Finite Group Databasebeta

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Cayley diagram

C4×#907 #78515

order384
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC4×#907

Splittings

FittingC3×C4×M16 ⋊ C2
SylowC3 ⋊ C4×#131

Subgroups

subgroupG/H
centerC42C22×D3
commutatorC2×C3C22×C42
FrattiniC2×C4C23×D3
FittingC3×C4×M16C2
socleC22×C3C23×C4
prime radicalC2×#235C22
cosocleC2×C3×C4C24
hypercenterC4×M16D3
hypocenterC3C4×#131
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C42 ◁ C4×M16

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3)(8,10,11,17,12,16,18,22)(9,14,15,21,13,19,20,23);(9,13)(10,16)(15,20)(17,22);(2,3)(8,9)(10,14)(11,15)(12,13)(16,19)(17,21)(18,20)(22,23);(4,5,6,7)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10685858, #10685863, #10685868, …
maximal normal subgroup of#10685858, #10685863, #10685868, …
maximal quotient ofC8×#907, #10760800, #10760850, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11091);