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Cayley diagram

C22×D32 #33975

order256
nilpotency class5
isoclinismisoclinic class of D32
automorphismAut(G) order 196608
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productC22×D32

Subgroups

subgroupG/H
center, socleC23D16
commutator, Frattini, cosocleC16C24
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G
LECSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

271 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2);(3,4);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36);(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,25)(18,24)(19,23)(20,22)

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofA4×D32, S4×D32, #72342895
maximal subgroup of#152521, #152526, #152969, …
maximal normal subgroup of#152521, #152526, #152969, …
maximal quotient ofC22×#1415, C22×#1423, C2×C4×D32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26963);