Finite Group Databasebeta

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Cayley diagram

C2×#6171 #88405

order432
isoclinismisoclinic class of Dic54
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×#6171

Splittings

FittingC23×C27 ⋊ C2
SylowC27 ⋊ C2×D4

Subgroups

subgroupG/H
centerC22C2×D27
commutator, cosocleC2×C27C23
FrattiniC2×C9C22×D3
FittingC23×C27C2
socleC22×C3C2×D9
hypercenterC23D27
hypocenterC27C2×D4
LCSG ▷ C2×C27 ▷ C27
UCSC1 ◁ C22 ◁ C23

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 216

A minimal faithful permutation representation — 3 generators on 33 points:

(4,6)(7,31,30,16,14,11,28,26,29,10,13,32,15,25,20,9,33,12,18,23,19,27,8,24,17,22,21);(3,4)(5,6)(8,11)(9,15)(10,18)(12,13)(14,24)(16,17)(19,26)(20,25)(21,31)(22,30)(23,29)(27,28)(32,33);(1,2)

Conjugacy classes & character table

114 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11720046, #6171⋊C4, #11720131, …
maximal normal subgroup of#11720046, #6171⋊C4, #11720131, …
maximal quotient ofC2×Dic27⋊C4, #11720173, C2×(C2×D27)⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 52);