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Cayley diagram

C2×Q32 #507

order64
nilpotency class4
isoclinismisoclinic class of D16
automorphismAut(G) = #72213937 (order 1024) · Inn(G) = D8 · Out(G) = #525
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×Q32

Subgroups

subgroupG/H
center, socle, prime radicalC22D8
commutator, Frattini, cosocleC8C23
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ G
LECSG ▷ C8 ▷ C4 ▷ C2 ▷ C1

41 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);(3,4,8,12)(5,13,16,32)(6,17,21,31)(7,22,19,11)(9,24,26,14)(10,27,29,20)(15,33,23,28)(18,30,34,25);(3,5,8,16)(4,9,12,26)(6,18,21,34)(7,23,19,15)(10,28,29,33)(11,30,22,25)(13,31,32,17)(14,27,24,20)

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#1416, #1420, C4×Q32, …
maximal normal subgroup of#1416, #1420, C4×Q32, …
maximal quotient ofC2×#358, C2×#366, C4×Q32, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 188);