C₂²×C₃²×#126 #11770487

order	1152
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃² · Schur multiplier (H₂) = C₂⁸×C₃

Aliases

direct product

C₂²×C₃²×#126

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃²
prime radical	C₂⁵×C₃²	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃² ◁ G

3960 conjugacy classes of subgroups; only the 2616 normal subgroups are shown.

normal subgroups only

This group has 3960 conjugacy classes of subgroups — too many to draw. Only its 2616 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) = 26
transitive degree	μ_t(G) = 576

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

504 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 4498);