C₂₇×#123 #11720212

order	864
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₂₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂₇×#123

Subgroups

	subgroup	G/H
center	C₂²×C₂₇	C₂³
commutator	C₂²	C₂³×C₂₇
Frattini, cosocle	C₂²×C₉	C₂³×C₃
socle	C₂²×C₃	C₂³×C₉
prime radical	C₂×C₃×D₄	C₂×C₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇ ◁ G

156 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 39
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 3 generators on 39 points:

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

Conjugacy classes & character table

378 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13134035, #13134558, #13136386, …
maximal subgroup of	#13134035, C₂₇×#352, #13134558, …
maximal normal subgroup of	#13134035, C₂₇×#352, #13134558, …
maximal quotient of	C₂₇×#381, C₂₇×#385, C₂₇×#386, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 179);