C₅×#73024 #14510748

order	1920
isoclinism	isoclinic class of #73011
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₅×#73024

Splittings

Fitting

C₅×C₈×Q₈ ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₄⋊S₄
commutator, cosocle	C₄×SL(2,𝔽₃)	C₂²×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₅×C₈×Q₈	D₃
prime radical	C₅×C₄⋊GL(2,𝔽₃)	C₂
hypercenter	C₂×C₅×C₈	S₄
hypocenter	SL(2,𝔽₃)	C₅×D₈
LCS	G ▷ C₄×SL(2,𝔽₃) ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈

214 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236446);