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Cayley diagram

C22×#106 #2907

order128
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) order 196608
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C26

Aliases

direct productC22×#106

Subgroups

subgroupG/H
center, socle, prime radicalC24D4
commutatorC4C23×C4
Frattini, cosocleC2×C4C24
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

300 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);(5,6,7,8)(10,13)(11,15)(14,16);(9,10,11,13,12,14,15,16)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofA4×#106, #11711553, #11711572, …
maximal subgroup of#10861, #10863, #10869, …
maximal normal subgroup of#10861, #10863, #10869, …
maximal quotient ofC22×#334, C2×#1566, C2×#1570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1639);