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Cayley diagram

#6897⋊C2 #92310

order486
isoclinismisoclinic class of #6897⋊C2
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect product#6897⋊C2

Splittings

Fitting, Sylow#6897 ⋊ C2
derivedC92 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC92C2×C3
Frattini, socleC32C3×C3⋊D3
Fitting#6897C2
LCSG ▷ C92
UCSC1

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 3 generators on 54 points:

(2,5)(3,19,21,9,10,20)(4,11)(6,8)(7,24,26,15,16,25)(12,27,14,17,22,23)(13,18)(29,39,42,32,33,41)(30,37,49,35,36,48)(31,38)(34,53,43,44,45,51)(40,47)(46,52)(50,54);(1,3,9)(2,7,17)(4,10,20)(5,12,15)(6,14,24)(8,16,23)(11,21,19)(13,22,25)(18,26,27)(28,49,35,38,36,37,31,30,48)(29,45,47,42,43,46,33,34,54)(32,50,44,39,52,51,41,40,53);(1,18,5,11,8,6,4,2,13)(3,26,12,21,16,14,10,7,22)(9,27,15,19,23,24,20,17,25)(28,29,32)(30,34,40)(31,33,41)(35,47,44)(36,43,52)(37,46,51)(38,42,39)(45,50,49)(48,54,53)

Polycyclic presentation:

2,3^5|c^3=e^2,d^3=f^2,c^a=c^2e,d^a=d^2f,e^a=e^2,f^a=f^2,c^b=cf,d^b=def

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11757682, C2×#6897⋊C2, C3×#6897⋊C2, …
maximal normal subgroup of#11757682, C2×#6897⋊C2, C3×#6897⋊C2, …
maximal quotient of#11757614, C2×#6897⋊C2, #10624551⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 155);