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Cayley diagram

C2×#5465 #85656

order384
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22

Aliases

direct productC2×#5465

Splittings

Fitting, SylowC2×H(3,𝔽4) ⋊ C3
derivedH(3,𝔽4) ⋊ C2×C3

Subgroups

subgroupG/H
center, socle, hypercenterC23C22⋊A4
commutator, cosocle, hypocenterH(3,𝔽4)C2×C3
FrattiniC22C2×C22⋊A4
FittingC2×H(3,𝔽4)C3
LCSG ▷ H(3,𝔽4)
UCSC1 ◁ C23

343 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72227615, #72228025, #72261182
maximal subgroup of#12774⋊C3, C2×#2202⋊C3, C4×#5465, …
maximal normal subgroup of#12774⋊C3, C2×#2202⋊C3, C4×#5465, …
maximal quotient ofC4×#5465, C2×#73293, #11710043, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18232);