C₃²×#1415 #11768464

order	1152
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#1415

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	D₁₆
commutator	C₁₆	C₂×C₃²×C₄
Frattini, cosocle	C₂×C₁₆	C₂²×C₃²
prime radical	C₂×C₃²×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×C₈ ◁ C₂×C₃²×C₁₆ ◁ G

246 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 120 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 — 2 generators on 42 points:

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

Conjugacy classes & character table

342 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₃⋊D₃×#1415, #72269625
maximal subgroup of	C₃²×#13654, C₃⋊D₃×#1415, #72269625
maximal normal subgroup of	C₃²×#13654, C₃⋊D₃×#1415, #72269625

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2475);