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Cayley diagram

C2×#870 #5469

order192
automorphismAut(G) order 49152
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C24

Aliases

direct productC2×#870

Splittings

SylowC3 ⋊ C2×#125

Subgroups

subgroupG/H
centerC23C22×D3
commutator, cosocleC22×C3C24
FrattiniC22C23×D3
FittingC2×C3×C42C2
socle, prime radicalC23×C3C23
hypercenterC2×C42D3
hypocenterC3C2×#125
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42

226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 192

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

(1,2)(3,4,5)(6,8,9,12)(7,11,10,13)(15,19)(18,21);(7,10)(11,13)(14,16,17,20)(15,18,19,21);(4,5)(6,7,9,10)(8,11,12,13)(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#68634, #68679, #68690, …
maximal normal subgroup of#68634, #68679, #68690, …
maximal quotient ofC4×#870, #73307, #73310, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1028);