#157×SL(2,𝔽₅) #72228173

order	4320
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product

#157×SL(2,𝔽₅)

Splittings

derived

C₃²×SL(2,𝔽₅) ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#157×A₅
commutator, hypocenter	C₃²×SL(2,𝔽₅)	C₄
Fitting, socle	C₂×C₃²	C₄×A₅
prime radical	C₃⋊D₃×SL(2,𝔽₅)	C₂
cosocle	C₂×C₃⋊D₃	C₂×A₅
LCS	G ▷ C₃²×SL(2,𝔽₅)
UCS	C₁ ◁ C₂

188 conjugacy classes of subgroups; only the 14 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 14 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) = 30
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

SL(2,𝔽₅)×A₆

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,12,10,4,15,3,14,5,9,8)(2,6)(7,24,16,17,18,13,22,11,19,20)(21,23)(25,30,26,27)(28,29), (1,20,7,5,21)(2,22,9,8,17)(3,18,13,10,23)(4,15,19,6,24)(25,30)(26,27,28,29));