C₅×SO⁺(4,𝔽₃) #72229349

order	2880
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₂×C₃

solvable

Aliases

direct product

C₅×SO⁺(4,𝔽₃)

Splittings

Fitting	C₅×H(5,𝔽₂) ⋊ C₃⋊D₃
derived	Ω⁺(4,𝔽₃) ⋊ C₂×C₅

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂×C₅	A₄⋊S₄
commutator, cosocle, hypocenter	Ω⁺(4,𝔽₃)	C₂×C₅
Frattini	C₂	C₅×A₄⋊S₄
Fitting	C₅×H(5,𝔽₂)	C₃⋊D₃
LCS	G ▷ Ω⁺(4,𝔽₃)
UCS	C₁ ◁ C₂×C₅

254 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 120

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×SO⁺(4,𝔽₃), C₂×C₅×SO⁺(4,𝔽₃)
maximal normal subgroup of	D₅×SO⁺(4,𝔽₃), C₂×C₅×SO⁺(4,𝔽₃)
maximal quotient of	C₂×C₅×SO⁺(4,𝔽₃)

GAP

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

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