C₃²×C₄×Q₃₂ #11769223

order	1152
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×C₄×Q₃₂

Subgroups

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

450 conjugacy classes of subgroups; only the 252 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 252 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) = 1152

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

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

Conjugacy classes & character table

396 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3234);