C₃²×#1392 #11768441

order	1152
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#1392

Subgroups

	subgroup	G/H
center	C₃²×C₄	#99
commutator	C₂³	C₃²×C₄²
Frattini, cosocle	C₂×#41	C₂²×C₃²
socle	C₂×C₃²	#342
prime radical	C₂×C₃²×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×#41 ◁ G

474 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 38
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 2 generators on 38 points:

(1,2,3)(7,8,11,16,13,18,12,17)(9,19,21,34,23,36,22,35)(10,24,25,33,26,15,27,31)(14,28,29,20,30,37,32,38);(4,5,6)(7,9,12,22,13,23,11,21)(8,14,17,29,18,30,16,32)(10,20,25,37,26,38,27,28)(15,19,31,35,24,36,33,34)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2452);