C₉×C₃₇×C₃²⋊C₉ #41004334

order	26973
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃×C₉²×C₃₇ · Schur multiplier (H₂) = C₃³×C₉

Aliases

direct product	C₉×C₃₇×C₃²⋊C₉
unambiguous semidirect product	C₃²⋊(C₉²×C₃₇)

Subgroups

	subgroup	G/H
center	C₃²×C₉×C₃₇	C₃²
commutator	C₃	C₃×C₉²×C₃₇
Frattini, cosocle	C₃³	C₃³×C₃₇
socle	C₃³×C₃₇	C₃³
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉×C₃₇ ◁ G

662 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 332 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) = 8991

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

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

Conjugacy classes & character table

10989 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 105);