C₂×C₃²×H(3,𝔽₄) #11770575

order	1152
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃² · Schur multiplier (H₂) = C₂⁶×C₃×C₄²

Aliases

direct product

C₂×C₃²×H(3,𝔽₄)

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

4224 conjugacy classes of subgroups; only the 2328 normal subgroups are shown.

normal subgroups only

This group has 4224 conjugacy classes of subgroups — too many to draw. Only its 2328 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) = 24
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 5 generators on 24 points:

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

Conjugacy classes & character table

342 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4586);