C₅×#6887 #11925960

order	1215
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₅×C₉ · Schur multiplier (H₂) = C₃×C₉

Aliases

direct product

C₅×#6887

Subgroups

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

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 41
transitive degree	μ_t(G) = 405

A minimal faithful permutation representation — 3 generators on 41 points:

(1,2,3,4,5)(6,7,9,8,10,12,11,13,14)(15,16,19)(17,21,27)(18,22,26)(20,28,25)(23,30,35)(24,31,34)(29,36,33)(32,38,40)(37,41,39);(16,20,22)(19,26,28)(21,29,31)(27,34,36)(30,37,38)(35,40,41);(15,17,23,18,24,32,25,33,39)(16,21,30,22,31,38,20,29,37)(19,27,35,26,34,40,28,36,41)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×#10624457, C₅×#10624458, C₅×#10624460, …
maximal normal subgroup of	C₅×#10624457, C₅×#10624458, C₅×#10624460, …
maximal quotient of	C₅×#10624544, C₅×C₉×C₃²⋊C₉, C₅×C₉×C₉⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 34);

C5×#6887 #11925960