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Cayley diagram

C11×Q16 #4335

order176
nilpotency class3
isoclinismisoclinic class of D8
automorphismAut(G) = C2×C5×AGL(1,ℤ/8) (order 320) · Inn(G) = D4 · Out(G) = C23×C5
homologyAbelianization (H1) = C22×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×Q16

Subgroups

subgroupG/H
center, socle, prime radicalC2×C11D4
commutator, Frattini, cosocleC4C22×C11
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C11 ◁ C4×C11 ◁ G

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 176

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19)(20,27,26,25,24,23,22,21);(12,20,16,24)(13,21,17,25)(14,22,18,26)(15,23,19,27)

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#66892, #66893, Q16×D11, …
maximal subgroup of#66892, #66893, C11×SD32, …
maximal normal subgroup of#66892, #66893, C11×SD32, …
maximal quotient ofC11×#103, C11×#107, C2×C11×Q16, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(176, 26);