,15,25,16,24,29,20,27,30)

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, 488);