Finite Group Databasebeta

← Prev (#60189) Next (#60191) →

Cayley diagram

C2×Q8×M16 #60190

order256
nilpotency class2
isoclinismisoclinic class of D42
automorphismAut(G) order 393216
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×Q8×M16

Subgroups

subgroupG/H
centerC22×C4C24
commutatorC22C24×C4
Frattini, cosocleC2×C4C25
socleC23C23×C4
prime radicalC24C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

1077 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofC2×M16×SL(2,𝔽3)
maximal subgroup of#105695, #105700, #105995, …
maximal normal subgroup of#105695, #105700, #105995, …
maximal quotient ofC2×C4⋊C4×M16, C2×#20520, C2×Q8×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53178);