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Cayley diagram

C4⋊Dic11 #4321

order176
isoclinismisoclinic class of Dic22
automorphismAut(G) = #120×AGL(1,𝔽11) (order 3520) · Inn(G) = C2×D11 · Out(G) = C2×C5×D4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC4⋊Dic11

Splittings

SylowC11 ⋊ C4⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C2×D11
commutatorC2×C11C2×C4
FittingC2×C4×C11C2
socle, prime radical, cosocleC22×C11C22
hypercenterC2×C4D11
hypocenterC11C4⋊C4
LCSG ▷ C2×C11 ▷ C11
UCSC1 ◁ C22 ◁ C2×C4

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 176

A minimal faithful permutation representation — 2 generators on 19 points:

(1,2,4,6,8,10,11,9,7,5,3)(16,17,18,19);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,14,15)(17,19)

Polycyclic presentation:

2^4,11|a^2=c,b^2=d,b^a=bd,e^a=e^10

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#66871, #66872, #66880, …
maximal normal subgroup of#66871, #66872, #66880, …
maximal quotient ofC4⋊(C11⋊C8), #66881, #66882, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(176, 12);