C₃²×#2645 #11769694

order	1152
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

C₃²×#2645

Subgroups

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

1056 conjugacy classes of subgroups; only the 552 normal subgroups are shown.

normal subgroups only

This group has 1056 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) = 1152

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

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

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, 3705);