C₉×#3025 #11767746

order	1152
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₉×#3025

Subgroups

	subgroup	G/H
center	C₄×C₉	#120
commutator	C₂³	C₂⁴×C₉
Frattini, cosocle	C₂³×C₃	C₂⁴×C₃
socle	C₂×C₃	C₂×C₃×#120
prime radical	C₃×#3025	C₃
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₉ ◁ C₂²×C₄×C₉ ◁ G

1149 conjugacy classes of subgroups; only the 318 normal subgroups are shown.

normal subgroups only

This group has 1149 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)(15,21)(22,25);(10,11)(12,15)(13,16)(14,17)(18,22)(19,21)(20,23)(24,25);(10,12)(11,15)(13,18)(14,19)(16,22)(17,21)(20,24)(23,25);(10,13,14,20)(11,16,17,23)(12,18,19,24)(15,22,21,25)

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, 1757);