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Cayley diagram

#2988⋊C3 #73255

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

Aliases

unambiguous semidirect product#2988⋊C3

Splittings

Fitting, Sylow#2988 ⋊ C3

Subgroups

subgroupG/H
center, FrattiniC2×C4C22×A4
commutator, hypocenterQ8C2×C3×C8
Fitting#2988C3
socleC22C2×C4×A4
prime radicalC2×#41⋊C3C4
cosocleC4×Q8C22×C3
hypercenterC4⋊C8A4
LCSG ▷ Q8
UCSC1 ◁ C2×C4 ◁ C4⋊C8

122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#17438⋊C3, #17439⋊C3, #17720⋊C3, …
maximal normal subgroup of#17438⋊C3, #17439⋊C3, #17720⋊C3, …
maximal quotient of#20569⋊C3, #22530⋊C3, SL(2,𝔽3)×C4⋊C8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5831);