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Cayley diagram

C2×#859 #5418

order192
isoclinismisoclinic class of #226
automorphismAut(G) = S4×#120 (order 768) · Inn(G) = S4 · Out(G) = #120
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×#859

Splittings

derivedSL(2,𝔽3) ⋊ C2×C4

Subgroups

subgroupG/H
center, socle, hypercenterC23S4
commutator, hypocenterSL(2,𝔽3)C2×C4
FrattiniC22C2×S4
FittingC22×Q8D3
prime radicalC22×SL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C22
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C23

97 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×#859, #73059, C4⋊#859, …
maximal normal subgroup ofC4×#859, #73059, C4⋊#859, …
maximal quotient ofC2×SL(2,𝔽3)⋊C8, #73055, C4×#859, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 977);