C₃²×#3293 #11770342

order	1152
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃² · Schur multiplier (H₂) = C₂⁵×C₃

Aliases

direct product

C₃²×#3293

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ G

1332 conjugacy classes of subgroups; only the 552 normal subgroups are shown.

normal subgroups only

This group has 1332 conjugacy classes of subgroups — too many to draw. Only its 552 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) = 38
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 4 generators on 38 points:

(1,2,3)(7,8,14,19)(9,20,23,25)(10,22,15,29)(11,16,28,34)(12,18,13,17)(21,36,38,32)(24,37,31,26)(27,35,30,33);(4,5,6)(7,9,14,23)(8,15,19,10)(11,26,28,37)(12,29,13,22)(16,32,34,36)(17,25,18,20)(21,30,38,27)(24,33,31,35);(7,10,14,15)(8,9,19,23)(11,24,28,31)(12,25,13,20)(16,21,34,38)(17,22,18,29)(26,35,37,33)(27,36,30,32);(7,11)(8,16)(9,21)(10,24)(12,30)(13,27)(14,28)(15,31)(17,35)(18,33)(19,34)(20,36)(22,37)(23,38)(25,32)(26,29)

Conjugacy classes & character table

261 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4353);