C₃²×#2140 #11769189

order	1152
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃×C₄

Aliases

direct product

C₃²×#2140

Subgroups

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

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

normal subgroups only

This group has 780 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) = 26
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

288 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 3200);