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Cayley diagram

C4×#383 #13774

order256
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 1572864
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4

Aliases

direct productC4×#383

Subgroups

subgroupG/H
centerC23×C4C23
commutatorC22C22×C42
Frattini, socle, prime radical, cosocleC24C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

783 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 24
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC4×#383⋊C3
maximal subgroup of#94983, #94984, #94987, …
maximal normal subgroup of#94983, #94984, #94987, …
maximal quotient ofC4×#1434, C4×#1443, #153758, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6762);