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Cayley diagram

#733⋊C4 #66333

order324
isoclinismisoclinic class of #733⋊C2
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#733⋊C4

Splittings

derived, Sylow#733 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2#733⋊C2
commutator, hypocenter#733C4
Frattini, prime radicalC2×C32C3⋊D3
Fitting, cosocleC2×#733C2
socleC2×C3#272
LCSG ▷ #733
UCSC1 ◁ C2

39 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,3,4)(6,10)(7,14)(8,9)(11,13)(12,23)(15,26)(16,18)(17,25)(19,21)(20,27)(22,29)(24,30)(28,31);(5,17,14,8,18,16,9,7,25)(6,28,27,19,15,24,13,12,29)(10,22,23,11,30,26,21,20,31);(5,19,10,8,13,11,9,6,21)(7,12,30,17,28,20,18,15,22)(14,29,26,16,27,31,25,24,23)

Polycyclic presentation:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#733⋊Q8, C4×#733⋊C2, C2×#733⋊C4, …
maximal normal subgroup of#733⋊Q8, C4×#733⋊C2, C2×#733⋊C4, …
maximal quotient of#733⋊C8, C2×#733⋊C4, #6858⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 25);