C₃²×#2158 #11769207

order	1152
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃×C₄

Aliases

direct product

C₃²×#2158

Subgroups

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

456 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 312 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)(7,8,9,10)(11,12)(13,22)(14,19)(15,29)(16,31)(17,21)(18,25)(20,23)(24,35)(26,41)(27,37)(28,34)(30,42)(32,39)(33,40)(36,38);(4,5,6)(11,13,23,38,16,26,30,40,17,27,28,39,14,24,15,25)(12,18,29,35,19,32,34,37,21,33,42,41,31,36,20,22);(11,14,17,16)(12,19,21,31)(13,24,27,26)(15,28,30,23)(18,32,33,36)(20,29,34,42)(22,35,37,41)(25,39,40,38)

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, 3218);