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Cayley diagram

C2×C22⋊(C42⋊C3) #85642

order384
isoclinismisoclinic class of C22⋊(C42⋊C3)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23×C4

Aliases

direct productC2×C22⋊(C42⋊C3)

Splittings

Fitting, SylowC23×C42 ⋊ C3
derivedC22×C42 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2C22⋊(C42⋊C3)
commutator, cosocle, hypocenterC22×C42C2×C3
FrattiniC22C2×C22⋊A4
FittingC23×C42C3
socleC25A4
LCSG ▷ C22×C42
UCSC1 ◁ C2

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 16 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72231237, C2×#72231237, #72239181, …
maximal subgroup of#7554⋊C3, C2×#1437⋊C3, C4×C22⋊(C42⋊C3), …
maximal normal subgroup of#7554⋊C3, C2×#1437⋊C3, C4×C22⋊(C42⋊C3), …
maximal quotient ofC4×C22⋊(C42⋊C3), C2×C22⋊(#95⋊C3), #11709737, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18218);