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Cayley diagram

C2×#4447 #69365

order384
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C22×C16 · Schur multiplier (H2) = C22

Aliases

direct productC2×#4447

Splittings

FittingC2×C3×C32 ⋊ C2
SylowC3 ⋊ C2×M64

Subgroups

subgroupG/H
centerC2×C16C2×D3
commutatorC2×C3C22×C16
FrattiniC16C22×D3
FittingC2×C3×C32C2
socleC22×C3C2×C16
prime radicalC22×D3C16
cosocleC3×C16C23
hypercenterC2×C32D3
hypocenterC3C2×M64
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C16 ◁ C2×C32

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 37
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10682997, C4×#4447, #10715363, …
maximal normal subgroup of#10682997, C4×#4447, #10715363, …
maximal quotient ofC2×#67589, C4×#4447, C2×(C2×D3)⋊C32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1941);