C₃²×C₇×C₄₁⋊C₅ #27166552

order	12915
isoclinism	isoclinic class of C₄₁⋊C₅
automorphism	?
homology	Abelianization (H₁) = C₃²×C₅×C₇ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃²×C₇×C₄₁⋊C₅
unambiguous semidirect product	C₄₁⋊(C₃²×C₅×C₇)
unambiguous semidirect product	(C₃×C₄₁)⋊(C₃×C₅×C₇)
unambiguous semidirect product	(C₃²×C₄₁)⋊(C₅×C₇)
unambiguous semidirect product	(C₃²×C₇×C₄₁)⋊C₅

Splittings

Fitting, socle, Hall	C₃²×C₇×C₄₁ ⋊ C₅
derived, Sylow	C₄₁ ⋊ C₃²×C₅×C₇
Hall	C₃²×C₄₁ ⋊ C₅×C₇
Hall	C₇×C₄₁ ⋊ C₃²×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₇	C₄₁⋊C₅
commutator, cosocle, hypocenter	C₄₁	C₃²×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₃²×C₇×C₄₁	C₅
LCS	G ▷ C₄₁
UCS	C₁ ◁ C₃²×C₇

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 2583

A minimal faithful permutation representation — 2 generators on 54 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃²×C₇×D₄₁⋊C₅, C₃²×C₇⋊(D₄₁⋊C₅), C₃²×D₇×C₄₁⋊C₅, …
maximal normal subgroup of	C₃²×C₇×D₄₁⋊C₅, C₃²×C₇⋊(D₄₁⋊C₅), C₃²×D₇×C₄₁⋊C₅, …
maximal quotient of	C₂×C₃²×C₇×C₄₁⋊C₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12915, 7);

C32×C7×C41⋊C5 #27166552