C₃²×#2249 #11769298

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃×C₄

Aliases

direct product

C₃²×#2249

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₂×C₃²×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ G

552 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 216 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) = 42
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2,3)(8,10)(11,12)(13,21)(14,26)(15,18)(16,29)(17,23)(19,22)(20,27)(24,33)(25,40)(28,41)(30,39)(31,42)(32,37)(34,36)(35,38);(4,5,6)(11,13,22,36,16,25,28,39,15,24,37,42,27,38,14,23)(12,17,26,35,20,31,32,33,18,30,41,40,29,34,19,21);(7,8,9,10)(12,18)(17,30)(19,32)(20,29)(21,33)(26,41)(31,34)(35,40)

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, 3309);