C₃²×#1544 #11768593

order	1152
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#1544

Subgroups

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

606 conjugacy classes of subgroups; only the 288 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 288 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) = 576

A minimal faithful permutation representation — 3 generators on 38 points:

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

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, 2604);