Finite Group Databasebeta
Cayley diagram

C2×#4978 #82148

order384
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#4978

Splittings

SylowC3 ⋊ C2×#387

Subgroups

subgroupG/H
centerC24C22×D3
commutatorC22×C3C23×C4
FrattiniC23C23×D3
FittingC22×C3×C4⋊C4C2
socle, prime radicalC24×C3C23
cosocleC23×C3C24
hypercenterC22×C4⋊C4D3
hypocenterC3C2×#387
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C4⋊C4

712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 384

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

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10690570, #10690590, C4×#4978, …
maximal normal subgroup of#10690570, #10690590, C4×#4978, …
maximal quotient ofC2×#69402, C2×#69406, C2×#71564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14724);