Finite Group Databasebeta

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Cayley diagram

C2×#408 #2878

order128
nilpotency class2
isoclinismisoclinic class of #120
automorphismAut(G) order 16384
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#408

Subgroups

subgroupG/H
centerC22×C4C23
commutatorC22C23×C4
Frattini, cosocleC2×C4C24
socleC23C22×C4
prime radicalC22×#41C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(3,4)(5,6)(7,8,10,13,11,14,17,20)(9,12,16,19,15,18,21,22);(4,6)(7,9)(8,12)(10,16)(11,15)(13,19)(14,18)(17,21)(20,22);(9,15)(12,18)(16,21)(19,22)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#94092, #94093, #94094, …
maximal subgroup of#10747, #10748, #10752, …
maximal normal subgroup of#10747, #10748, #10752, …
maximal quotient ofC2×#1468, C2×#1469, C2×#1470, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1610);