C₃²×#1348 #11768397

order	1152
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#1348

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#102
commutator	C₂×C₈	C₂×C₃²×C₄
Frattini, cosocle	C₂²×C₈	C₂²×C₃²
prime radical	C₂³×C₃²	D₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂²×C₃²×C₄ ◁ C₂²×C₃²×C₈ ◁ G

348 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 30
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2408);