C₉×C₁₇×C₃²⋊C₉ #27136709

order	12393
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₉²×C₁₇ · Schur multiplier (H₂) = C₃³×C₉

Aliases

direct product	C₉×C₁₇×C₃²⋊C₉
unambiguous semidirect product	C₃²⋊(C₉²×C₁₇)
unambiguous semidirect product	(C₃²×C₁₇)⋊C₉²

Subgroups

	subgroup	G/H
center	C₃²×C₉×C₁₇	C₃²
commutator	C₃	C₃×C₉²×C₁₇
Frattini, cosocle	C₃³	C₃³×C₁₇
socle	C₃³×C₁₇	C₃³
prime radical	C₃⁴×C₁₇	C₃²
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉×C₁₇ ◁ 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) = 44
transitive degree	μ_t(G) = 4131

A minimal faithful permutation representation — 3 generators on 44 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);(10,11,13,12,14,16,15,17,18)(19,20,22)(21,24,26)(23,27,25);(20,23,24)(22,26,27)

Conjugacy classes & character table

5049 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12393, 105);