Finite Group Databasebeta

← Prev (#89080) Next (#89082) →

Cayley diagram

C22×C3⋊(C3⋊Dic3) #89081

order432
isoclinismisoclinic class of C3⋊(C3⋊D3)
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C33

Aliases

direct productC22×C3⋊(C3⋊Dic3)

Splittings

derived, SylowC33 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC23C3⋊(C3⋊D3)
commutator, hypocenterC33C22×C4
FrattiniC2C22×C3⋊(C3⋊D3)
Fitting, socle, prime radicalC23×C33C2
cosocleC2×C33C23
LCSG ▷ C33
UCSC1 ◁ C23

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 432

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11722061, #11722201, C2×Dic3×C3⋊Dic3, …
maximal normal subgroup of#11722061, #11722201, C2×Dic3×C3⋊Dic3, …
maximal quotient ofC22×C3⋊(C3⋊(C3⋊C8)), C2×#88855, C2×C4×C3⋊(C3⋊Dic3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 728);