C₃²×#2380 #11769429

order	1152
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃²×C₄ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

direct product

C₃²×#2380

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃²	C₂⁴
commutator	C₂²	C₂³×C₃²×C₄
Frattini, cosocle	C₂³	C₂⁴×C₃²
prime radical	C₂×C₃²×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

1944 conjugacy classes of subgroups; only the 792 normal subgroups are shown.

normal subgroups only

This group has 1944 conjugacy classes of subgroups — too many to draw. Only its 792 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) = 576

A minimal faithful permutation representation — 4 generators on 26 points:

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

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, 3440);