C₃²×#2245 #11769294

order	1152
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₃×C₄

Aliases

direct product

C₃²×#2245

Subgroups

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

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

normal subgroups only

This group has 396 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) = 46
transitive degree	μ_t(G) = 1152

A minimal faithful permutation representation — 3 generators on 46 points:

(1,2,3)(7,8)(9,12)(10,11)(13,14)(15,16,22,28)(17,24,32,43)(18,23,34,41)(19,26,36,38)(20,25,29,44)(21,37,35,27)(30,42,45,40)(31,46,33,39);(4,5,6)(8,11)(12,14)(15,17,29,45,21,31,19,18,22,32,20,30,35,33,36,34)(16,23,38,46,27,40,25,24,28,41,26,39,37,42,44,43);(7,9,10,13)(8,12,11,14)(15,18,19,33,35,45,29,32,22,34,36,31,21,30,20,17)(16,24,25,42,37,46,38,41,28,43,44,40,27,39,26,23)

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, 3305);