C₉×#2022 #11766743

order	1152
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂³×C₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₉×#2022

Subgroups

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂⁴	C₂³×C₉
Frattini, cosocle	C₂⁴×C₃	C₂³×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₃×#534	C₂×C₃
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂⁴×C₉ ◁ G

645 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 138 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)(16,19)(17,20)(18,21)(22,25)(23,26)(24,28)(27,29);(14,16,18,23)(15,19,21,26)(17,22,24,29)(20,27,28,25);(11,13)(14,17)(15,20)(16,22)(18,24)(19,25)(21,28)(23,29)(26,27)

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 754);