Finite Group Databasebeta

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Cayley diagram

C22×C22⋊C8 #2876

order128
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) order 98304
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C27

Aliases

direct productC22×C22⋊C8

Subgroups

subgroupG/H
centerC23×C4C22
commutatorC2C23×C8
Frattini, cosocleC2×C4C24
socleC24C2×C4
prime radicalC25C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

536 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);(5,6)(7,8)(9,10,11,13,12,14,15,16);(6,8)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#92963, #92965, #92971, …
Fitting subgroup ofA4×C22⋊C8, #11711146, #11711147, …
maximal subgroup ofC2×#1280, #7907, #7913, …
maximal normal subgroup ofC2×#1280, #7907, #7913, …
maximal quotient ofC22×#323, C22×#324, C2×#1458, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1608);