C₄×#6864 #11757349

order	972
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₄×C₉² · Schur multiplier (H₂) = C₃

Aliases

direct product

C₄×#6864

Subgroups

	subgroup	G/H
center	C₃×C₄×C₉	C₃²
commutator	C₃	C₄×C₉²
Frattini, cosocle	C₂×C₃×C₉	C₂×C₃²
socle, prime radical	C₂×C₃²	C₂×C₃×C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₉ ◁ G

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 972

A minimal faithful permutation representation — 2 generators on 40 points:

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

Conjugacy classes & character table

396 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#6864⋊C₈, #6864⋊Q₈, C₄×#6864⋊C₂, …
maximal subgroup of	#6864⋊C₈, C₈×#6864, #6864⋊Q₈, …
maximal normal subgroup of	#6864⋊C₈, C₈×#6864, #6864⋊Q₈, …
maximal quotient of	C₈×#6864, C₂×C₄×#6864

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(972, 74);

C4×#6864 #11757349