Finite Group Databasebeta

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C22⋊(C22⋊(C22⋊A4)) #11710940

order768
isoclinismisoclinic class of C22⋊(C22⋊(C22⋊A4))
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C216

Aliases

unambiguous semidirect productC22⋊(C22⋊(C22⋊A4))

Splittings

Fitting, socle, derived, SylowC28 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC28C3
LCSG ▷ C28
UCSC1

139948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 48

Polycyclic presentation:

3,2^8|b^a=cdfi,c^a=ceh,d^a=cfgh,e^a=bcgh,f^a=bcei,g^a=cdfhi,h^a=bgh,i^a=dg

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72240592, C2×#72215789, #72258300, …
maximal subgroup of#72209700, #72209702, C2×C22⋊(C22⋊(C22⋊A4)), …
maximal normal subgroup of#72209700, C2×C22⋊(C22⋊(C22⋊A4)), #72215614, …
maximal quotient of#72210027, C2×C22⋊(C22⋊(C22⋊A4)), #72257803

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1085321);