Finite Group Databasebeta

← Prev (#62626) Next (#62628) →

Cayley diagram

C23×Q32 #62627

order256
nilpotency class4
isoclinismisoclinic class of D16
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC23×Q32

Subgroups

subgroupG/H
center, socle, prime radicalC24D8
commutator, Frattini, cosocleC8C25
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C8 ▷ C4 ▷ C2 ▷ C1

1039 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 38
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 38 points:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×A4×Q32, Q32×AGL(1,𝔽8)
maximal subgroup of#147903, #147910, #147915, …
maximal normal subgroup of#147903, #147910, #147915, …
maximal quotient ofC23×#358, C23×#366, C22×C4×Q32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55615);