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Cayley diagram

C22×Dic24 #82004

order384
isoclinismisoclinic class of D48
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC22×Dic24

Splittings

SylowC3 ⋊ C22×Q32

Subgroups

subgroupG/H
centerC23D24
commutator, cosocleC3×C8C24
FrattiniC8C23×D3
FittingC22×C3×C16C2
socle, prime radicalC23×C3D8
hypercenterC22×C16D3
hypocenterC3C22×Q32
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16

360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 39
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 4 generators on 39 points:

(5,6,7)(8,10,19,35,12,21,25,27,13,22,30,39,24,36,11,20)(9,14,23,33,16,28,31,34,17,29,38,37,26,32,15,18);(1,2)(3,4)(6,7)(8,9,13,17)(10,18,22,34)(11,23,25,38)(12,26,24,16)(14,27,29,20)(15,30,31,19)(21,37,36,33)(28,35,32,39);(3,4);(1,2)

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10712264, #10712266, #10712299, …
maximal normal subgroup of#10712264, #10712266, #10712299, …
maximal quotient ofC22×#4503, C22×#4504, C2×C4×Dic24, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14580);