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Cayley diagram

C2×#876 #5478

order192
automorphismAut(G) order 36864
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24

Aliases

direct productC2×#876

Splittings

FittingC2×C3×C42 ⋊ C2
SylowC3 ⋊ C2×#126

Subgroups

subgroupG/H
centerC23C22×D3
commutator, cosocleC22×C3C24
FrattiniC22C23×D3
FittingC2×C3×C42C2
socleC23×C3C23
prime radicalC23×D3C22
hypercenterC2×C42D3
hypocenterC3C2×#126
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42

246 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)(6,8,9,13)(7,11,10,12)(15,19)(18,21);(7,10)(11,12)(14,16,17,20)(15,18,19,21);(4,5)(6,7)(8,12)(9,10)(11,13)(14,15)(16,18)(17,19)(20,21);(1,2)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#70785, #70829, C4×#876, …
maximal normal subgroup of#70785, #70829, C4×#876, …
maximal quotient ofC4×#876, #73309, #73314, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1037);