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Cayley diagram

#41⋊C9 #3693

order144
isoclinismisoclinic class of SL(2,𝔽3)
automorphismAut(G) = C2×C3×S4 (order 144) · Inn(G) = A4 · Out(G) = C22×C3
holomorphG ⋊ Aut(G) = #72268997
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect product#41⋊C9

Splittings

Sylow#41 ⋊ C9

Subgroups

subgroupG/H
center, hypercenterC3×C4A4
commutator, hypocenterQ8C2×C9
Frattini, socleC2×C3C2×A4
Fitting, prime radicalC3×#41C3
cosocleC3×Q8C2×C3
LCSG ▷ Q8
UCSC1 ◁ C3×C4

25 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 25 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

derived subgroup of#10589505, #10589508, #10589509, …
maximal subgroup of#63422, #131⋊C9, #63690, …
maximal normal subgroup of#63422, #131⋊C9, #63690, …
maximal quotient ofC4×Q8⋊C9, C2×#41⋊C9, #41⋊C27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 36);