C₅×#3568⋊C₃ #14511401

order	1920
isoclinism	isoclinic class of #3568⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product	C₅×#3568⋊C₃
unambiguous semidirect product	#3568⋊(C₃×C₅)
unambiguous semidirect product	(C₅×#3568)⋊C₃

Splittings

Fitting, Hall	C₅×#3568 ⋊ C₃
derived	H(3,𝔽₄) ⋊ C₂×C₃×C₅
Sylow	#3568 ⋊ C₃×C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₅	#3568⋊C₃
commutator, cosocle, hypocenter	H(3,𝔽₄)	C₂×C₃×C₅
Frattini	C₂²	C₂×C₅×C₂²⋊A₄
Fitting	C₅×#3568	C₃
socle	C₂²×C₅	C₂×C₂²⋊A₄
LCS	G ▷ H(3,𝔽₄)
UCS	C₁ ◁ C₅

386 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237099);