Finite Group Databasebeta

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Cayley diagram

C2×#5145 #83791

order384
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×#5145

Splittings

FittingC2×C3×#131 ⋊ C2
SylowC3 ⋊ C2×#444

Subgroups

subgroupG/H
centerC2×C4C2×#67
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC2×C3×#131C2
socleC22×C3C4×D4
prime radicalC2×#950C2
cosocleC2×C3×C4C24
hypercenterC2×#131D3
hypocenterC3C2×#444
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#131

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 21 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10712093, #10712094, #10712096, …
maximal normal subgroup of#10712093, #10712094, #10712096, …
maximal quotient ofC2×#70937, C2×#70839, C2×#70932, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16367);