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Cayley diagram

C2×#5774 #87190

order384
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#5774

Splittings

FittingC2×C3×AGL(1,ℤ/8) ⋊ C2
SylowC3 ⋊ C2×#576

Subgroups

subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC2×C3×AGL(1,ℤ/8)C2
socleC22×C3C22×D4
hypercenterC2×AGL(1,ℤ/8)D3
hypocenterC3C2×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

1512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 5 generators on 21 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10811088, #10811115, #10811164, …
maximal normal subgroup of#10811088, #10811115, #10811164, …
maximal quotient ofC2×#78882, C2×#78925, C2×#79210, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19766);