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Cayley diagram

C2×#441 #2941

order128
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) order 16384
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×#441

Subgroups

subgroupG/H
center, socle, prime radicalC23C2×D4
commutatorC4C23×C4
Frattini, cosocleC2×C4C24
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#9088, #9092, #12648, …
maximal normal subgroup of#9088, #9092, #12648, …
maximal quotient ofC2×#1579, C2×#1587, C2×#1593, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1673);