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Cayley diagram

#2835⋊C3 #73209

order384
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product#2835⋊C3

Splittings

Fitting, Sylow#2835 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#5450
commutator, cosocle, hypocenterC22×Q8C22×C3
Frattini, socleC23C22×A4
Fitting#2835C3
prime radical#393⋊C3C2
LCSG ▷ C22×Q8
UCSC1 ◁ C2

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 32 points:

(1,3)(2,6)(4,11,12,10,5,9)(7,13,8,14,16,15)(19,25,28)(20,26,27)(22,29,32)(23,30,31);(1,8,4,2)(3,14,10,13)(5,16,12,7)(6,9,15,11)(17,29)(18,27)(19,32)(20,22)(21,31)(23,26)(24,25)(28,30)

Polycyclic presentation:

2^2,3,2^5|a^2=h,d^2=h,e^2=h,d^a=dfg,e^a=ef,d^b=df,e^b=eg,d^c=defg,e^c=dgh,f^c=fg,g^c=f,e^d=eh

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#2835⋊C3, #11710042, #52914⋊C3, …
maximal normal subgroup ofC2×#2835⋊C3, #11710042, #52914⋊C3, …
maximal quotient of#14635⋊C3, #14655⋊C3, #17218⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5785);