C₃₇×UT(4,𝔽₃) #41004536

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃³×C₃₇ · Schur multiplier (H₂) = C₃⁵

Aliases

direct product

C₃₇×UT(4,𝔽₃)

Subgroups

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

560 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 92 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)(3,9,14)(4,8,15)(6,16,12)(7,19,13)(10,22,25)(11,18,21)(17,27,24)(20,26,23)(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);(1,3,10)(2,6,17)(4,12,23)(5,14,25)(7,9,20)(8,18,22)(11,24,13)(15,16,26)(19,21,27);(1,2,5)(3,6,14)(4,8,15)(7,19,13)(9,21,11)(10,17,25)(12,18,16)(20,27,24)(22,26,23)

Conjugacy classes & character table

2109 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 307);