Finite Group Databasebeta

← Prev (#5948) Next (#5950) →

Cayley diagram

C2×#997 #5949

order192
automorphismAut(G) = C2×(#1020⋊C3)⋊C2 (order 1152) · Inn(G) = C22⋊A4 · Out(G) = C22×D3
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×#997

Splittings

Fitting, SylowC2×H(5,𝔽2) ⋊ C3
derivedH(5,𝔽2) ⋊ C2×C3

Subgroups

subgroupG/H
center, socle, hypercenterC22C22⋊A4
commutator, cosocle, hypocenterH(5,𝔽2)C2×C3
FrattiniC2C2×C22⋊A4
FittingC2×H(5,𝔽2)C3
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C22

193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 16

A minimal faithful permutation representation — 3 generators on 10 points:

(1,2)(4,5,8)(7,9,10);(3,8)(4,5)(6,10)(7,9);(4,7)(5,9)

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711626, #11711627, #11711628, …
maximal subgroup of#1790⋊C3, C2×B(4,𝔽2)⋊C3, C2×#458⋊C3, …
maximal normal subgroup of#1790⋊C3, C2×B(4,𝔽2)⋊C3, C2×#458⋊C3, …
maximal quotient ofC4×#997, C2×#543⋊C3, #3459⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1508);