Finite Group Databasebeta

← Prev (#88393) Next (#88395) →

Cayley diagram

C2×Q8⋊C27 #88394

order432
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×C9×S4 (order 432) · Inn(G) = A4 · Out(G) = C22×C9
homologyAbelianization (H1) = C2×C27 · Schur multiplier (H2) = C1

Aliases

direct productC2×Q8⋊C27
unambiguous semidirect product(C2×Q8)⋊C27

Splittings

derivedQ8 ⋊ C2×C27
SylowC2×Q8 ⋊ C27

Subgroups

subgroupG/H
center, hypercenterC22×C9A4
commutator, hypocenterQ8C2×C27
FrattiniC2×C9C2×A4
FittingC2×Q8×C9C3
socle, prime radicalC22×C3C22⋊C9
cosocleQ8×C9C2×C3
LCSG ▷ Q8
UCSC1 ◁ C22×C9

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 432

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2)(4,5,7)(8,10,9)(11,16,28,19,31,30,33,12,21,14,24,34,27,36,15,18,17,29,20,32,37,13,23,22,26,25,35);(3,7,6,9)(4,10,8,5)

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13136124, #13136127, C4⋊#88390, …
maximal subgroup of#11720108, C4×Q8⋊C27, C2×#88390, …
maximal normal subgroup of#11720108, C4×Q8⋊C27, C2×#88390, …
maximal quotient ofC4×Q8⋊C27, C22×Q8⋊C27, C2×Q8⋊C81, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 41);