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Cayley diagram

C4×#901 #78452

order384
isoclinismisoclinic class of Dic6
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4

Aliases

direct productC4×#901

Splittings

FittingC2×C3×C4×C8 ⋊ C2
SylowC3 ⋊ C4×#131

Subgroups

subgroupG/H
centerC4×C8C2×D3
commutatorC2×C3C22×C42
FrattiniC2×C4C23×D3
FittingC2×C3×C4×C8C2
socleC22×C3C23×C4
prime radicalC2×#235C22
cosocleC2×C3×C4C24
hypercenterC2×C4×C8D3
hypocenterC3C4×#131
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C8 ◁ C2×C4×C8

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

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

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10685860, #10685862, #10685865, …
maximal normal subgroup of#10685860, #10685862, #10685865, …
maximal quotient ofC8×#901, #10760796, #10760847, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11028);