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Cayley diagram

C3×C4⋊C4 #220

order48
nilpotency class2
isoclinismisoclinic class of D4
automorphismAut(G) = C2×#120 (order 64) · Inn(G) = C22 · Out(G) = C2×D4
holomorphG ⋊ Aut(G) = D3×#346446
homologyAbelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productC3×C4⋊C4
unambiguous semidirect productC4⋊(C3×C4)

Subgroups

subgroupG/H
center, socle, prime radicalC22×C3C22
commutatorC2C2×C3×C4
Frattini, cosocleC22C22×C3
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ G

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2,3,4)(6,8)(9,10,11);(5,6,7,8)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#807, #808, #809, …
maximal subgroup of#807, #808, #809, …
maximal normal subgroup of#807, #808, #809, …
maximal quotient ofC3×#95, C3×C4⋊C8, C3×#106, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 22);