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H(7,𝔽3) #14555449

order2187
nilpotency class2
isoclinismisoclinic class of H(7,𝔽3)
automorphism?
homologyAbelianization (H1) = C36 · Schur multiplier (H2) = C314

Aliases

extraspecial3+1+6
HeisenbergH(7,𝔽3)

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC3C36
LCSG ▷ C3 ▷ C1
UCSC1 ◁ C3 ◁ G
LECSG ▷ C3 ▷ C1

61757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 81
transitive degreeμt(G) = 81

Polycyclic presentation:

3^7|b^a=bg,d^c=dg,f^e=fg

Conjugacy classes & character table

731 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×H(7,𝔽3), #72264967, #72265230, …
maximal normal subgroup ofC3×H(7,𝔽3), #72264967, #72265230, …
maximal quotient of#72228849, C3×H(7,𝔽3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2187, 9308);