Finite Group Databasebeta

← Prev (#88742) Next (#88744) →

Cayley diagram

D4×#272 #88743

order432
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productD4×#272

Splittings

FittingD4×H(3,𝔽3) ⋊ C2
SylowH(3,𝔽3) ⋊ C2×D4

Subgroups

subgroupG/H
center, Frattini, socleC2×C3C22×C3⋊D3
commutator, cosocleC2×H(3,𝔽3)C23
FittingD4×H(3,𝔽3)C2
hypercenterC3×D4C3⋊D3
hypocenterH(3,𝔽3)C2×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×D4

297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11720788, #11721188, #11721190, …
maximal normal subgroup of#11720788, #11721188, #11721190, …
maximal quotient of#11721734, C22⋊C4×#272, #11721737, …
automorphism group ofC2×C4×3−1+2, D4×3−1+2
outer automorphism group ofC2×C3⋊Dic9

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 390);