C₃²×#1719 #11768768

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₃×C₄

Aliases

direct product

C₃²×#1719

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
prime radical	C₃²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

558 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 216 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 — 3 generators on 30 points:

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

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2779);