C₉×#2380 #11767101

order	1152
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₉×#2380

Subgroups

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, socle, cosocle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂×C₃×#127	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

972 conjugacy classes of subgroups; only the 396 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 396 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) = 29
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 4 generators on 29 points:

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

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, 1112);