Finite Group Databasebeta

← Prev (#4633) Next (#4635) →

Cayley diagram

#384⋊C3 #4634

order192
isoclinismisoclinic class of #127⋊C3
automorphismAut(G) = C2×#4626⋊C2 (order 768) · Inn(G) = #127⋊C3 · Out(G) = C23
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C4

Aliases

unambiguous semidirect product#384⋊C3

Splittings

Fitting, Sylow#384 ⋊ C3
derivedC42 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC2#127⋊C3
commutator, hypocenterC42C3×C4
Frattini, socleC23C2×A4
Fitting#384C3
prime radicalC2×C42⋊C3C2
cosocleC2×C42C2×C3
LCSG ▷ C42
UCSC1 ◁ C2

46 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2,3,2^5|a^2=c,d^2=fg,e^2=f,d^a=dfg,e^a=ef,d^b=e,e^b=deg,f^b=g,g^b=fg

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711398, #11711399, #11711434, …
maximal subgroup of#67999, #72992, #857⋊C4, …
maximal normal subgroup of#67999, #72992, #857⋊C4, …
maximal quotient of#1444⋊C3, #1842⋊C3, #1843⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 193);