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Cayley diagram

#2982⋊C3 #73253

order384
automorphism?
homologyAbelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#2982⋊C3

Splittings

Fitting, Sylow#2982 ⋊ C3

Subgroups

subgroupG/H
center, FrattiniC2×C4C22×A4
commutator, hypocenterQ8C2×C3×C8
Fitting#2982C3
socleC22C2×C4×A4
prime radicalC22×SL(2,𝔽3)C4
cosocleC4×Q8C22×C3
hypercenterC22⋊C8A4
LCSG ▷ Q8
UCSC1 ◁ C2×C4 ◁ C22⋊C8

126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 24 points:

(10,14)(11,17,12)(15,24,16,21,23,22)(18,20,19);(1,2,3,5,4,6,7,8)(9,23,13,24)(10,17,14,18)(11,16,19,22)(12,21,20,15)

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#17352⋊C3, #17353⋊C3, #17618⋊C3, …
maximal normal subgroup of#17352⋊C3, #17353⋊C3, #17618⋊C3, …
maximal quotient of#20569⋊C3, SL(2,𝔽3)×C22⋊C8, #22734⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5829);