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Cayley diagram

C42⋊C3 #201

order48
isoclinismisoclinic class of C42⋊C3
automorphismAut(G) = #4626⋊C2 (order 384) · Inn(G) = G · Out(G) = D4
holomorphG ⋊ Aut(G) = #72219768
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C4

Aliases

unambiguous semidirect productC42⋊C3

Splittings

Fitting, derived, SylowC42 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenterC42C3
Frattini, socleC22A4
LCSG ▷ C42
UCSC1

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 2 generators on 12 points:

(1,2,5)(3,7,12)(4,8,11)(6,10,9);(1,3,4,9)(2,8)(5,12,11,10)(6,7)

Polycyclic presentation:

3,2^4|b^2=de,c^2=d,b^a=c,c^a=bce,d^a=e,e^a=de

Conjugacy classes & character table

C1C2C3C4C5C6C7C8
size1333331616
order12444433
χ111111111
χ2111111−1−i32−1+i32
χ3111111−1+i32−1−i32
χ433-1-1-1-100
χ53-111−1+2i−1−2i00
χ63-111−1−2i−1+2i00
χ73-1−1+2i−1−2i1100
χ83-1−1−2i−1+2i1100

Here i is the imaginary unit (i² = −1).

Relationships

derived subgroup of#857, #4623, #4626, …
maximal subgroup of#857, C2×C42⋊C3, #126⋊C3, …
maximal normal subgroup of#857, C2×C42⋊C3, #126⋊C3, …
maximal quotient of#95⋊C3, C2×C42⋊C3, C42⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 3);