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Cayley diagram

C2×#442 #2942

order128
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) order 16384
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×#442

Subgroups

subgroupG/H
center, socleC23C2×D4
commutatorC4C23×C4
Frattini, cosocleC2×C4C24
prime radicalC22×D8C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

304 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);(3,4,6,5)(7,8,10,12)(9,11,14,13);(8,11)(9,14)(12,13);(3,5,6,4)(8,12)(11,13)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#9089, #9093, #12647, …
maximal normal subgroup of#9089, #9093, #12647, …
maximal quotient ofC2×#1580, C2×#1586, C2×#1594, …
automorphism group of#359, C3×SD32

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1674);