Finite Group Databasebeta

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Cayley diagram

C2×#340 #1734

order128
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) order 16384
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C25

Aliases

direct productC2×#340

Subgroups

subgroupG/H
center, socle, prime radicalC24D4
commutatorC4C2×C42
Frattini, cosocleC22×C4C23
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#7258, #7264, C4×#340, …
maximal normal subgroup of#7258, #7264, C4×#340, …
maximal quotient ofC2×#1277, #7874, #7877, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 466);