C₂₇×C₂²⋊Q₈ #11720211

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

Aliases

direct product

C₂₇×C₂²⋊Q₈

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	C₂³×C₃	C₂²×C₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇ ◁ G

148 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 96 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)(32,33,35,37)(34,38,39,36);(32,34,35,39)(33,36,37,38);(28,29)(30,31)(32,33)(34,36)(35,37)(38,39)

Conjugacy classes & character table

378 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13134034, #13134548, #13134549, …
maximal subgroup of	#13134034, C₂₇×#328, #13134548, …
maximal normal subgroup of	#13134034, C₂₇×#328, #13134548, …
maximal quotient of	C₂₇×#380, C₂₇×#385, C₂₇×#387, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 178);