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Cayley diagram

C22×Q32 #3410

order128
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) order 49152
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productC22×Q32

Subgroups

subgroupG/H
center, socle, prime radicalC23D8
commutator, Frattini, cosocleC8C24
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ G
LECSG ▷ C8 ▷ C4 ▷ C2 ▷ C1

180 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)(21,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22);(5,21,13,29)(6,22,14,30)(7,23,15,31)(8,24,16,32)(9,25,17,33)(10,26,18,34)(11,27,19,35)(12,28,20,36)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

Fitting subgroup ofA4×Q32, #11711663, #11711665, …
maximal subgroup of#13015, #13020, #13236, …
maximal normal subgroup of#13015, #13020, #13236, …
maximal quotient ofC22×#358, C22×#366, C2×C4×Q32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2142);