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Cayley diagram

C2×#131⋊C3 #5453

order192
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×D4×S4 (order 384) · Inn(G) = A4 · Out(G) = C22×D4
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×#131⋊C3
unambiguous semidirect product(C2×#131)⋊C3

Splittings

Fitting, SylowC2×#131 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2×C8A4
commutator, hypocenterQ8C2×C3×C4
FrattiniC4C22×A4
FittingC2×#131C3
socleC22C4×A4
prime radicalC2×#41⋊C3C2
cosocle#41C22×C3
LCSG ▷ Q8
UCSC1 ◁ C2×C8

76 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#4628, #73119, #73120, …
maximal normal subgroup ofC2×#4628, #73119, #73120, …
maximal quotient ofC4×#131⋊C3, #2880⋊C3, C4⋊(#131⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1012);