Finite Group Databasebeta

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Cayley diagram

C2×#809 #4965

order192
isoclinismisoclinic class of #213
automorphismAut(G) order 6144
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

direct productC2×#809

Splittings

FittingC2×C3×C4⋊C4 ⋊ C2
SylowC3 ⋊ C2×#102

Subgroups

subgroupG/H
centerC23#67
commutatorC3×C4C22×C4
FrattiniC2×C4C22×D3
FittingC2×C3×C4⋊C4C2
socleC23×C3D4
prime radicalC22×D12C2
cosocleC2×C3×C4C23
hypercenterC2×C4⋊C4D3
hypocenterC3C2×#102
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#68780, #68782, #68791, …
maximal normal subgroup of#68780, #68782, #68791, …
maximal quotient ofC2×#4451, #69469, C4×#809, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 524);