C₅×#73152 #14510876

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

solvable

Aliases

direct product

C₅×#73152

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#5432
commutator	C₂×SL(2,𝔽₃)	C₂×C₄×C₅
Frattini	C₂×C₄	C₂×C₅×S₄
Fitting	C₂×C₅×H(5,𝔽₂)	D₃
prime radical	C₂×C₅×#994	C₂
cosocle	C₂×#41⋊C₃	C₂²×C₅
hypercenter	C₂×C₅×Q₈	S₄
hypocenter	SL(2,𝔽₃)	C₅×C₂²⋊C₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×Q₈

384 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 2 generators on 53 points:

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236574);