Finite Group Databasebeta

← Prev (#2963) Next (#2965) →

Cayley diagram

C8×#41 #2964

order128
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 6144
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C25

Aliases

direct productC8×#41

Subgroups

subgroupG/H
centerC4×C8C22
commutatorC2C23×C8
Frattini, cosocleC2×C4C24
socleC22C23×C4
prime radicalC2×#41C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C4×C8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC8×#41⋊C3, C8×GU(2,𝔽3), #11710995, …
maximal subgroup ofC8×#104, #8191, #8192, …
maximal normal subgroup ofC8×#104, #8191, #8192, …
maximal quotient ofC8×#117, C4×C8×D4, C4×C8×Q8, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1696);