Finite Group Databasebeta

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C3×#87639 #11924226

order1176
isoclinismisoclinic class of D72
automorphism?
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×#87639

Splittings

derived, SylowC72 ⋊ C2×C3×C4
HallC3×C72 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC2×C3D72
commutator, hypocenterC72C2×C3×C4
FrattiniC2C3×D72
Fitting, socleC2×C3×C72C22
prime radicalC2×C3×C7⋊D7C2
cosocleC2×C72C22×C3
LCSG ▷ C72
UCSC1 ◁ C2×C3

94 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 2 generators on 21 points:

(1,2,3)(4,6,7,5)(9,10)(11,12)(13,14)(15,16,18,20,21,19,17);(4,5,7,6)(8,9,11,13,14,12,10)(16,17)(18,19)(20,21)

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#11924174, D3×#87639, C3×#11716115, …
maximal normal subgroup ofC3×#11924174, D3×#87639, C3×#11716115, …
maximal quotient ofC3×#11924252, C2×C3×#87639, C3×#11716029, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 141);