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Cayley diagram

C2×D96 #69367

order384
isoclinismisoclinic class of D96
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×D96

Splittings

FittingC2×C3×C32 ⋊ C2
SylowC3 ⋊ C2×D32

Subgroups

subgroupG/H
centerC22D48
commutator, cosocleC3×C16C23
FrattiniC16C22×D3
FittingC2×C3×C32C2
socleC22×C3D16
hypercenterC2×C32D3
hypocenterC3C2×D32
LCSG ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32

120 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)(6,7,13,23,9,14,26,32,20,30,34,37,33,35,19,25,11,16,28,36,21,31,18,24,10,15,27,29,17,22,8,12);(1,2)(4,5)(7,12)(8,13)(9,17)(10,20)(14,29)(15,32)(16,25)(18,34)(19,28)(21,33)(22,23)(24,30)(26,27)(31,37)(35,36);(1,2)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10683064, #10683066, #10683076, …
maximal normal subgroup of#10683064, #10683066, #10683076, …
maximal quotient ofC2×#67596, C2×#67591, C4×D96, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1943);