Finite Group Databasebeta

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Cayley diagram

C13×C4⋊C4 #6111

order208
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C3×C4×#120 (order 384) · Inn(G) = C22 · Out(G) = C3×C4×D4
homologyAbelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2

Aliases

direct productC13×C4⋊C4
unambiguous semidirect productC4⋊(C4×C13)

Subgroups

subgroupG/H
center, socle, prime radicalC22×C13C22
commutatorC2C2×C4×C13
Frattini, cosocleC22C22×C13
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C13 ◁ G

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 208

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#88047, #88048, #88049, …
maximal subgroup of#88047, #88048, #88049, …
maximal normal subgroup of#88047, #88048, #88049, …
maximal quotient ofC13×#95, C13×C4⋊C8, C13×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(208, 22);