C₅×C₈₁⋊C₃ #11925950

order	1215
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₅×C₂₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×C₈₁⋊C₃
unambiguous semidirect product	C₈₁⋊(C₃×C₅)
unambiguous semidirect product	(C₅×C₈₁)⋊C₃

Subgroups

	subgroup	G/H
center	C₅×C₂₇	C₃²
commutator	C₃	C₃×C₅×C₂₇
Frattini, cosocle	C₂₇	C₃²×C₅
socle	C₃×C₅	C₃×C₂₇
prime radical	C₃²×C₅	C₂₇
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₅×C₂₇ ◁ G

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 86
transitive degree	μ_t(G) = 405

A minimal faithful permutation representation — 2 generators on 86 points:

(1,2,3,4,5)(6,7,11,8,13,23,16,28,42,9,14,24,17,29,43,33,50,64,19,31,44,35,52,65,55,71,79,10,15,22,18,30,40,34,51,61,20,32,41,36,53,62,56,72,77,38,54,63,58,73,78,74,83,85,21,12,25,37,26,45,57,47,66,39,27,46,59,48,67,75,69,80,60,49,68,76,70,81,84,82,86);(7,12,15)(11,22,25)(13,26,30)(14,27,32)(23,40,45)(24,41,46)(28,47,51)(29,48,53)(31,49,54)(42,61,66)(43,62,67)(44,63,68)(50,69,72)(52,70,73)(64,77,80)(65,78,81)(71,82,83)(79,85,86)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×#10624500, C₅×#10624502, C₅×#10624503, …
maximal normal subgroup of	C₅×#10624500, C₅×#10624502, C₅×#10624503, …
maximal quotient of	C₅×#10624499, C₅×C₃²⋊C₈₁, C₅×C₉⋊C₈₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1215, 24);

C5×C81⋊C3 #11925950