#6898

order	243
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	Aut(G) = C₃×#72277538 (order 11664) · Inn(G) = C₃³ · Out(G) = C₃×#3775
homology	Abelianization (H₁) = C₃³ · Schur multiplier (H₂) = C₉

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:

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

Polycyclic presentation:

3^5|b^3=e,c^3=d^2,b^a=bd,c^a=ce

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11757691, #92311⋊C₂, #14516734, …
Fitting subgroup of	#92311, #92312, #11757683, …
maximal subgroup of	#92311, #92312, C₂×#6898, …
maximal normal subgroup of	#92311, #92312, C₂×#6898, …
maximal quotient of	C₂×#6898, #10624552, #10624557, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(243, 45);