Finite Group Databasebeta

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Cayley diagram

C4×#127 #2306

order128
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 98304
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4

Aliases

direct productC4×#127

Subgroups

subgroupG/H
centerC22×C4C23
commutatorC22C23×C4
Frattini, socle, cosocleC23C24
prime radicalC2×#127C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C23 ▷ C1

462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 12 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72280184
Frattini subgroup of#94130, #94134, #94143, …
Fitting subgroup ofC4×#127⋊C3, #11711110, #11711111, …
maximal subgroup of#7569, #7587, #7603, …
maximal normal subgroup of#7569, #7587, #7603, …
maximal quotient ofC4×#390, C4×#384, C4×#392, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1038);