C₃×C₃₇×#6869 #41004482

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃²×C₉×C₃₇ · Schur multiplier (H₂) = C₃³

Aliases

direct product

C₃×C₃₇×#6869

Subgroups

	subgroup	G/H
center	C₃×C₉×C₃₇	H(3,𝔽₃)
commutator	C₃²	C₃²×C₉×C₃₇
Frattini, cosocle	C₃×C₉	C₃³×C₃₇
socle	C₃²×C₃₇	C₃²⋊C₉
prime radical	C₃⁴×C₃₇	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

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

normal subgroups only

This group has 416 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) = 67
transitive degree	μ_t(G) = 2997

A minimal faithful permutation representation — 3 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);(4,5,8,6,11,16,12,20,24,7,9,15,13,17,23,21,25,29,14,10,18,22,19,26,28,27,30);(5,9,10)(11,17,19)(20,25,27)

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, 253);