#6897

order	243
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	Aut(G) = C₃²×#10620191 (order 5832) · Inn(G) = C₃³ · Out(G) = C₃×#622
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₃

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃²	C₃³
prime radical	C₃³	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ G
LECS	G ▷ C₃² ▷ C₁

50 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 81

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

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

Polycyclic presentation:

3^5|b^3=de,c^3=d,b^a=bd,c^a=ce

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#6897⋊C₂, #11757682, #14517905
maximal subgroup of	#6897⋊C₂, C₂×#6897, C₃×#6897, …
maximal normal subgroup of	#6897⋊C₂, C₂×#6897, C₃×#6897, …
maximal quotient of	C₂×#6897, #10624551, #10624557, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(243, 44);