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Cayley diagram

C2×#984 #5920

order192
automorphismAut(G) = S4×B(4,𝔽2) (order 1536) · Inn(G) = C2×S4 · Out(G) = #120
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×#984

Subgroups

subgroupG/H
center, socleC22C2×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C23
FrattiniC2C22×S4
FittingC2×#41D3
prime radicalC2×#41⋊C3C2
hypercenterC2×C4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 34 points:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#73077, #73124, #73130, …
maximal normal subgroup of#73077, #73124, #73130, …
maximal quotient ofC2×#5386, C2×#5388, #85319, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1479);