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Cayley diagram

C2×#4727 #78599

order384
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#4727

Splittings

FittingC2×C3×C22⋊C8 ⋊ C2
SylowC3 ⋊ C2×#408

Subgroups

subgroupG/H
centerC22×C4C22×D3
commutatorC22×C3C23×C4
FrattiniC2×C4C23×D3
FittingC2×C3×C22⋊C8C2
socleC23×C3C22×C4
prime radicalC22×#235C2
cosocleC2×C3×C4C24
hypercenterC2×C22⋊C8D3
hypocenterC3C2×#408
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 25 points:

(3,4,5)(7,9)(10,12,13,18,14,19,20,24)(11,16,17,23,15,21,22,25);(6,7)(8,9)(10,11)(12,16)(13,17)(14,15)(18,23)(19,21)(20,22)(24,25);(1,2)(4,5)(11,15)(16,21)(17,22)(23,25);(1,2)

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10704250, C4×#4727, #10792714, …
maximal normal subgroup of#10704250, C4×#4727, #10792714, …
maximal quotient ofC2×#68122, C2×#68146, C2×#68104, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11175);