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Cayley diagram

C2×C22⋊SL(2,𝔽3) #5947

order192
automorphismAut(G) = C2×#11921467 (order 2304) · Inn(G) = C22⋊A4 · Out(G) = C2×S4
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×C22⋊SL(2,𝔽3)

Splittings

Fitting, SylowC23×Q8 ⋊ C3
derivedC22×Q8 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC22C22⋊A4
commutator, cosocle, hypocenterC22×Q8C2×C3
FrattiniC2C2×C22⋊A4
FittingC23×Q8C3
socleC24A4
LCSG ▷ C22×Q8
UCSC1 ◁ C22

175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711237, #11711238, #11711241, …
maximal subgroup of#1446⋊C3, C2×#393⋊C3, C2×#395⋊C3, …
maximal normal subgroup of#1446⋊C3, C2×#393⋊C3, C2×#395⋊C3, …
maximal quotient ofC4×C22⋊SL(2,𝔽3), C2×#543⋊C3, #3458⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1506);