C₃²×#1391 #11768440

order	1152
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄² · Schur multiplier (H₂) = C₂×C₃×C₄

Aliases

direct product

C₃²×#1391

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

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

normal subgroups only

This group has 498 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)(9,17)(10,21,22,14)(11,15)(12,16)(13,18,25,31)(19,29)(20,30)(23,34,35,27)(24,28)(26,32,36,38)(33,37);(4,5,6)(7,9,11,19,12,20,24,33)(8,13,15,26,16,25,28,36)(10,18,23,32,22,31,35,38)(14,17,27,29,21,30,34,37)

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, 2451);