C₃×C₃₇×#6859 #41004370

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

Aliases

direct product

C₃×C₃₇×#6859

Subgroups

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

488 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 76
transitive degree	μ_t(G) = 8991

A minimal faithful permutation representation — 3 generators on 76 points:

(1,2,3)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76);(5,7,9)(13,14,17)(15,21,25)(16,20,26)(18,27,23)(19,30,24)(22,32,35)(28,38,39)(29,37,34)(31,36,33);(4,5,8)(6,9,12)(7,11,10)(13,15,22,24,34,39,16,23,33)(14,18,28,19,21,31,20,29,32)(17,25,35,30,37,38,26,27,36)

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, 141);