C₃×C₅×C₂₇⋊C₃ #11925975

order	1215
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₅×C₉ · Schur multiplier (H₂) = C₃²

Aliases

direct product	C₃×C₅×C₂₇⋊C₃
unambiguous semidirect product	C₂₇⋊(C₃²×C₅)
unambiguous semidirect product	(C₅×C₂₇)⋊C₃²

Subgroups

	subgroup	G/H
center	C₃×C₅×C₉	C₃²
commutator	C₃	C₃²×C₅×C₉
Frattini, cosocle	C₉	C₃³×C₅
socle	C₃²×C₅	C₃×C₉
prime radical	C₃³×C₅	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₅×C₉ ◁ G

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 405

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(31,32,33,34,35);(4,5,8,6,10,16,12,19,24,7,11,15,13,20,23,21,27,29,14,9,17,22,18,25,28,26,30);(5,9,11)(8,15,17)(10,18,20)(16,23,25)(19,26,27)(24,29,30)

Conjugacy classes & character table

495 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×#10624453, C₅×#10624454, C₅×#10624455, …
maximal normal subgroup of	C₅×#10624453, C₅×#10624454, C₅×#10624455, …
maximal quotient of	C₃×C₅×#6864, C₅×C₉×C₂₇⋊C₃, C₃×C₅×C₃²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 49);

C3×C5×C27⋊C3 #11925975