C₅×C₇×C₂²⋊A₄ #13132976

order	1680
isoclinism	isoclinic class of C₂²⋊A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₅×C₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	C₅×C₇×C₂²⋊A₄
unambiguous semidirect product	C₂²⋊(C₅×C₇×A₄)
unambiguous semidirect product	(C₂²×C₅)⋊(C₇×A₄)

	subgroup	G/H
center, hypercenter	C₅×C₇	C₂²⋊A₄
commutator, cosocle, hypocenter	C₂⁴	C₃×C₅×C₇
Frattini	C₁	G
Fitting, socle	C₂⁴×C₅×C₇	C₃
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₅×C₇

136 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 420

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,3,4,5)(6,7,8,9,10,11,12)(14,15,16)(18,19,20);(17,20)(18,19);(13,16)(14,15)

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 996);