C₅×#87493 #14511427

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

solvable

Aliases

direct product

C₅×#87493

Splittings

Fitting

C₂×C₅×H(5,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂³×S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂⁴×C₅
Frattini	C₂	C₂³×C₅×S₄
Fitting	C₂×C₅×H(5,𝔽₂)	D₃
hypercenter	C₂×C₅×Q₈	S₄
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₅×Q₈

1468 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1468 conjugacy classes of subgroups — too many to draw. Only its 200 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) = 53
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 4 generators on 53 points:

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

Conjugacy classes & character table

215 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#983⋊A₄, D₅×#87493
maximal normal subgroup of	C₅×#983⋊A₄, D₅×#87493

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 237125);