C₂×C₉×#458 #11767745

order	1152
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×C₉×#458

Subgroups

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂³	C₂⁴×C₉
Frattini, cosocle	C₂³×C₃	C₂⁴×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₂×C₃×H(5,𝔽₂)	C₂×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂⁴×C₉ ◁ G

1191 conjugacy classes of subgroups; only the 324 normal subgroups are shown.

normal subgroups only

This group has 1191 conjugacy classes of subgroups — too many to draw. Only its 324 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 — 4 generators on 27 points:

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

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 1756);