Finite Group Databasebeta

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Cayley diagram

C11×C22⋊C4 #4329

order176
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×C5×#120 (order 320) · Inn(G) = C22 · Out(G) = C2×C5×D4
homologyAbelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C22

Aliases

direct productC11×C22⋊C4
unambiguous semidirect productC22⋊(C4×C11)

Subgroups

subgroupG/H
center, socleC22×C11C22
commutatorC2C2×C4×C11
Frattini, cosocleC22C22×C11
prime radicalC23×C11C2
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C11 ◁ G

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 19
transitive degreeμt(G) = 88

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

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#66870, #66930, #66931, …
maximal subgroup of#66870, C11×#99, #66930, …
maximal normal subgroup of#66870, C11×#99, #66930, …
maximal quotient ofC11×#95, C11×C22⋊C8, C11×#99, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(176, 20);