C₉×#1782 #11766503

order	1152
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₉×#1782

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₉
Frattini, cosocle	C₂⁴×C₃	C₂³×C₃
socle	C₂²×C₃	C₂×C₃×C₂²⋊C₄
prime radical	C₂²×C₃×#41	C₂×C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂³×C₄×C₉ ◁ G

918 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 204 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) = 288

A minimal faithful permutation representation — 3 generators on 29 points:

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

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, 514);