Finite Group Databasebeta

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Cayley diagram

C23×#97 #60051

order256
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 88080384
homologyAbelianization (H1) = C23×C42 · Schur multiplier (H2) = C210

Aliases

direct productC23×#97

Subgroups

subgroupG/H
centerC24×C4C22
commutatorC2C23×C42
Frattini, cosocleC2×C4C25
socle, prime radicalC25C2×C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

2063 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×A4×#97, #97×AGL(1,𝔽8)
maximal subgroup of#105486, #105517, #106470, …
maximal normal subgroup of#105486, #105517, #106470, …
maximal quotient ofC23×#322, C22×#1451, C22×#1452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53039);