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Cayley diagram

Q8×M16 #2963

order128
nilpotency class2
isoclinismisoclinic class of D42
automorphismAut(G) order 6144
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C24

Aliases

direct productQ8×M16

Subgroups

subgroupG/H
center, Frattini, cosocleC2×C4C24
commutator, socleC22C23×C4
prime radicalC23C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

194 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:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofM16×SL(2,𝔽3), #11711102, #11711103, …
maximal subgroup of#8907, #8909, #8990, …
maximal normal subgroup of#8907, #8909, #8990, …
maximal quotient ofC4⋊C4×M16, #20520, Q8×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1695);