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Cayley diagram

C2×D32 #2259

order128
nilpotency class5
isoclinismisoclinic class of D32
automorphismAut(G) = #72225134 (order 4096) · Inn(G) = D16 · Out(G) = #2929
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC2×D32

Subgroups

subgroupG/H
center, socleC22D16
commutator, Frattini, cosocleC16C23
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G
LECSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 34 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);(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,25)(14,24)(15,23)(16,22)(17,21)(18,20)

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#7537, #7541, C4×D32, …
maximal normal subgroup of#7537, #7541, C4×D32, …
maximal quotient ofC2×#1415, C2×#1423, C4×D32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 991);