Finite Group Databasebeta

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Cayley diagram

C2×#830 #5206

order192
isoclinismisoclinic class of Dic6
automorphismAut(G) = C2×D3×#347042 (order 6144) · Inn(G) = C2×D3 · Out(G) = C2×#32898
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C24

Aliases

direct productC2×#830

Splittings

SylowC3 ⋊ C2×C22⋊C8

Subgroups

subgroupG/H
centerC22×C4C2×D3
commutatorC2×C3C22×C8
FrattiniC2×C4C22×D3
FittingC23×C3×C4C2
socleC23×C3C2×C4
prime radicalC24×C3C4
cosocleC2×C3×C4C23
hypercenterC23×C4D3
hypocenterC3C2×C22⋊C8
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C23×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);(4,5)(7,9)(10,11,12,14,13,15,16,17);(1,2)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#67612, #67613, #67617, …
maximal normal subgroup of#67612, #67613, #67617, …
maximal quotient ofC2×#4524, C4×#830, #71530, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 765);