Finite Group Databasebeta

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Cayley diagram

C22×#859 #85500

order384
isoclinismisoclinic class of #226
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC22×#859

Splittings

derivedSL(2,𝔽3) ⋊ C22×C4

Subgroups

subgroupG/H
center, socle, hypercenterC24S4
commutator, hypocenterSL(2,𝔽3)C22×C4
FrattiniC22C22×S4
FittingC23×Q8D3
prime radicalC23×SL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C23
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C24

439 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11711319, C2×C4×#859, C2×#73063, …
maximal normal subgroup of#11711319, C2×C4×#859, C2×#73063, …
maximal quotient ofC22×SL(2,𝔽3)⋊C8, C2×#73055, #11711803, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18076);