Finite Group Databasebeta

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Cayley diagram

C2×#5006 #82266

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

Aliases

direct productC2×#5006

Splittings

FittingC2×C3×#117 ⋊ C2
SylowC3 ⋊ C2×#410

Subgroups

subgroupG/H
centerC2×C4(C2×D3)⋊C4
commutatorC22×C3C23×C4
FrattiniC23C23×D3
FittingC2×C3×#117C2
socleC22×C3C2×C22⋊C4
prime radicalC22×#235C2
cosocleC23×C3C24
hypercenterC2×#117D3
hypocenterC3C2×#410
LCSG ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

788 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:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10700513, #10700516, #10700518, …
maximal normal subgroup of#10700513, #10700516, #10700518, …
maximal quotient ofC2×#70782, C2×#70783, C2×#70786, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14842);