Finite Group Databasebeta

← Prev (#2901) Next (#2903) →

Cayley diagram

C22×C4⋊C8 #2902

order128
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 98304
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C26

Aliases

direct productC22×C4⋊C8

Subgroups

subgroupG/H
centerC23×C4C22
commutatorC2C23×C8
Frattini, cosocleC2×C4C24
socle, prime radicalC24C2×C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#92967, #92969, #92975, …
Fitting subgroup ofA4×C4⋊C8, #11711080, #11711081, …
maximal subgroup ofC2×#1277, #7873, #7877, …
maximal normal subgroup ofC2×#1277, #7873, #7877, …
maximal quotient ofC22×#334, C22×#335, C2×#1564, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1634);