Finite Group Databasebeta

← Prev (#73224) Next (#73226) →

Cayley diagram

C4×#131⋊C3 #73225

order384
isoclinismisoclinic class of SL(2,𝔽3)
automorphism?
homologyAbelianization (H1) = C3×C42 · Schur multiplier (H2) = C4

Aliases

direct productC4×#131⋊C3
unambiguous semidirect product(C4×#131)⋊C3

Splittings

Fitting, SylowC4×#131 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4×C8A4
commutator, hypocenterQ8C3×C42
FrattiniC2×C4C22×A4
FittingC4×#131C3
socleC22C2×C4×A4
prime radicalC2×#41⋊C3C22
cosocleC2×#41C22×C3
LCSG ▷ Q8
UCSC1 ◁ C4×C8

157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 2 generators on 36 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC8×#131⋊C3, #17316⋊C3, #17320⋊C3, …
maximal normal subgroup ofC8×#131⋊C3, #17316⋊C3, #17320⋊C3, …
maximal quotient ofC8×#131⋊C3, #17320⋊C3, #20336⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5801);