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Cayley diagram

C2×#4689 #78451

order384
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#4689

Splittings

FittingC2×C3×C4×C8 ⋊ C2
SylowC3 ⋊ C2×#405

Subgroups

subgroupG/H
centerC22×C8C2×D3
commutatorC2×C3C22×C42
FrattiniC2×C4C23×D3
FittingC2×C3×C4×C8C2
socleC23×C3C22×C4
prime radicalC23×D3C2×C4
cosocleC2×C3×C4C24
hypercenterC2×C4×C8D3
hypocenterC3C2×#405
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C8 ◁ C2×C4×C8

592 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)(10,12,13,18,14,19,20,24)(11,16,17,23,15,21,22,25);(6,7,8,9)(11,15)(16,21)(17,22)(23,25);(1,2)(4,5)(10,11)(12,16)(13,17)(14,15)(18,23)(19,21)(20,22)(24,25);(1,2)

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10685716, #10685720, #10688045, …
maximal normal subgroup of#10685716, #10685720, #10688045, …
maximal quotient ofC2×C8×ΓL(1,𝔽25), C2×#68047, C2×#68070, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11027);