Finite Group Databasebeta

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Cayley diagram

C2×#423 #2903

order128
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 32768
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×#423

Subgroups

subgroupG/H
centerC22×C4C23
commutatorC22C23×C4
Frattini, cosocleC2×C4C24
socleC23C22×C4
prime radicalC24C2×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 64

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72338337, #72338338, #72338361, …
Frattini subgroup of#94108, #94109, #94110, …
maximal subgroup of#7874, #7876, #7877, …
maximal normal subgroup of#7874, #7876, #7877, …
maximal quotient ofC2×#1566, C2×#1567, C2×#1568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1635);