C₅×A₄×C₄₇ #15667705

order	2820
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₅×C₄₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×A₄×C₄₇
unambiguous semidirect product	C₂²⋊(C₃×C₅×C₄₇)
unambiguous semidirect product	(C₂²×C₅)⋊(C₃×C₄₇)
unambiguous semidirect product	(C₂²×C₄₇)⋊(C₃×C₅)
unambiguous semidirect product	(C₂²×C₅×C₄₇)⋊C₃

Splittings

Fitting, socle, Hall	C₂²×C₅×C₄₇ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₅×C₄₇
Hall	C₂²×C₄₇ ⋊ C₃×C₅
Hall	C₂²×C₅ ⋊ C₃×C₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₄₇	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₅×C₄₇
Frattini	C₁	G
Fitting, socle	C₂²×C₅×C₄₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₅×C₄₇

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 940

A minimal faithful permutation representation — 2 generators on 56 points:

(1,2,3,4,5)(7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56);(6,7,8)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	A₄×C₂₅×C₄₇, C₅²×A₄×C₄₇, C₅×C₄₇×A₅, …
maximal normal subgroup of	A₄×C₂₅×C₄₇, C₅²×A₄×C₄₇, C₅×C₇×A₄×C₄₇, …
maximal quotient of	A₄×C₂₅×C₄₇, C₅²×A₄×C₄₇, C₅×C₄₇×C₂²⋊ΓL(1,𝔽₈), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2820, 21);

C5×A4×C47 #15667705