C₉×#3026 #11767747

order	1152
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₉×#3026

Subgroups

	subgroup	G/H
center	C₂×C₉	C₂×#120
commutator	C₂³	C₂⁴×C₉
Frattini, cosocle	C₂³×C₃	C₂⁴×C₃
socle	C₂×C₃	C₂×C₃×#120
prime radical	C₃×#3026	C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂⁴×C₉ ◁ G

1200 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 318 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) = 25
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 25 points:

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

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₉×#3026
maximal subgroup of	D₉×#3026
maximal normal subgroup of	D₉×#3026
maximal quotient of	C₉×#32898

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 1758);