Finite Group Databasebeta

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Cayley diagram

C2×SD64 #2260

order128
nilpotency class5
isoclinismisoclinic class of D32
automorphismAut(G) = #72217166 (order 2048) · Inn(G) = D16 · Out(G) = C2×#131
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×SD64

Subgroups

subgroupG/H
center, socleC22D16
commutator, Frattini, cosocleC16C23
prime radicalC2×D16C2
LCSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ G
LECSG ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

56 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);(4,18)(5,33)(6,16)(7,31)(8,14)(9,29)(10,12)(11,27)(13,25)(15,23)(17,21)(20,34)(22,32)(24,30)(26,28)

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×SD64, #13652, #13662, …
maximal normal subgroup ofC4×SD64, #13652, #13662, …
maximal quotient ofC2×#1415, C2×#1416, C2×#1424, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 992);