C₃²×#1709 #11768758

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂×C₃×C₈

Aliases

direct product

C₃²×#1709

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
prime radical	C₃²×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

636 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 240 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)(8,9)(10,11)(13,14)(15,16)(17,23)(18,26)(19,21)(20,24)(22,27)(25,29)(28,30);(4,5,6)(7,8,11,13,12,14,10,9)(15,17,18,24,19,25,27,30)(16,20,22,23,21,28,26,29);(7,9,10,14,12,13,11,8)(16,21)(20,28)(22,26)(23,29)

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, 2769);