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Cayley diagram

C22×M32 #3405

order128
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#72233475 (order 12288) · Inn(G) = C22 · Out(G) = C22×#87588⋊C2
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C25

Aliases

direct productC22×M32

Subgroups

subgroupG/H
centerC22×C8C22
commutatorC2C23×C8
Frattini, cosocleC8C24
socleC23C2×C8
prime radicalC24C8
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C8 ◁ G
LECSG ▷ C8 ▷ C4 ▷ C2 ▷ C1

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,14)(8,16)(10,18)(12,20)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of#94064, #94065, #94066, …
Fitting subgroup ofA4×M32, #11711490, #11711534, …
maximal subgroup of#12544, #12545, C2×#1375, …
maximal normal subgroup of#12544, #12545, C2×#1375, …
maximal quotient ofC22×#346, C2×C4×M32, C22×C22⋊C16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2137);