C₃²×#2644 #11769693

order	1152
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

direct product

C₃²×#2644

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃²	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃²
prime radical	C₂²×C₃²×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

1260 conjugacy classes of subgroups; only the 552 normal subgroups are shown.

normal subgroups only

This group has 1260 conjugacy classes of subgroups — too many to draw. Only its 552 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) = 30
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 4 generators on 30 points:

(1,2,3)(8,11)(12,13)(15,16)(17,20)(18,21)(19,22)(23,24,27,29)(25,30,26,28);(4,5,6)(7,8,10,11)(9,12,14,13)(15,17)(16,18)(19,21)(20,22)(23,25,27,26)(24,28,29,30);(7,8)(9,13)(10,11)(12,14)(15,17)(16,20)(18,22)(19,21)(23,24)(25,28)(26,30)(27,29);(7,9,10,14)(8,12,11,13)(15,18,19,20)(16,21,22,17)(23,26,27,25)(24,28,29,30)

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, 3704);