Finite Group Databasebeta

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Cayley diagram

C11×C4⋊C4 #4330

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) = C2

Aliases

direct productC11×C4⋊C4
unambiguous semidirect productC4⋊(C4×C11)

Subgroups

subgroupG/H
center, socle, prime radicalC22×C11C22
commutatorC2C2×C4×C11
Frattini, cosocleC22C22×C11
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C11 ◁ G

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:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#66871, #66872, #66873, …
maximal subgroup of#66871, #66872, #66873, …
maximal normal subgroup of#66871, #66872, #66873, …
maximal quotient ofC11×#95, C11×C4⋊C8, C11×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(176, 21);