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Cayley diagram

C2×#4720 #78583

order384
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4

Aliases

direct productC2×#4720

Splittings

SylowC3 ⋊ C2×#433

Subgroups

subgroupG/H
centerC22×C4C22×D3
commutatorC22×C3C23×C4
FrattiniC2×C4C23×D3
FittingC2×C3×C22⋊C8C2
socleC23×C3C22×C4
prime radicalC24×C3C2×C4
cosocleC2×C3×C4C24
hypercenterC2×C22⋊C8D3
hypocenterC3C2×#433
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10704390, C4×#4720, #10793335, …
maximal normal subgroup of#10704390, C4×#4720, #10793335, …
maximal quotient ofC2×#68598, C2×#69907, C2×#68604, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11159);