Finite Group Databasebeta

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Cayley diagram

C22×#102 #2890

order128
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) order 98304
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC22×#102

Subgroups

subgroupG/H
center, socleC24D4
commutatorC4C23×C4
Frattini, cosocleC2×C4C24
prime radicalC23×D4C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ 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)(12,15)(14,16);(10,13)(11,16)(14,15)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofA4×#102, S4×#102, #11712900, …
maximal subgroup of#10788, #10925, #10927, …
maximal normal subgroup of#10788, #10925, #10927, …
maximal quotient ofC22×D4⋊C8, C2×#1486, C2×#1480, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1622);