C₃²×#1414 #11768463

order	1152
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#1414

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#354
commutator	C₄²	C₂×C₃²×C₄
Frattini, cosocle	#124	C₂²×C₃²
prime radical	C₂×C₃²×D₄	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂²×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×#124 ◁ G

252 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 2 generators on 38 points:

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

Conjugacy classes & character table

153 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 2474);