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Cayley diagram

C2×#5837 #87286

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

Aliases

direct productC2×#5837

Splittings

FittingC2×C3×2−1+4 ⋊ C2
SylowC3 ⋊ C2×#577

Subgroups

subgroupG/H
centerC22C22×#67
commutator, cosocleC3×C4C25
FrattiniC4C24×D3
FittingC2×C3×2−1+4C2
socleC22×C3C22×D4
hypercenterC2×2−1+4D3
hypocenterC3C2×#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

1460 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:

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

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10811094, #10811102, #10811121, …
maximal normal subgroup of#10811094, #10811102, #10811121, …
maximal quotient ofC2×#82563, C2×#84111, C2×#82561, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19862);