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Cayley diagram

C4×C42⋊C3 #4629

order192
isoclinismisoclinic class of C42⋊C3
automorphismAut(G) = C2×#4626⋊C2 (order 768) · Inn(G) = C42⋊C3 · Out(G) = C2×D4
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C4

Aliases

direct productC4×C42⋊C3
unambiguous semidirect productC43⋊C3

Splittings

Fitting, SylowC43 ⋊ C3
derivedC42 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC4C42⋊C3
commutator, hypocenterC42C3×C4
Frattini, socleC23C2×A4
FittingC43C3
prime radicalC2×C42⋊C3C2
cosocleC2×C42C2×C3
LCSG ▷ C42
UCSC1 ◁ C4

58 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:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711184, #11711185, #11711187, …
maximal subgroup of#67995, C8×C42⋊C3, #1839⋊C3, …
maximal normal subgroup of#67995, C8×C42⋊C3, #1839⋊C3, …
maximal quotient ofC4×#95⋊C3, C8×C42⋊C3, #1726⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 188);