C₃²×#2210 #11769259

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂²×C₃

Aliases

direct product

C₃²×#2210

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃²	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃²
prime radical	C₂×C₃²×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ G

510 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 192 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 — 3 generators on 26 points:

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

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, 3270);