C₂×C₉×#543 #11768177

order	1152
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₉ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×C₉×#543

Subgroups

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂⁵×C₃	C₂²×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

1944 conjugacy classes of subgroups; only the 1308 normal subgroups are shown.

normal subgroups only

This group has 1944 conjugacy classes of subgroups — too many to draw. Only its 1308 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) = 27
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 5 generators on 27 points:

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

Conjugacy classes & character table

396 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2188);