Finite Group Databasebeta

← Prev (#5817) Next (#5819) →

Cayley diagram

C2×#948 #5818

order192
automorphismAut(G) = C2×D3×#11713200 (order 9216) · Inn(G) = C22×D3 · Out(G) = C2×D4×S4
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×#948

Splittings

SylowC3 ⋊ C2×#131

Subgroups

subgroupG/H
centerC2×C4C22×D3
commutatorC2×C3C23×C4
FrattiniC4C23×D3
Fitting, prime radicalC2×C3×#41C2
socleC22×C3C22×C4
cosocleC3×C4C24
hypercenterC2×#41D3
hypocenterC3C2×#131
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×#41

266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 4 generators on 21 points:

(1,2)(3,4,5)(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21);(8,14)(11,17)(15,20)(18,21);(4,5)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21);(1,2)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#70040, #70041, #70133, …
maximal normal subgroup of#70040, #70041, #70133, …
maximal quotient ofC2×#4997, C2×#4999, C2×#5000, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1377);