C₃²×#1835 #11768884

order	1152
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃×C₄

Aliases

direct product

C₃²×#1835

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃²	C₂×D₄
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ C₂×C₃²×C₄² ◁ G

648 conjugacy classes of subgroups; only the 360 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 360 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) = 34
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 3 generators on 34 points:

(1,2,3)(7,8,9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,29)(22,25)(23,33)(24,28)(26,31)(27,30)(32,34);(4,5,6)(12,15)(16,18)(19,21,30,32,24,31,23,22)(20,25,33,26,28,34,27,29);(11,13,14,17)(12,16,15,18)(19,22,23,31,24,32,30,21)(20,26,27,25,28,29,33,34)

Conjugacy classes & character table

396 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2895);