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C4⋊#6249 #11722290

order864
automorphism?
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC4⋊#6249

Splittings

FittingC4×H(3,𝔽3) ⋊ C8
SylowH(3,𝔽3) ⋊ C4⋊C8

Subgroups

subgroupG/H
centerC2C2×#6249
commutatorC2×H(3,𝔽3)C2×C8
Frattini, socleC2×C3C2×AGL(1,𝔽9)
FittingC4×H(3,𝔽3)C8
prime radicalC2×#272C2×C4
cosocleC2×#1088C22
hypercenterC4#6249
hypocenterH(3,𝔽3)C4⋊C8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4

73 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 108

Polycyclic presentation:

2^5,3^3|a^2=c,b^2=d,c^2=e,b^a=bd,f^a=f^2g^2,g^a=f^2h^2,h^a=h^2,f^c=f^2g,g^c=fgh^2,f^e=f^2,g^e=g^2h^2,g^f=gh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13146168, #13146170, #13146176, …
maximal normal subgroup of#13146168, #13146170, #13146176, …
maximal quotient of#13146430, #13146431, #13146434, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 2257);