Finite Group Databasebeta

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Cayley diagram

C2×#407 #2877

order128
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 32768
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#407

Subgroups

subgroupG/H
centerC22×C4C23
commutatorC22C23×C4
Frattini, cosocleC2×C4C24
socleC23C22×C4
prime radicalC25C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

504 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72338352, #72338357, #72338359, …
Frattini subgroup of#94076, #94077, #94078, …
maximal subgroup of#7909, #7911, #7913, …
maximal normal subgroup of#7909, #7911, #7913, …
maximal quotient ofC2×#1462, C2×#1463, C2×#1464, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1609);