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Cayley diagram

C2×#912 #5756

order192
automorphismAut(G) = D3×#72214076 (order 6144) · Inn(G) = D3×D4 · Out(G) = #2403
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC2×#912

Splittings

FittingC2×C3×D8 ⋊ C2
SylowC3 ⋊ C2×#135

Subgroups

subgroupG/H
centerC22D3×D4
commutator, cosocleC3×C4C24
FrattiniC4C23×D3
FittingC2×C3×D8C2
socleC22×C3C2×D4
hypercenterC2×D8D3
hypocenterC3C2×#135
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#69188, #69189, #69192, …
maximal normal subgroup of#69188, #69189, #69192, …
maximal quotient ofC2×#4758, C2×#4766, C2×#4768, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1315);