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Cayley diagram

C22×M16 #566

order64
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#72209832 (order 6144) · Inn(G) = C22 · Out(G) = C2×#87588⋊C2
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC22×M16

Subgroups

subgroupG/H
centerC22×C4C22
commutatorC2C23×C4
Frattini, cosocleC4C24
socleC23C2×C4
prime radicalC24C4
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C4 ▷ C2 ▷ C1

149 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 12 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#142256, #142257, #142258, …
Frattini subgroup of#7191, #7192, #7193, …
Fitting subgroup ofA4×M16, #73114, #73115, …
maximal subgroup of#1728, #1729, C2×H(3,ℤ/4), …
maximal normal subgroup of#1728, #1729, C2×H(3,ℤ/4), …
maximal quotient ofC22×#97, C2×C4×M16, C22×C22⋊C8, …
outer automorphism group of#2164, #7065, #7066, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 247);